ADVANCES IN ECONOMETRICS
VOLUME 17
MAXIMUM LIKELIHOOD ESTIMATION OF MISSPECIFIED MODELS: TWENTY YEARS LATER EDITED BY
THOMAS B. FOMBY Department of Economics, Southern Methodist University, USA
R. CARTER HILL Economics Department, Louisiana State University, USA
2003
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CONTENTS LIST OF CONTRIBUTORS
vii
INTRODUCTION Thomas B. Fomby and R. Carter Hill
ix
A COMPARATIVE STUDY OF PURE AND PRETEST ESTIMATORS FOR A POSSIBLY MISSPECIFIED TWO-WAY ERROR COMPONENT MODEL Badi H. Baltagi, Georges Bresson and Alain Pirotte
1
TESTS OF COMMON DETERMINISTIC TREND SLOPES APPLIED TO QUARTERLY GLOBAL TEMPERATURE DATA Thomas B. Fomby and Timothy J. Vogelsang
29
THE SANDWICH ESTIMATE OF VARIANCE James W. Hardin
45
TEST STATISTICS AND CRITICAL VALUES IN SELECTIVITY MODELS R. Carter Hill, Lee C. Adkins and Keith A. Bender
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ESTIMATION, INFERENCE, AND SPECIFICATION TESTING FOR POSSIBLY MISSPECIFIED QUANTILE REGRESSION Tae-Hwan Kim and Halbert White
107
QUASI-MAXIMUM LIKELIHOOD ESTIMATION WITH BOUNDED SYMMETRIC ERRORS Douglas Miller, James Eales and Paul Preckel
133
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CONSISTENT QUASI-MAXIMUM LIKELIHOOD ESTIMATION WITH LIMITED INFORMATION Douglas Miller and Sang-Hak Lee
149
AN EXAMINATION OF THE SIGN AND VOLATILITY SWITCHING ARCH MODELS UNDER ALTERNATIVE DISTRIBUTIONAL ASSUMPTIONS Mohamed F. Omran and Florin Avram
165
ESTIMATING A LINEAR EXPONENTIAL DENSITY WHEN THE WEIGHTING MATRIX AND MEAN PARAMETER VECTOR ARE FUNCTIONALLY RELATED Chor-yiu Sin
177
TESTING IN GMM MODELS WITHOUT TRUNCATION Timothy J. Vogelsang
199
BAYESIAN ANALYSIS OF MISSPECIFIED MODELS WITH FIXED EFFECTS Tiemen Woutersen
235
LIST OF CONTRIBUTORS Lee C. Adkins
Oklahoma State University, Stillwater, USA
Florin Avram
University de Pau, France
Badi H. Baltagi
Texas A&M University, College Station, USA
Keith A. Bender
University of Wisconsin-Milwaukee, Milwaukee, USA
Georges Bresson
Universit´e Paris II, Paris, France
James Eales
Purdue University, West Lafayette, USA
Thomas B. Fomby
Southern Methodist University, Dallas, USA
James W. Hardin
University of South Carolina, Columbia, USA
R. Carter Hill
Louisiana State University, Baton Rouge, USA
Tae-Hwan Kim
University of Nottingham, Nottingham, UK
Sang-Hak Lee
Purdue University, West Lafayette, USA
Douglas Miller
Purdue University, West Lafayette, USA
Mohammed F. Omran
University of Sharjah, UAE
Alain Pirotte
Universit´e de Valenciennes and Universit´e Paris II, Paris, France
Paul Preckel
Purdue University, West Lafayette, USA
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viii
Dhor-yiu Sin
Hong Kong Baptist University, Hong Kong
Timothy J. Vogelsang
Cornell Univerisity, Ithaca, USA
Halbert White
University of California, San Diego, USA
Tiemen Woutersen
University of Western Ontario, Ontario, Canada
INTRODUCTION It is our pleasure to bring you a volume of papers which follow in the tradition of the seminal work of Halbert White, especially his work in Econometrica (1980, 1982) and his Econometric Society monograph no. 22 Estimation, Inference and Specification Analysis (1994). Approximately 20 years have passed since White’s initial work on heteroskedasticity-consistent covariance matrix estimation and maximum likelihood estimation in the presence of misspecified models, so-called quasi-maximum likelihood (QMLE) estimation. Over this time, much has been written on these and related topics, many contributions being by Hal himself. For example, following Hal’s pure heteroskedasticity robust estimation work, Newey and West (1987) extended robust estimation to autocorrelated data. Extensions and refinements of these themes continue today. There is no econometric package that we know of today that does not have some provision for robust standard errors in most of the estimation methods offered. All of these innovations can be credited to the germinating ideas produced by Hal in his econometric research. Thus, we offer this volume in recognition of the pioneering work that he has done in the past and that has proved to be so wonderfully useful in empirical research. We look forward to seeing Hal’s work continuing well into the future, yielding, we are sure, many more useful econometric techniques that will be robust to misspecifications of the sort we often face in empirical problems in economics and elsewhere. Now let us turn to a brief review of the contents of this volume. In the spirit of White (1982), Baltagi, Bresson and Pirotte in their paper entitled “A Comparative Study of Pure and Pretest Estimators for a Possibly Misspecified Two-way Error Component Model” examine the consequences of model misspecification using a panel data regression model. Maximum likelihood, random and fixed effects estimators are compared using Monte Carlo experiments under normality of the disturbances but with possibly misspecified variance-covariance matrix. In the presence of perfect foresight on the form of the variance-covariance matrix, GLS (maximum likelihood) is always the best in MSE terms. However, in the absence of perfect foresight (the more typical case), the authors show that a pre-test estimator is a viable alternative given that its performance is a close second to correct GLS whether the true specification is a two-way, one-way error component or a pooled regression model. The authors further show that incorrect GLS, ix
x
maximum likelihood, or fixed effects estimators may lead to a big loss in mean square error. In their paper “Tests of Common Deterministic Trend Slopes Applied to Quarterly Global Temperature Data” Fomby and Vogelsang apply the multivariate deterministic trend-testing framework of Franses and Vogelsang (2002) to compare global warming trends both within and across the hemispheres of the globe. They find that globally and within hemispheres the seasons appear not to be warming equally fast. In particular, winters appear to be warming faster than summers. Across hemispheres, it appears that the winters in the northern and southern hemispheres are warming equally fast whereas the remaining seasons appear to have unequal warming rates. In his paper “The Sandwich Estimate of Variance” Hardin examines the history, development, and application of the sandwich estimate of variance. In describing this estimator he pays attention to applications that have appeared in the literature and examines the nature of the problems for which this estimator is used. He also describes various adjustments to the estimate for use with small samples and illustrates the estimator’s construction for a variety of models. In their paper “Test Statistics and Critical Values in Selectivity Models” Hill, Adkins, and Bender examine the finite sample properties of alternative covariance matrix estimators of the Heckman (1979) two-step estimator (Heckit) for the selectivity model so widely used in Economics and other social sciences. The authors find that, in terms of how the alternative versions of asymptotic variancecovariance matrices used in selectivity models capture the finite sample variability of the Heckit two-step estimator, the answer depends on the degree of censoring and on whether the explanatory variables in the selection and regression equation differ or not. With severe censoring and if the explanatory variables in the two equations are identical, then none of the asymptotic standard error formulations is reliable in small samples. In larger samples the bootstrap does a good job in reflecting estimator variability as does a version of the White heteroskedasticity-consistent estimator. With respect to finite sample inference the bootstrap standard errors seem to match the nominal standard errors computed from asymptotic covariance matrices unless censoring is severe and there is not much difference in the explanatory variables in the selection and regression equations. Most importantly the critical values of the pivotal bootstrap t-statistics lead to better test size than those based on usual asymptotic theory. To date the literature on quantile regression and least absolute deviation regression has assumed either explicitly or implicitly that the conditional quantile regression model is correctly specified. In their paper “Estimation, Inference, and Specification Testing for Possibly Misspecified Quantile Regression” Kim and White allow for possible misspecification of a linear conditional quantile
xi
regression model. They obtain consistency of the quantile estimator for certain “pseudo-true” parameter values and asymptotic normality of the quantile estimator when the model is misspecified. In this case, the asymptotic covariance matrix has a novel form, not seen in earlier work, and they provide a consistent estimator of the asymptotic covariance matrix. They also propose a quick and simple test for conditional quantile misspecification based on the quantile residuals. Miller, Eales, and Preckel propose in their paper “Quasi-Maximum Likelihood Estimation with Bounded Symmetric Errors” a QMLE estimator for the location parameters of a linear regression model with bounded and symmetrically distributed errors. The errors outcomes are restated as the convex combination of the bounds, and they use the method of maximum entropy to derive the quasi-log likelihood function. Under the stated model assumptions, they show that the proposed estimator is unbiased, consistent, and asymptotically normal. Miller, Eales, and Preckel then conduct a series of Monte Carlo exercises designed to illustrate the sampling properties of QMLE to the least squares estimator. Although the least squares estimator has smaller quadratic risk under normal and skewed error processes, the proposed QML estimator dominates least squares for the bounded and symmetric error distribution considered in their paper. In their paper “Consistent Quasi-Maximum Likelihood Estimation with Limited Information” Miller and Lee use the minimum cross-entropy method to derive an approximate joint probability model for a multivariate economic process based on limited information about the marginal quasi-density functions and the joint moment conditions. The modeling approach is related to joint probability models derived from copula functions. They note, however, that the entropy approach has some practical advantages over copula-based models. Under suitable regularity conditions, the authors show that the quasi-maximum likelihood estimator (QMLE) of the model parameters is consistent and asymptotically normal. They demonstrate the procedure with an application to the joint probability model of trading volume and price variability for the Chicago Board of Trade soybean futures contract. There is growing evidence in the financial economics literature that the response of current volatility in financial data to past shocks is asymmetric with negative shocks having more impact on current volatility than positive shocks. In their paper “An Examination of the Sign and Volatility Switching ARCH Models Under Alternative Distributional Assumptions” Omran and Avram investigate the asymmetric ARCH models of Glosten et al. (1993) and Fornari and Mele (1997) and the sensitivity of their models to the assumption of normality in the innovations. Omran and Avram hedge against the possibility of misspecification by basing the inferences on the robust variance-covariance matrix suggested by White (1982). Their results
xii
suggest that using more flexible distributional assumptions on financial data can have a significant impact on the inferences drawn from asymmetric ARCH models. Gourieroux et al. (1984) investigate the consistency of the parameters in the conditional mean, ignoring or misspecifying other features of the true conditional density. They show that it suffices to have a QMLE of a density from the linear exponential family (LEF). Conversely, a necessary condition for a QMLE being consistent for the parameters in the conditional mean is that the likelihood function belongs to the LEF. As a natural extension in Chapter 5 of his book, White (1994) shows that the Gourieroux et al. (1984) results carry over to dynamic models with possibly serially correlated and/or heteroskedastic errors. In his paper “Estimating a Linear Exponential Density when the Weighting Matrix and Mean Parameter Vector are Functionally Related” Sin shows that the above results do not hold when the weighting matrix of the density and the mean parameter vector are functionally related. A prominent example is an autoregressive moving-average (ARMA) model with generalized autoregressive conditional heteroscedasticity (GARCH) error. However, correct specification of the conditional variance adds conditional moment conditions for estimating the parameters of the conditional mean. Based on the recent literature of efficient instrumental variables estimator (IVE) or generalized method of moments (GMM), the author proposes an estimator that is based on the QMLE of a density from the quadratic exponential family (QEF). The asymptotic variance of this modified QMLE attains the lower bound for minimax risk. In this modeling approach the GARCH-M is also allowed. In his paper “Testing in GMM Models Without Truncation” Vogelsang proposes a new approach to testing in the generalized method of moments (GMM) framework. The new tests are constructed using heteroskedasticity autocorrelation (HAC) robust standard errors computed using nonparametric spectral density estimators without truncation. While such standard errors are not consistent, a new asymptotic theory shows that they lead to valid tests nonetheless. In an over-identified linear instrumental variables model, simulations suggest that the new tests and the associated limiting distribution theory provide a more accurate first order asymptotic null approximation than both standard nonparametric HAC robust tests and VAR-based parametric HAC robust tests. Finite sample power of the new tests is shown to be comparable to standard tests. In applied work, economists analyze individuals or firms that differ in observed and unobserved ways. These unobserved differences are usually referred to as heterogeneity and one can control for the heterogeneity in panel data by allowing for time-invariant, individual-specific parameters. This fixed effect approach introduces many parameters into the model that causes the “incidental parameter problem”: the maximum likelihood estimator is in general inconsistent. Woutersen (2001) shows how to approximately separate the parameters of interest from
xiii
the fixed effects using a reparameterization. He then shows how a Bayesian method gives a general solution to the incidental parameter for correctly specified models. In his paper in this volume “Bayesian Analysis of Misspecified Models with Fixed Effects” Woutersen extends his 2001 work to misspecified models. Following White (1982) he assumes that the expectation of the score of the integrated likelihood is zero at the true values of the parameters. He then derives √ the conditions under which a Bayesian estimator converges at the rate of N, where N is the number of individuals. Under these conditions, Woutersen shows that the variance-covariance matrix of the Bayesian estimator has the form of White (1982). He goes on to illustrate the approach by analyzing the dynamic linear model with fixed effects and a duration model with fixed effects. Thomas B. Fomby and R. Carter Hill Co-editors
REFERENCES Fornari, F., & Mele, A. (1997). Sign- and volatility-switching ARCH models: Theory and applications to international stock markets. Journal of Applied Econometrics, 12, 49–65. Franses, P. H., & Vogelsang, T. J. (2002). Testing for common deterministic trend slopes, Center for Analytic Economics Working Paper 01–15, Cornell University. Glosten, L. R., Jagannathan, R., & Runkle, D. (1993). On the relationship between expected value and the volatility of nominal excess returns on stocks. Journal of Finance, 43, 1779–1801. Gourieroux, C., Monfort, A., & Trognon, A. (1984). Pseudo-maximum likelihood methods: Theory. Econometrica, 52, 681–700. Heckman, J. J. (1979). Sample selection bias as a specification error. Econometrica, 47, 153–161. Newey, W., & West, K. (1987). A simple positive semi-definite heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55, 703–708. White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48, 817–838. White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50, 1–25. White, H. (1994). Estimation, inference and specification analysis. Econometric Society Monograph No. 22. Cambridge, UK: Cambridge University Press. Woutersen, T. M. (2001). Robustness against incidental parameters and mixing distributions, Working Paper, Department of Economics, University of Western Ontario.
A COMPARATIVE STUDY OF PURE AND PRETEST ESTIMATORS FOR A POSSIBLY MISSPECIFIED TWO-WAY ERROR COMPONENT MODEL Badi H. Baltagi, Georges Bresson and Alain Pirotte ABSTRACT In the spirit of White’s (1982) paper, this paper examines the consequences of model misspecification using a panel data regression model. Maximum likelihood, random and fixed effects estimators are compared using Monte Carlo experiments under normality of the disturbances but with a possibly misspecified variance-covariance matrix. We show that the correct GLS (ML) procedure is always the best according to MSE performance, but the researcher does not have perfect foresight on the true form of the variance covariance matrix. In this case, we show that a pretest estimator is a viable alternative given that its performance is a close second to correct GLS (ML) whether the true specification is a two-way, a one-way error component model or a pooled regression model. Incorrect GLS, ML or fixed effects estimators may lead to a big loss in MSE. A fundamental assumption underlying classical results on the properties of the maximum likelihood estimator . . . is that the stochastic law which determines the behavior of the phenomena investigated (the “true” structure) is known to lie within a specified parametric family of probability distributions (the model). In other words, the probability model is Maximum Likelihood Estimation of Misspecified Models: Twenty Years Later Advances in Econometrics, Volume 17, 1–27 Copyright © 2003 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(03)17001-6
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BADI H. BALTAGI, GEORGES BRESSON AND ALAIN PIROTTE assumed to be “correctly specified”. In many (if not most) circumstances, one may not have complete confidence that this is so. [e.g. White, 1982, p. 1]
1. INTRODUCTION Due to the non-experimental nature of econometrics, researchers use the same data set to select the model and to estimate the parameters in the selected model. Often, we employ a preliminary test estimator (or pretest estimator for short) whenever we test some aspect of a model’s specification and then decide, on the basis of the test results, what version of the model to estimate or what estimation method to use. Unfortunately, the statistical properties of pretest estimators are, in practice, very difficult to derive. The literature on pretest estimators is well surveyed in Judge and Bock (1978, 1983) and in the special issue of the Journal of Econometrics (1984) edited by Judge. More recently, asymptotic aspects have been considered and different selection strategies (especially general-to-specific and specific-to-general) have been discussed. For a summary of the latest developments, see Giles and Giles (1993) and Magnus (1999). In the panel data literature, few pretest studies have been conducted. These include Ziemer and Wetzstein (1983) on the pooling problem and Baltagi and Li (1997) on the estimation of error component models with autocorrelated disturbances. The Baltagi and Li (1997) Monte-Carlo study compares the finite sample performance of a number of pure and pretest estimators for an error component model with first-order AR or MA remainder disturbances. They show that the correct GLS procedure is always the best but the researcher does not have perfect foresight on which one it is (GLS assuming random error component with the remainder term following an AR(1) or MA(1) process). In that case, Baltagi and Li show that the pretest estimator is a viable alternative given that its performance is a close second to correct GLS whether the true serial correlation process is AR(1) or MA(1). It is the aim of this paper to study the small sample performance of pure and pretest estimators in a panel data regression model when the two-way error component structure for the disturbances may be misspecified. More specifically, this study performs Monte-Carlo experiments to compare the properties of 11 alternative estimators when the true model may be a two-way error component model with both individual and time effects, a one-way error component model with only time or individual effects, or simply a pooled regression model with no time or individual effects. The estimators considered are: Ordinary Least Squares (OLS), Fixed effects (Within), Feasible Generalized Least Squares (FGLS) and Maximum Likelihood (ML) assuming normality of the disturbances. The only
A Comparative Study of Pure and Pretest Estimators
3
type of misspecification considered is where one or both variance components are actually equal to zero. The pretest estimator is based on the results of two tests. The first test is the Kuhn-Tucker test suggested by Gouri´eroux, Holly and Monfort (1982) which tests whether both random effects are zero versus the alternative that at least one of them is positive. If the null hypothesis that both time and individual effects are jointly zero is not rejected, the pretest estimator reduces to OLS. If it is rejected, a second test proposed by Baltagi, Chang and Li (1992) is implemented. This is a conditional LM test for whether one of the variance components is zero given that the other variance component is positive. Either hypothesis can be tested first, the order does not matter. If both hypotheses are rejected, the pretest estimator reduces to the two-way FGLS estimator. If one of these hypotheses is rejected while the other is not, the pretest estimator reduces to a one-way FGLS estimator. Of course, if both hypotheses are not rejected, the pretest estimator reduces to the OLS estimator. This is described in the flow chart given below, see Fig. 1. The
Fig. 1. Flow Chart for the Pretest Estimator. Note: GHM: Gouri´eroux, Holly and Monfort test. BCL: Baltagi, Chang and Li test. Relative MSE = MSE(estimator)/MSE(True GLS).
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BADI H. BALTAGI, GEORGES BRESSON AND ALAIN PIROTTE
SAS computer code underlying the pretest estimator in the flow chart is given in the Appendix. Using Monte Carlo experiments, we compare the performance of this pretest estimator as well as the standard one-way and two-way error component estimators using the relative mean squared error (MSE) criterion. The basis for comparison is the MSE of the true GLS estimator which is based on the true variance components. In fact, the relative MSE of each estimator is obtained by dividing its MSE by that of true GLS. Section 2 describes the model and the pretest estimator. Section 3 presents the Monte-Carlo design and the results of experiments. Section 4 provides our conclusion.
2. PURE AND PRETEST ESTIMATORS FOR A TWO-WAY ERROR COMPONENT MODEL Consider the two-way error component model written in matrix notation: y = ␥NT + X + u
(1)
where y is an NT × 1 vector denoting the dependent variable y i,t stacked such that the slower index is i = 1, 2, . . ., N, and the faster index is t = 1, 2, . . ., T. ␥ is a scalar, NT is a vector of ones of dimension NT × 1, X is an NT × k matrix of independent variables x j,i,t , with j = 1, 2, . . ., k. The disturbances follow a two-way error component model u = Z + Z +
(2)
with Z = I N ⊗ T and Z = N ⊗ I T . I N and I T are identity matrices of dimension N ×N and T × T. N and T are vectors of ones of dimension N × 1 and T × 1. The variance-covariance matrix of u is given by = E[uu ] = 2u [(I N ⊗ J T ) + (J N ⊗ I T ) + (1 − − )I NT ]
(3)
where J T = T T , J N = N N , 2u = 2 + 2 + 2 , = 2 /2u and = 2 /2u , see Baltagi (2001). The GLS estimator of  is given by ˆ TGLS = (21 B i,xx + 22 B t,xx + W it,xx )−1 (21 B i,xy + 22 B t,xy + W it,xy ) where 21 =
2 , T2 + 2
22 =
2 N2 + 2
(4)
A Comparative Study of Pure and Pretest Estimators
5
JT JN ⊗ X IN − B i,xx = X N T JT JN B t,xx = X ⊗ IT − X N T JN JT W it,xx = X IN − ⊗ IT − X N T
with B denoting the Between variation and W denoting the Within variation. We compute the MSE resulting from blind estimation of (i) a two-way error component (TWEC) model, (ii) a one-way individual error component (OWIEC) model, (iii) a one-way time error component (OWTEC) model or (iv) a pooled regression model with no individual or time effects. Note that if both variance components are positive, i.e. 2 > 0 and 2 > 0, estimating this TWEC model by OLS or assuming it is a OWEC model yields to possible loss in efficiency. This loss will depend on the size of the panel, the magnitude of the variance components and the Between and Within variation in X. Based on the sequence of hypotheses tests described above on the variance components, the pretest estimator is either OLS, the OWEC (individual or time) FGLS, or the TWEC estimator, see the flow chart given in Fig. 1. This pretest estimator is based on two Lagrange Multiplier (LM) tests. The first one is based on modifying the Breusch and Pagan (1980) joint LM test for the null hypothesis that both the individual and time variance components are zero (see Baltagi, 2001): H 0,A :
2 = 0
and
2 = 0
(5)
This is done using the Kuhn-Tucker test suggested by Gouri´eroux, Holly and Monfort (1982) allowing for the fact that the alternative hypothesis is one-sided given that variances cannot be negative: 2 A + B2 A2 2m = B2 0
if A > 0, B > 0 if A > 0, B ≤ 0 if A ≤ 0, B > 0 if A ≤ 0, B ≤ 0
where 2m denotes the mixed 2 distribution and A=
NT u˜ (I N ⊗ J T )˜u −1 2(T − 1) u˜ u˜
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BADI H. BALTAGI, GEORGES BRESSON AND ALAIN PIROTTE
and
B=
u˜ (J N ⊗ I T )˜u NT −1 2(N − 1) u˜ u˜
and u˜ denote the OLS residuals. Under the null hypothesis:
2m ∼ 41 2 (0) + 21 2 (1) + 41 2 (2) If we do not reject the null hypothesis, the pretest estimator is the OLS estimator. If we reject H 0,A , we compute two conditional LM tests proposed by Baltagi, Chang and Li (1992). These test the following two null hypotheses with the order being immaterial: H 0,B :
2 = 0
given that
2 > 0
(6)
H 0,C :
2 = 0
given that
2 > 0
(7)
and
The corresponding LM test for testing H 0,B : 2 = 0 (assuming 2 > 0) is given by: √ 2 2 2˜ 2 ˜ ˜ LM = D T (T − 1)[˜ 4 + (N − 1)˜ 42 ] where
u˜ ((J N /N) ⊗ (J T /T))˜u −1 ˜ 22 N − 1 u˜ ((I N − J N /N) ⊗ (J T /T))˜u + −1 ˜ 2 (N − 1)˜ 2
˜ = T D 2
1 ˜ 22
with ˜ 22 = u˜ ((J N /N) ⊗ I T )˜u/T and ˜ 2 = u˜ ((I N − (J N /N)) ⊗ I T )˜u/T(N − 1). The estimated disturbances u˜ denote the one-way time effects GLS residuals using the maximum likelihood estimates ˜ 22 and ˜ 2 . Similarly, the alternative LM test statistics for testing H 0,C : 2 = 0 (assuming 2 > 0) can be obtained as follows: √ 2 2 2˜ 1 ˜ ˜ D LM = N(N − 1)[˜ 4 + (T − 1)˜ 41 ]
A Comparative Study of Pure and Pretest Estimators
where
7
u˜ ((J N /N) ⊗ (J T /T))˜u −1 ˜ 21 T − 1 u˜ ((J N /N) ⊗ (I T − (J T /T)))˜u + − 1 ˜ 2 (T − 1)˜ 2
˜ = N D 2
1 ˜ 21
with ˜ 21 = u˜ (I N ⊗ (J T /T))˜u/N and ˜ 2 = u˜ (I N ⊗ (I T − (J T /T)))˜u/N(T − 1). The estimated disturbances u˜ denote the one-way individual effects GLS residuals using the maximum likelihood estimates ˜ 21 and ˜ 2 . If both hypotheses H 0,B and H 0,C are rejected, the pretest estimator is the TWEC FGLS estimator. If we reject H 0,C but we do not reject H 0,B , the pretest estimator is the OWEC time FGLS estimator. Similarly, if we reject H 0,B but we do not reject H 0,C , the pretest estimator is the OWEC individual FGLS estimator. If both hypotheses H 0,B and H 0,C are not rejected, the pretest estimator is OLS. Hence, if both H0,B and H0,C are not rejected ˆ OLS ˆ OWT-FGLS if H0,B is not rejected, while H0,C is rejected ˆ pre = ˆ OWI-FGLS if H0,C is not rejected, while H0,B is rejected ˆ TW-FGLS if both H0,B and H0,C are rejected This pretest estimator is compared with the pure one-way and two-way estimators using Monte Carlo experiments.
3. THE MONTE CARLO RESULTS First, we describe the Monte Carlo design. We use the following simple regression model: y i,t = ␥ + x i,t + u i,t
i = 1, . . . , N
and
t = 1, . . . , T
(8)
with u i,t = i + t + i,t
(9)
Throughout the experiment1 ␥ = 5,  = 0.5 and 2 = 20. The latter means that the total variance of the disturbances is fixed at 20 for all experiments. We consider three values for the number of individuals in the panel (N = 25, 50 and 500); two values for the time dimension (T = 10 and 20). The exogenous variable is generated by choosing a random variable i,t , uniformly distributed on the
8
BADI H. BALTAGI, GEORGES BRESSON AND ALAIN PIROTTE
interval [−5, 5] and forming: x i,t = ␦1 x i,t−1 + i,t ,
␦1 = 0.5
For each cross-sectional unit T + 10 observations are generated starting with x i,0 = 0. The first ten observations are then dropped in order to reduce the dependency on initial values. The parameters and are varied over the set (0, 0.01, 0.2, 0.4, 0.6, 0.8) such that (1 − − ) is always positive. For each experiment, 5,000 replications were performed. For every replication, (NT + N + T) IIN(0, 1) random numbers were generated. The first N numbers were used to generate the i ’s as IIN(0, 2 ). The second T numbers were used to generate the t ’s as IIN(0, 2 ), and the last NT numbers were used to generate the i,t ’s as IIN(0, 2 ). Next, the u i,t ’s were formed from (9) and the y’s were calculated from (8). Twenty-six experiments were performed for each pair of values of N and T. This means that a total of 156 designs were run, and for each design, 11 estimators were computed 5,000 times.
3.1. Results of the LM Tests The Kuhn-Tucker test of Gouri´eroux, Holly and Monfort (GHM) performs well in testing the null of H 0,A : 2 = 2 = 0. When one of the individual or time variance components is more than 1% of the total variance, this test has high power rejecting the null in over 98% of the cases for T = 10 and N = 25. In fact, this power becomes almost 100% when T = 20 and/or N = 50, 500. The power performance improves as T increases for fixed N or as N increases for a fixed T. Also, the power of this test increases as the other variance component increases. For example, for (N, T) = (25, 10) and for ( = 0.01 and = 0, 0.01, 0.2, 0.4 and 0.6), the rate for rejection of the null increases very quickly from 7% to 12.7% to 97.9% to 99.9% to 100%. When both variances components are zero, the size of the test is around 5% for all values of N and T. In fact the frequency of type I error is 4.8%, 5.1% for N = 25 and T = (10, 20) and 4.4%, 5.5% for N = 50 and T = (10, 20) and 5.4%, 5.1% for N = 500 and T = (10, 20). The conditional LM test of Baltagi, Chang and Li (BCL) for testing H 0,B : 2 = 0 implicitly assumes that 2 > 0. Conversely, the other conditional LM test of BCL for testing H 0,C : 2 = 0 implicitly assumes that 2 > 0. These two conditional LM tests perform well. When = 0, and if 2 represents more than 1% of the total variance, the H 0,B test has high power rejecting the null in over 99.9% of the cases when N = 25 and T = 10. Similarly, when = 0, and if 2 represents more than 1% of the total variance, the H 0,C test has also high power rejecting the null in
A Comparative Study of Pure and Pretest Estimators
9
more than 99.9% of the cases when N = 25 and T = 10. This power performance improves as T increases for fixed N or as N increases for a fixed T. Finally, when both variance components are higher than 20% of the total variance, the non rejection rate for the TWEC estimator is more than 99.3% for N = 25 and T = 10 and increases to 100% for the other values of N and T considered. Using the flow chart given above, the pretest estimator is determined for each replication depending on the outcome of these tests. For 5,000 replications and (N, [T]) = (25, [10, 20]), (50, [10, 20]), (500, [10, 20]), this choice of the pretest estimator is given by Tables 1–3. For example, for N = 25 and T = 10, and = = 0.4, the pretest estimator is identical to the two-way FGLS estimator for all 5,000 replications. If = 0.01, and = 0.8, the pretest estimator is a two-way FGLS estimator in 2,259 replications and a one-way time effects FGLS estimator in the remaining 2,741 replications. We checked the sensitivity of our pretest estimator to the level of significance used by increasing it from 5 to 10%. The results are given in Table 1. For example, for N = 25 and T = 10 and = = 0.4, the 10% pretest estimator is still identical to the two-way FGLS estimator for all 5,000 replications. If = 0.01 and = 0.8, the 10% pretest estimator is a two-way FGLS estimator in 2,754 replications and a one-way time effects FGLS estimator in the remaining 2,246 replications. This is an improvement over the 5% pretest estimator in that we are selecting the right estimator more often.
3.2. Performance of the Estimators Now let us turn to the relative MSE comparisons of pure and pretest estimators for (N, [T]) = (25, [10, 20]), (50, [10, 20]) and (500, [10, 20]). Tables 4–6 report our findings and Figs 2–4 plot some of these results based on 5,000 replications. From these tables and figures, we observe the following: (1) When the true model is a pooled regression with no individual or time effects, i.e. = = 0, the two-way fixed effects estimator (Within) performs the worst. Its relative MSE with respect to true GLS, which in this case is OLS, yields 1.456 for (N = 25, T = 10) and 1.264 for (N = 25, T = 20). The Within estimator wipes out the Between variation whereas the OLS estimator weights both the Between and Within variation equally. When the Between variation is important, this can lead to considerable loss in efficiency. This performance improves as the variance components increase. The one-way fixed effects estimators perform better than the two-way Within estimator yielding a relative MSE of 1.041 and 1.384 for (N = 25, T = 10) and 1.056 and 1.190 for
10
Table 1. Number of Non-Rejections of H 0,A , H 0,B and H 0,C for 5,000 Replications N = 25, T = 10 and T = 20. Pretest GHM H 0,A T = 10
BCL OLS
T = 20
T = 10
BCL TIME
T = 20
T = 10
BCL INDIV
T = 20
T = 10
BCL TW
T = 20
T = 10
T = 20
5%
10%
5%
10%
5%
10%
5%
10%
5%
10%
5%
10%
5%
10%
5%
10%
5%
10%
5%
10%
0 0 0 0 0 0
0 0.01 0.2 0.4 0.6 0.8
4758 4648 109 0 0 0
4579 4411 62 0 0 0
4747 4375 2 0 0 0
4571 4068 1 0 0 0
6 18 18 2 0 4
19 16 12 2 0 3
0 4 0 0 0 0
1 1 0 0 0 0
116 101 4 0 0 0
174 156 4 0 0 0
117 117 0 0 0 0
189 182 0 0 0 0
102 203 4601 4691 4685 4675
191 354 4481 4491 4465 4486
112 452 4683 4676 4652 4708
191 628 4487 4447 4443 4481
18 30 268 307 315 321
37 63 441 507 535 511
24 52 315 324 348 292
48 121 512 553 557 519
0.01 0.01 0.01 0.01 0.01 0.01
0 0.01 0.2 0.4 0.6 0.8
4479 4366 88 0 0 0
4223 4069 58 0 0 0
4296 3958 0 0 0 0
3961 3532 0 0 0 0
14 9 3 0 0 0
8 16 4 0 0 0
3 1 0 0 0 0
3 0 0 0 0 0
377 341 4 0 0 0
519 460 4 0 0 0
563 457 0 0 0 0
760 587 0 0 0 0
101 200 3983 3818 3219 1841
174 269 3655 3411 2779 1490
84 421 3701 3201 2453 766
141 550 3256 2727 1962 556
29 84 922 1182 1781 3159
76 186 1279 1589 2221 3510
54 163 1299 1799 2547 4234
135 331 1744 2273 3038 4444
0.2 0.2 0.2 0.2 0.2
0 0.01 0.2 0.4 0.6
135 106 0 0 0
90 68 0 0 0
1 1 0 0 0
0 0 0 0 0
7 1 0 0 0
5 1 0 0 0
0 0 0 0 0
0 0 0 0 0
4596 4310 12 0 0
4436 4046 8 0 0
4714 3901 0 0 0
4499 3438 0 0 0
5 3 24 4 0
4 5 21 2 0
0 0 0 0 0
0 0 0 0 0
257 580 4964 4996 5000
465 880 4971 4998 5000
285 1098 5000 5000 5000
501 1562 5000 5000 5000
0.4 0.4 0.4 0.4
0 0.01 0.2 0.4
4 4 0 0
1 3 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
4721 4211 0 0
4524 3840 0 0
4716 3519 0 0
4520 3003 0 0
0 0 2 0
0 0 1 0
0 0 0 0
0 0 0 0
275 785 4998 5000
475 1157 4999 5000
284 1481 5000 5000
480 1997 5000 5000
0.6 0.6 0.6
0 0.01 0.2
0 1 0
0 1 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
4726 3905 0
4526 3442 0
4727 2640 0
4514 2109 0
0 0 0
0 0 0
0 0 0
0 0 0
274 1094 5000
474 1557 5000
273 2360 5000
486 2891 5000
0.8 0.8
0 0.01
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
4683 2741
4477 2246
4708 938
4498 659
0 0
0 0
0 0
0 0
317 2259
523 2754
292 4062
502 4341
GHM H 0,A : Gouri´eroux, Holly and Monfort test (number of non-rejections). If GHM H 0,A is rejected then BCL tests: BCL OLS : both H 0,B and H 0,C are not rejected; BCL TIME : H 0,B is not rejected while H 0,C is rejected BCL INDIV : H 0,C is not rejected while H 0,B is rejected; BCL TW : both H 0,B and H 0,C are rejected.
BADI H. BALTAGI, GEORGES BRESSON AND ALAIN PIROTTE
A Comparative Study of Pure and Pretest Estimators
11
Table 2. Number of Non-Rejections of H 0,A , H 0,B and H 0,C for 5,000 Replications N = 50, T = 10 and T = 20. Pretest GHM H 0,A
BCL OLS
BCL TIME
BCL INDIV
BCL TW
T = 10
T = 20
T = 10
T = 20
T = 10
T = 20
T = 10
T = 20
T = 10
T = 20
0 0 0 0 0 0
0 0.01 0.2 0.4 0.6 0.8
4782 4529 2 0 0 0
4726 4086 0 0 0 0
0 7 0 0 0 0
1 3 0 0 0 0
110 105 0 0 0 0
139 99 0 0 0 0
91 316 4711 4693 4687 4663
109 741 4695 4685 4703 4705
17 43 287 307 313 337
25 71 305 315 297 295
0.01 0.01 0.01 0.01 0.01 0.01
0 0.01 0.2 0.4 0.6 0.8
4013 3816 0 0 0 0
3424 2726 0 0 0 0
3 1 0 0 0 0
3 3 0 0 0 0
821 749 0 0 0 0
1369 1153 0 0 0 0
90 253 3325 2820 1871 658
80 567 2447 1688 732 66
73 181 1675 2180 3129 4342
124 551 2553 3312 4268 4934
0.2 0.2 0.2 0.2 0.2
0 0.01 0.2 0.4 0.6
7 18 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
4721 4197 0 0 0
4712 3335 0 0 0
0 0 0 1 0
0 0 0 0 0
272 785 5000 4999 5000
288 1665 5000 5000 5000
0.4 0.4 0.4 0.4
0 0.01 0.2 0.4
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
4725 4005 0 0
4697 2688 0 0
0 0 0 0
0 0 0 0
275 995 5000 5000
303 2312 5000 5000
0.6 0.6 0.6
0 0.01 0.2
0 0 0
0 0 0
0 0 0
0 0 0
4726 3399 0
4705 1610 0
0 0 0
0 0 0
274 1601 5000
295 3390 5000
0.8 0.8
0 0.01
0 0
0 0
0 0
0 0
4713 1610
4684 161
0 0
0 0
287 3390
316 4839
(N = 25, T = 20). Two-way FGLS is slightly worse than OLS yielding a relative MSE of 1.007 for (N = 25, T = 10) and 1.010 for (N = 25, T = 20). One-way FGLS performs better than two-way FGLS yielding a relative MSE of 1.000 and 1.005 for (N = 25, T = 10) and 1.002 and 1.005 for (N = 25, T = 20). The ML estimators perform well yielding a relative MSE that is no more than 1.004 for (N = 25, T = 10), while the 5 and 10% pretest estimator yield a relative MSE no more than 1.003 for (N = 25, T = 10) and no more than 1.002 for (N = 25, T = 20). (2) When the true specification is a one-way individual error component model, OWIEC ( = 0), the two-way Within estimator continues to perform badly relative to true GLS yielding a relative MSE of 1.066 for = 0.8 and N = 25,
12
BADI H. BALTAGI, GEORGES BRESSON AND ALAIN PIROTTE
Table 3. Number of Non-Rejections of H 0,A , H 0,B and H 0,C for 5,000 Replications N = 500, T = 10 and T = 20. Pretest GHM H 0,A
BCL OLS
BCL TIME
BCL INDIV
BCL TW
T = 10
T = 20
T = 10
T = 20
T = 10
T = 20
T = 10
T = 20
T = 10
T = 20
0 0 0 0 0 0
0 0.01 0.2 0.4 0.6 0.8
4729 3425 0 0 0 0
4743 923 0 0 0 0
1 7 0 0 0 0
1 0 0 0 0 0
152 85 0 0 0 0
128 14 0 0 0 0
103 1384 4676 4683 4692 4653
111 3865 4712 4688 4698 4710
15 99 324 317 308 347
17 198 288 312 302 290
0.01 0.01 0.01 0.01 0.01 0.01
0 0.01 0.2 0.4 0.6 0.8
198 109 0 0 0 0
2 0 0 0 0 0
2 2 0 0 0 0
0 0 0 0 0 0
4549 2643 0 0 0 0
4753 530 0 0 0 0
6 50 64 21 7 1
1 0 2 1 0 0
245 2196 4936 4979 4993 4999
244 4470 4998 4999 5000 5000
0.2 0.2 0.2 0.2 0.2
0 0.01 0.2 0.4 0.6
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
4762 2171 0 0 0
4746 138 0 0 0
0 0 0 0 0
0 0 0 0 0
238 2829 5000 5000 5000
254 4862 5000 5000 5000
0.4 0.4 0.4 0.4
0 0.01 0.2 0.4
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
4773 1068 0 0
4738 14 0 0
0 0 0 0
0 0 0 0
227 3932 5000 5000
262 4986 5000 5000
0.6 0.6 0.6
0 0.01 0.2
0 0 0
0 0 0
0 0 0
0 0 0
4759 169 0
4761 0 0
0 0 0
0 0 0
241 4831 5000
239 5000 5000
0.8 0.8
0 0.01
0 0
0 0
0 0
0 0
4720 0
4745 0
0 0
0 0
280 5000
255 5000
T = 10 (see Table 4). The corresponding relative MSE for OLS is 9.090. The correct one-way fixed effects estimator has a relative MSE of 1.007 while the wrong one-way fixed effects estimator has a relative MSE of 9.772. These results are quite stable as N increases from 25 to 50 to 500. The two-way FGLS estimator yields a relative MSE of 1.004 for = 0.8, while the correct one-way FGLS estimator yields a relative MSE of 1.002. The wrong one-way FGLS yields a relative MSE of 9.090. The performance of two-way ML is similar to that of FGLS yielding a relative MSE of 1.003. This is compared to 1.002 for the correct one-way ML and 9.093 for the wrong one-way ML estimator. The pretest estimator yields a relative MSE of 1.007 for the 5% level
Table 4. Relative MSE of  with Respect to True GLS for 5,000 Replications N = 25, T = 10 and T = 20. Two-Way Error Component OLS
Within
FGLS
One-Way Time Error Component ML
Within
FGLS
One-Way Individual Error Component
ML
Within
FGLS
ML
0 0 0 0 0 0
0 0.01 0.2 0.4 0.6 0.8
1.000 1.000 1.003 1.005 1.405 1.553 2.138 2.617 3.763 4.852 9.090 11.161
1.456 1.440 1.180 1.119 1.087 1.066
1.264 1.215 1.104 1.080 1.073 1.065
1.007 1.017 1.016 1.009 1.003 1.004
1.010 1.009 1.006 1.003 1.006 1.002
1.003 1.011 1.017 1.008 1.003 1.003
1.006 1.007 1.006 1.003 1.006 1.001
1.041 1.056 1.050 1.059 1.488 1.672 2.273 2.860 4.022 5.320 9.772 12.275
1.000 1.002 1.007 1.008 1.409 1.556 2.142 2.619 3.763 4.852 9.090 11.161
1.000 1.002 1.006 1.008 1.409 1.556 2.142 2.619 3.764 4.854 9.093 11.166
1.384 1.353 1.104 1.062 1.028 1.007
1.190 1.143 1.031 1.014 1.005 1.001
1.005 1.010 1.011 1.004 1.000 1.002
1.005 1.004 1.002 1.001 1.001 1.000
1.004 1.007 1.013 1.004 1.000 1.002
1.004 1.004 1.003 1.001 1.002 1.000
0.01 0.01 0.01 0.01 0.01 0.01
0 0.01 0.2 0.4 0.6 0.8
1.002 1.001 1.001 1.006 1.354 1.571 2.156 2.625 3.890 4.772 9.430 11.051
1.501 1.463 1.155 1.102 1.058 1.029
1.257 1.225 1.073 1.060 1.046 1.024
1.017 1.014 1.012 1.011 1.004 1.007
1.010 1.009 1.005 1.001 1.005 1.003
1.011 1.010 1.013 1.011 1.005 1.009
1.007 1.034 1.042 1.007 1.042 1.045 1.006 1.412 1.673 1.001 2.296 2.852 1.006 4.149 5.223 1.004 10.127 12.102
1.002 1.002 1.004 1.005 1.361 1.579 2.166 2.634 3.897 4.780 9.437 11.052
1.002 1.002 1.004 1.005 1.361 1.579 2.165 2.633 3.897 4.781 9.440 11.056
1.434 1.391 1.108 1.061 1.030 1.056
1.205 1.172 1.031 1.025 1.014 1.045
1.013 1.008 1.011 1.009 1.012 1.045
1.007 1.008 1.007 1.007 1.011 1.044
1.010 1.007 1.013 1.010 1.012 1.045
0.2 0.2 0.2 0.2 0.2
0 0.01 0.2 0.4 0.6
1.183 1.181 1.725 2.927 7.169
1.252 1.260 2.056 3.799 9.422
1.427 1.403 1.110 1.050 1.020
1.211 1.162 1.038 1.017 1.006
1.015 1.011 1.007 1.003 1.002
1.006 1.009 1.001 1.002 1.000
1.011 1.008 1.008 1.004 1.002
1.005 1.009 1.001 1.002 1.000
1.005 1.010 1.562 2.771 7.114
1.007 1.013 1.827 3.535 9.160
1.003 1.005 1.552 2.747 7.033
1.000 1.008 1.811 3.502 9.050
1.003 1.005 1.552 2.747 7.033
1.000 1.008 1.811 3.502 9.049
1.789 1.737 1.563 1.614 2.168
1.553 1.513 1.429 1.650 2.299
1.192 1.189 1.368 1.522 2.092
1.256 1.270 1.390 1.631 2.289
0.4 0.4 0.4 0.4
0 0.01 0.2 0.4
1.543 1.598 2.480 6.084
1.702 1.753 3.157 7.878
1.471 1.346 1.063 1.029
1.196 1.142 1.016 1.008
1.016 1.007 1.006 1.003
1.005 1.008 1.004 1.000
1.011 1.006 1.007 1.003
1.004 1.009 1.004 1.000
1.002 1.010 1.914 5.029
1.005 1.017 2.257 6.315
1.000 1.011 1.908 5.019
1.000 1.015 2.254 6.294
1.000 1.011 1.908 5.019
1.000 1.015 2.254 6.294
2.555 2.525 2.360 3.668
2.171 2.161 2.363 3.745
1.546 1.603 2.051 3.421
0.6 0 2.354 0.6 0.01 2.360 0.6 0.2 4.775
2.718 1.457 2.716 1.373 6.168 1.041
1.180 1.123 1.012
1.015 1.011 1.005
1.005 1.006 1.000
1.010 1.009 1.005
1.005 1.007 1.000
1.001 1.009 2.934
1.001 1.001 1.030 1.009 3.531 2.932
1.001 1.001 1.029 1.009 3.526 2.932
1.001 4.107 1.029 4.065 3.525 4.867
3.585 3.446 4.973
0.8 0 4.492 0.8 0.01 4.444
5.679 1.443 5.278 1.292
1.191 1.094
1.013 1.012
1.007 1.004
1.005 1.009
1.005 1.007
1.001 1.039
1.000 0.999 1.069 1.038
1.000 0.999 1.068 1.038
1.000 8.266 1.068 8.047
7.692 7.016
Pretest T = 10
T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 5%
T = 20
10%
5%
10%
1.003 1.009 1.019 1.008 1.002 1.007
1.003 1.009 1.016 1.009 1.002 1.004
1.002 1.007 1.006 1.002 1.003 1.001
1.002 1.008 1.007 1.003 1.005 1.002
1.007 1.008 1.007 1.007 1.011 1.044
1.008 1.007 1.013 1.010 1.007 1.014
1.010 1.009 1.013 1.012 1.006 1.011
1.004 1.006 1.008 1.002 1.008 1.005
1.005 1.006 1.008 1.001 1.008 1.004
1.188 1.187 1.368 1.522 2.091
1.256 1.269 1.390 1.631 2.289
1.008 1.012 1.007 1.003 1.002
1.011 1.011 1.007 1.003 1.002
1.003 1.014 1.001 1.002 1.000
1.003 1.012 1.001 1.002 1.000
1.703 1.760 2.299 3.688
1.547 1.601 2.051 3.421
1.704 1.759 2.299 3.688
1.005 1.015 1.006 1.003
1.007 1.014 1.006 1.003
1.002 1.016 1.004 1.000
1.003 1.014 1.004 1.000
2.355 2.364 4.144
2.718 2.718 4.768
2.359 2.366 4.141
2.721 1.005 1.007 1.003 1.004 2.719 1.016 1.017 1.015 1.012 4.767 1.005 1.005 1.000 1.000
4.492 4.444
5.679 5.279
4.502 4.453
5.686 1.002 1.007 1.003 1.004 5.284 1.024 1.026 1.010 1.006
Table 5. Relative MSE of  with Respect to True GLS for 5,000 Replications N = 50, T = 10 and T = 20. Two-Way Error Component OLS
Within
FGLS
One-Way Time Error Component ML
Within
FGLS
One-Way Individual Error Component
ML
Within
FGLS
ML
Pretest
T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20
0 0 0 0 0 0
0 0.01 0.2 0.4 0.6 0.8
1.000 1.001 1.345 2.231 3.850 9.091
1.000 1.004 1.462 2.425 4.381 10.433
1.457 1.441 1.167 1.103 1.071 1.071
1.159 1.133 1.037 1.013 1.015 1.012
1.005 1.010 1.004 1.007 1.004 1.006
1.006 1.004 1.002 0.999 1.001 1.001
1.002 1.008 1.004 1.006 1.004 1.005
1.005 1.004 1.002 0.999 1.001 1.000
1.038 1.038 1.418 2.386 4.103 9.766
1.010 1.012 1.480 2.449 4.446 10.599
1.001 1.003 1.349 2.234 3.850 9.091
1.001 1.005 1.463 2.425 4.381 10.433
1.001 1.003 1.349 2.235 3.851 9.094
1.001 1.005 1.463 2.425 4.382 10.434
1.385 1.370 1.115 1.046 1.021 1.006
1.146 1.121 1.027 1.010 1.005 1.001
1.002 1.007 1.001 1.003 1.002 1.001
1.005 1.003 1.003 1.000 1.000 1.000
1.001 1.006 1.001 1.003 1.003 1.001
1.004 1.003 1.003 1.000 1.000 1.000
1.002 1.005 1.003 1.006 1.004 1.003
1.003 1.005 1.002 1.000 1.000 1.000
0.01 0.01 0.01 0.01 0.01 0.01
0 0.01 0.2 0.4 0.6 0.8
1.009 1.008 1.350 2.123 4.012 8.915
1.001 1.004 1.452 2.483 4.340 10.698
1.407 1.428 1.164 1.072 1.045 1.027
1.160 1.130 1.036 1.008 1.005 1.007
1.006 1.008 1.013 1.005 1.005 1.000
1.003 1.007 1.000 1.000 1.000 1.000
1.005 1.007 1.012 1.005 1.005 0.999
1.002 1.006 1.000 1.000 1.001 1.000
1.019 1.024 1.411 2.223 4.269 9.550
1.007 1.012 1.462 2.507 4.394 10.871
1.004 1.004 1.359 2.135 4.036 8.944
1.000 1.004 1.451 2.485 4.346 10.702
1.004 1.004 1.359 2.134 4.035 8.944
1.000 1.004 1.451 2.485 4.346 10.702
1.385 1.393 1.143 1.082 1.061 1.104
1.151 1.123 1.038 1.010 1.013 1.016
1.011 1.014 1.019 1.030 1.037 1.089
1.003 1.006 1.005 1.005 1.009 1.013
1.010 1.013 1.019 1.030 1.037 1.089
1.002 1.005 1.005 1.005 1.009 1.013
1.007 1.012 1.016 1.015 1.010 1.002
1.001 1.008 1.000 1.000 1.001 1.000
0.2 0.2 0.2 0.2 0.2
0 0.01 0.2 0.4 0.6
1.417 1.408 1.916 3.361 7.924
1.110 1.110 1.770 3.310 8.281
1.401 1.353 1.080 1.027 1.013
1.130 1.103 1.016 1.010 1.002
1.004 1.007 1.003 1.003 1.000
1.000 1.003 1.002 1.001 1.000
1.004 1.006 1.003 1.003 1.000
1.000 1.003 1.002 1.001 1.000
1.003 1.004 1.538 2.960 7.178
0.999 1.008 1.679 3.141 8.010
1.000 1.004 1.534 2.942 7.137
1.000 1.008 1.678 3.141 8.002
1.000 1.004 1.534 2.942 7.137
1.000 1.008 1.678 3.141 8.002
2.230 2.131 1.912 2.213 3.503
1.272 1.236 1.160 1.244 1.478
1.423 1.419 1.671 2.064 3.361
1.110 1.109 1.142 1.234 1.477
1.422 1.417 1.669 2.063 3.362
1.110 1.109 1.142 1.234 1.477
1.003 1.009 1.003 1.003 1.000
1.000 1.007 1.002 1.001 1.000
0.4 0.4 0.4 0.4
0 0.01 0.2 0.4
2.152 2.121 3.108 7.124
1.272 1.282 2.363 6.250
1.442 1.359 1.092 1.021
1.154 1.103 1.012 1.000
1.007 1.008 1.006 1.003
1.003 1.005 1.000 1.000
1.005 1.008 1.006 1.003
1.002 1.005 1.000 1.000
1.002 1.007 1.839 5.027
1.000 1.012 2.035 5.678
1.000 1.005 1.835 5.014
1.000 1.012 2.034 5.675
1.000 1.005 1.835 5.014
1.000 1.012 2.034 5.675
3.693 3.504 3.750 5.770
1.513 1.462 1.471 1.946
2.154 2.126 3.072 5.276
1.272 1.282 1.451 1.939
2.157 2.127 3.066 5.273
1.273 1.282 1.451 1.939
1.002 1.011 1.006 1.003
1.002 1.011 1.000 1.000
0.6 0.6 0.6
0 3.717 0.01 3.563 0.2 7.027
1.608 1.584 4.220
1.434 1.332 1.028
1.150 1.101 1.006
1.004 1.009 1.002
1.002 1.000 1.000
1.002 1.007 1.002
1.002 1.000 1.000
1.002 1.014 2.992
1.000 1.019 3.166
1.000 1.014 2.990
1.000 1.019 3.167
1.000 1.014 2.990
1.000 1.019 3.166
6.812 6.275 9.060
1.959 1.841 2.358
3.717 3.565 7.296
1.608 1.583 2.363
3.725 3.571 7.275
1.609 1.584 2.363
1.000 1.016 1.002
1.000 1.003 1.000
0.8 0.8
0 8.309 0.01 7.560
2.605 2.572
1.434 1.257
1.156 1.068
1.005 1.008
1.003 1.002
1.002 1.010
1.002 1.005
0.999 1.040
1.000 1.068
1.000 1.040
1.000 1.068
1.000 1.040
1.000 15.556 1.068 13.760
3.280 3.097
8.309 7.561
2.605 2.572
8.330 7.578
2.608 2.573
1.002 1.016
1.002 1.003
Table 6. Relative MSE of  with Respect to True GLS for 5,000 Replications N = 500, T = 10 and T = 20. Two-Way Error Component OLS
Within
FGLS
One-Way Time Error Component ML
Within
FGLS
One-Way Individual Error Component
ML
Within
FGLS
ML
Pretest
T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20
0 0 0 0 0 0
0 0.01 0.2 0.4 0.6 0.8
1.0000 1.0000 1.0003 1.0042 1.3294 1.4564 2.2113 2.5550 3.7842 4.3887 8.9208 10.2392
1.3847 1.3620 1.1117 1.0434 1.0238 1.0070
1.1634 1.1278 1.0318 1.0033 1.0030 1.0026
0.9995 0.9995 1.0006 1.0004 1.0012 0.9999
1.0005 1.0013 1.0001 0.9997 1.0000 1.0004
0.9995 0.9994 1.0005 1.0006 1.0009 1.0000
1.0004 1.0013 1.0000 0.9998 0.9999 1.0005
1.0005 1.0012 1.0020 1.0057 1.3319 1.4573 2.2143 2.5601 3.7886 4.3937 8.9306 10.2588
0.9999 1.0001 1.0005 1.0044 1.3296 1.4563 2.2114 2.5551 3.7842 4.3887 8.9208 10.2392
0.9999 1.0001 1.0005 1.0044 1.3296 1.4563 2.2114 2.5551 3.7842 4.3887 8.9208 10.2393
1.3839 1.3602 1.1098 1.0429 1.0229 1.0075
1.1618 1.1264 1.0311 1.0020 1.0043 1.0010
0.9996 0.9992 1.0003 1.0004 1.0009 1.0000
1.0003 1.0011 1.0003 0.9999 1.0002 1.0000
0.9997 0.9992 1.0003 1.0006 1.0007 1.0000
1.0003 1.0011 1.0003 1.0000 1.0001 1.0001
0.9991 1.0000 1.0005 1.0005 1.0010 1.0000
1.0005 1.0022 1.0001 0.9998 1.0001 1.0002
0.01 0.01 0.01 0.01 0.01 0.01
0 0.01 0.2 0.4 0.6 0.8
1.0035 1.0082 1.0034 1.0102 1.3410 1.4964 2.1063 2.4500 3.9391 4.4676 9.5283 10.5948
1.3681 1.3583 1.1122 1.0558 1.0253 1.0044
1.1420 1.1288 1.0258 1.0080 1.0053 1.0033
1.0007 1.0018 1.0004 0.9998 0.9999 1.0000
1.0003 1.0012 1.0001 0.9998 1.0001 1.0002
1.0007 1.0018 1.0003 0.9998 0.9999 1.0000
1.0003 1.0012 1.0000 0.9998 1.0000 1.0004
1.0000 0.9993 1.0007 1.0044 1.3378 1.4916 2.1022 2.4504 3.9364 4.4620 9.5312 10.5994
1.0001 1.0001 1.0004 1.0047 1.3382 1.4916 2.1025 2.4489 3.9363 4.4608 9.5283 10.5945
1.0001 1.0001 1.0004 1.0048 1.3382 1.4916 2.1025 2.4489 3.9363 4.4608 9.5283 10.5944
1.3723 1.3613 1.1197 1.0632 1.0386 1.0316
1.1520 1.1357 1.0334 1.0150 1.0185 1.0308
1.0040 1.0047 1.0059 1.0069 1.0125 1.0265
1.0083 1.0069 1.0072 1.0068 1.0133 1.0275
1.0040 1.0047 1.0059 1.0069 1.0126 1.0266
1.0082 1.0069 1.0072 1.0067 1.0134 1.0276
1.0006 1.0018 1.0004 0.9998 0.9999 1.0000
1.0000 1.0018 1.0001 0.9998 1.0001 1.0002
0.2 0.2 0.2 0.2 0.2
0 0.01 0.2 0.4 0.6
1.0681 1.0782 1.5772 2.8667 6.8041
1.1324 1.1347 1.8159 3.4774 8.2858
1.3704 1.3086 1.0814 1.0388 1.0112
1.1673 1.1217 1.0223 1.0076 0.9993
1.0004 1.0005 1.0009 1.0002 0.9999
1.0004 1.0008 1.0000 1.0001 1.0000
1.0003 1.0005 1.0009 1.0003 0.9997
1.0003 1.0008 0.9999 1.0001 1.0002
1.0001 1.0052 1.4799 2.7436 6.6017
0.9999 1.0071 1.6590 3.2521 7.8137
1.0000 1.0052 1.4800 2.7435 6.6014
1.0000 1.0071 1.6591 3.2523 7.8141
1.0000 1.0051 1.4801 2.7435 6.6014
0.9999 1.0071 1.6591 3.2524 7.8142
1.4919 1.4636 1.2527 1.2694 1.4786
1.3394 1.2871 1.2219 1.2677 1.6046
1.0681 1.0780 1.1494 1.2150 1.4515
1.1324 1.1333 1.1927 1.2592 1.6063
1.0683 1.0779 1.1495 1.2152 1.4517
1.1326 1.1333 1.1927 1.2591 1.6062
1.0000 1.0027 1.0009 1.0002 0.9999
1.0001 1.0009 1.0000 1.0001 1.0000
0.4 0.4 0.4 0.4
0 0.01 0.2 0.4
1.2314 1.2073 2.1520 5.0555
1.3080 1.3702 2.4451 6.3740
1.3726 1.2901 1.0437 1.0152
1.1448 1.1010 1.0163 1.0053
1.0014 1.0010 1.0007 1.0000
1.0005 0.9997 1.0002 0.9999
1.0013 1.0010 1.0008 0.9999
1.0005 0.9997 1.0003 0.9999
1.0000 1.0079 1.8988 4.6241
1.0000 1.0127 2.0180 5.4440
1.0000 1.0079 1.8988 4.6240
1.0000 1.0128 2.0180 5.4441
1.0000 1.0079 1.8988 4.6239
1.0000 1.0128 2.0180 5.4443
1.8041 1.6786 1.5496 1.9361
1.5472 1.5734 1.5928 2.1714
1.2314 1.2070 1.4534 1.8801
1.3080 1.3701 1.5575 2.1567
1.2321 1.2072 1.4535 1.8803
1.3083 1.3702 1.5575 2.1569
1.0008 1.0022 1.0007 1.0000
1.0001 0.9997 1.0002 0.9999
0.6 0.6 0.6
0 1.5187 0.01 1.5049 0.2 3.6441
1.7458 1.3640 1.1626 1.0007 1.0013 1.0006 1.0012 1.0000 1.7692 1.3013 1.1152 1.0000 0.9994 1.0000 0.9995 1.0086 4.5223 1.0403 1.0054 1.0002 1.0000 0.9999 0.9999 2.8528
1.0000 1.0000 1.0146 1.0086 3.2219 2.8528
1.0000 1.0001 1.0146 1.0087 3.2219 2.8529
1.0001 2.3217 2.1538 1.5187 1.7458 1.5200 1.7468 1.0002 1.0004 1.0147 2.1883 2.1151 1.5050 1.7691 1.5058 1.7696 1.0000 0.9994 3.2218 2.4157 2.7416 2.3414 2.6995 2.3421 2.6994 1.0002 1.0000
0.8 0.8
0 2.2152 0.01 2.3828
2.8935 1.3399 1.1702 1.0002 1.0014 0.9997 1.0008 1.0000 2.9377 1.2511 1.0701 1.0020 1.0010 1.0045 1.0065 1.0391
1.0000 1.0000 1.0674 1.0392
1.0000 0.9996 1.0673 1.0395
0.9997 3.6829 3.7162 2.2152 2.8935 2.2185 2.8962 1.0004 1.0008 1.0669 3.7392 3.5924 2.3828 2.9377 2.3853 2.9392 1.0020 1.0010
16 BADI H. BALTAGI, GEORGES BRESSON AND ALAIN PIROTTE
Fig. 2. Relative MSE of  for N = 25 and T = 10 for 5,000 Replications ( = 0).
A Comparative Study of Pure and Pretest Estimators
Fig. 3. Relative MSE of  for the N = 25 and T = 10 for 5,000 Replications ( = 0). 17
18 BADI H. BALTAGI, GEORGES BRESSON AND ALAIN PIROTTE
Fig. 4. Relative MSE of  for N = 25 and T = 10 for 5,000 Replications ( = 0.2).
A Comparative Study of Pure and Pretest Estimators
19
and 1.004 for the 10% level. Figure 2, corresponding to N = 25 and T = 10, confirms that for = 0, the wrong one-way Within estimator performs the worst in terms of relative MSE. For high values of (0.5–0.8), its relative MSE explodes. Similarly, OLS, the wrong one-way FGLS and ML estimators have the same increasing shape for their relative MSE as gets large. Inside the box, the correct one-way ML and FGLS perform the best followed by two-way ML and FGLS. The 5% pretest estimator is in that box with relative MSE not higher than 1.041 for N = 25, T = 10, = 0 and = 0.1. (3) Similarly, when the true specification is a one-way time effect error component model, OWTEC ( = 0), the two-way Within estimator continues to perform badly relative to true GLS yielding a relative MSE of 1.443 for = 0.8 and N = 25, T = 10 (see Table 4). The corresponding relative MSE for OLS is 4.492. The correct one-way fixed effects estimator has a relative MSE of 1.001 while the wrong one-way fixed effects estimator has a relative MSE of 8.266. The two-way FGLS estimator yields a relative MSE of 1.013 for = 0.8, while the correct one-way FGLS estimator yields a relative MSE of 0.999. The wrong one-way FGLS yields a relative MSE of 4.492. The performance of two-way ML is similar to that of FGLS yielding a relative MSE of 1.005. This is compared to 0.999 for the correct one-way ML and 4.502 for the wrong one-way ML estimator. The 5% pretest estimator yields a relative MSE of 1.002 while the 10% pretest estimator yields a relative MSE of 1.007. Figure 3, corresponding to N = 25 and T = 10, confirms that for = 0, the wrong one-way Within estimator performs the worst in terms of relative MSE. For high values of (0.5–0.8), its relative MSE explodes. Similarly, OLS, the wrong one-way FGLS and ML estimators have the same increasing shape for their relative MSE as gets large. Inside the box, the correct one-way ML and FGLS perform the best followed by two-way ML and FGLS. The 5% pretest estimator is in that box with relative MSE not higher than 1.015 for N = 25, T = 10, = 0 and = 0.1. (4) When the true specification is a two-way error component model, TWEC, say = = 0.4, the two-way Within estimator performs well with a relative MSE of 1.029 for N = 25, T = 10 compared to 6.084 for OLS, see Table 4. The two-way FGLS and ML estimators have a relative MSE of 1.003. The wrong one-way time effects FGLS and ML estimators have a relative MSE of 5.019, while the wrong one-way individual effects FGLS and ML estimators have a relative MSE of 3.421. The wrong one-way fixed effects estimators have a relative MSE of 5.029 for time and 3.668 for individual effects. The pretest estimator matches the two-way FGLS estimator in MSE performance. This is not surprising given the results of the tests in Table 1. The larger the variance components the more likely it is for the pretest to match the two-way
20
BADI H. BALTAGI, GEORGES BRESSON AND ALAIN PIROTTE
FGLS estimator in 100% of the cases. Figure 4, corresponding to N = 25 and T = 10, confirms that for = 0.2, OLS performs the worst, followed by the wrong one-way time effects estimators and the wrong one-way individual effects estimators, whether this is Within, FGLS or ML estimators. Inside the box, the correct two-way ML and FGLS estimators perform the best, followed closely by the 5% pretest estimator. The latter has relative MSE no higher than 1.022 for N = 25, T = 10, = 0.2 and = 0.1. In summary, it is clear from Figs 2–4 that the wrong Within, FGLS or ML estimators can result in a big loss in MSE. The pretest estimator comes out with a clear bill of health being no more than 3% above the MSE of true GLS. This performance improves as N or T increases. It is also not that sensitive to doubling the size of the test from 5 to 10%. 3.3. Robustness to Non-Normality So far, we have been assuming that the error components have been generated by the normal distribution. In this section, we check the sensitivity of our results to non-normal disturbances. In particular, we generate the i ’s and t ’s from 2 distributions and we let the remainder disturbances follow the normal distribution. In all experiments, we fix the total variance to be 20. Table 7 gives the choice of the pretest estimator under non-normality of the random effects for N = 25 and T = 10, 20 using the 5 and 10% significance levels, respectively. Comparing the results to the normal case in Table 1, we find that the GHM test committs a higher probability of type II error under non-normality. For example, when = 0 and = 0.2, GHM does not reject the null when false in 34% of the cases at the 5% level of significance and 29% of the cases at the 10% level of significance as compared to 2.7 and 1.8% of the cases under normality. The BCL tests are affected in the same way too. For = 0.2 and = 0.4, the choice of the pretest as a two-way estimator is only 89.2% at the 5% level and 92.6% at the 10% level. This is compared to almost 100% of the cases under normality no matter what significance level is chosen. Despite this slight deterioration in the correct choice of pretest estimator under non-normality, the resulting relative MSE performance reported in Table 8 seems to be unaffected. This indicates that, at least for our limited experiments, the pretest estimator seems to be robust to non-normality of the disturbances of the 2 type. We have also presented results for N = 25 and T = 20 at both the 5 and 10% significance levels. The choice of the pretest estimator as well as its relative MSE improves as T doubles from 10 to 20 for N = 25.
Pretest GHM H 0,A T = 10
BCL OLS
T = 20
T = 10
BCL TIME
T = 20
T = 10
BCL INDIV
T = 20
T = 10
BCL TW
T = 20
T = 10
T = 20
5%
10%
5%
10%
5%
10%
5%
10%
5%
10%
5%
10%
5%
10%
5%
10%
5%
10%
5%
10%
0 0 0 0 0 0
0 0.1 0.2 0.4 0.6 0.8
4753 2055 1735 162 4 0
4578 1689 1405 115 3 0
4758 641 564 9 0 0
4559 483 419 5 0 0
11 53 56 15 1 1
16 39 44 14 0 1
3 2 4 0 0 0
5 0 1 0 0 0
110 28 22 3 0 0
179 28 40 2 0 0
117 13 9 0 0 0
200 17 14 0 0 0
109 2658 2966 4519 4691 4639
186 2900 3136 4375 4508 4454
111 4070 4181 4684 4686 4682
191 4029 4137 4466 4445 4480
17 206 221 301 304 360
41 344 375 494 489 545
11 274 242 307 314 318
45 471 429 529 555 520
0.1 0.1 0.1 0.1 0.1
0 0.1 0.2 0.4 0.6
1837 624 492 27 1
1598 437 360 15 0
703 56 44 0 0
529 36 28 0 0
25 0 1 0 0
21 2 3 0 0
3 0 0 0 0
2 0 0 0 0
2930 852 759 44 0
2991 685 631 27 0
4037 375 255 1 0
3994 281 196 1 0
35 822 793 830 452
57 745 700 666 360
16 353 322 145 41
20 266 240 112 27
173 2702 2955 4099 4547
333 3131 3306 4292 4640
241 4216 4379 4854 4959
455 4417 4536 4887 4973
0.2 0.2 0.2 0.2 0.2
0 0.1 0.2 0.4 0.6
1704 449 344 12 0
1458 331 249 5 0
559 35 19 0 0
449 21 12 0 0
25 6 1 0 0
13 5 0 0 0
2 0 0 0 0
1 0 0 0 0
3020 801 662 21 0
3124 639 518 13 0
4225 276 204 2 0
4155 196 135 1 0
34 812 741 623 283
39 747 640 512 220
5 294 245 111 9
8 226 184 87 8
217 2932 3252 4344 4717
366 3278 3593 4470 4780
209 4395 4532 4887 4991
387 4557 4669 4912 4992
0.4 0.4 0.4 0.4
0 0.1 0.2 0.4
266 33 13 0
206 23 8 0
10 0 0 0
6 0 0 0
5 0 0 0
4 0 0 0
0 0 0 0
0 0 0 0
4466 628 445 0
4331 445 300 0
4695 131 79 0
4532 96 55 0
3 114 82 13
6 93 61 10
0 3 3 0
0 1 3 0
260 4225 4460 4987
453 4439 4631 4990
295 4866 4918 5000
462 4903 4942 5000
0.6 0.6 0.6
0 0.1 0.2
26 0 0
18 0 0
0 0 0
0 0 0
2 0 0
2 0 0
0 0 0
0 0 0
4682 211 92
4479 158 62
4706 38 7
4474 32 4
0 6 2
0 6 1
0 0 0
0 0 0
290 4783 4906
501 4836 4937
294 4962 4993
526 4968 4996
0.8 0.8
0 0.1
1 0
1 0
0 0
0 0
0 0
0 0
0 0
0 0
4706 13
4494 6
4702 4
4473 3
0 0
0 0
0 0
0 0
293 4987
505 4994
298 4996
527 4997
A Comparative Study of Pure and Pretest Estimators
Table 7. Number of Non-Rejections of H 0,A , H 0,B and H 0,C for 5,000 Replications Under Non-Normality of the Random Effects N = 25, T = 10 and T = 20.
21
Table 8. Relative MSE of  with Respect to True GLS for 5,000 Replications Under Non-Normality of the Random Effects N = 25, T = 10 and T = 20. Two-Way Error Component OLS
Within
FGLS
One-Way Time Error Component ML
Within
FGLS
One-Way Individual Error Component
ML
Within
FGLS
ML
Pretest T = 10
T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 T = 10 T = 20 5%
T = 20
10%
5%
10%
0 0 0 0 0 0
0 0.1 0.2 0.4 0.6 0.8
1.000 1.094 1.130 1.528 2.297 4.823
1.000 1.211 1.196 1.740 2.783 6.050
1.478 1.277 1.211 1.143 1.079 1.073
1.285 1.114 1.103 1.079 1.075 1.063
1.016 0.998 0.985 0.987 0.988 0.996
1.010 0.999 0.994 0.994 1.003 0.998
1.010 0.996 0.985 0.987 0.989 0.996
1.007 1.000 0.994 0.994 1.003 0.998
1.037 1.144 1.187 1.621 2.441 5.175
1.055 1.292 1.270 1.868 3.014 6.636
1.001 1.096 1.133 1.532 2.298 4.823
1.002 1.212 1.199 1.742 2.785 6.050
1.001 1.096 1.133 1.532 2.299 4.825
1.002 1.212 1.198 1.743 2.786 6.052
1.407 1.222 1.154 1.079 1.037 1.018
1.207 1.050 1.046 1.018 1.007 1.005
1.011 0.994 0.981 0.981 0.989 0.993
1.005 0.999 0.992 0.992 0.999 0.996
1.008 0.994 0.983 0.983 0.989 0.993
1.005 0.999 0.993 0.992 0.999 0.997
1.003 1.009 0.999 0.991 0.989 0.994
1.004 1.005 0.995 0.988 0.990 0.994
1.002 1.006 0.997 0.993 1.001 0.997
1.002 1.003 0.996 0.994 1.001 0.998
0.1 0.1 0.1 0.1 0.1
0 0.1 0.2 0.4 0.6
1.076 1.209 1.192 1.676 2.958
1.104 1.323 1.355 2.058 3.681
1.448 1.208 1.171 1.075 1.038
1.204 1.068 1.056 1.026 1.021
1.009 0.995 0.974 0.977 0.985
1.009 0.987 0.985 0.992 0.994
1.005 0.996 0.974 0.978 0.986
1.008 0.987 0.985 0.992 0.994
1.009 1.169 1.139 1.633 2.952
1.011 1.269 1.296 2.022 3.681
1.001 1.149 1.123 1.601 2.885
1.002 1.242 1.272 1.971 3.576
1.001 1.149 1.123 1.601 2.885
1.002 1.242 1.272 1.971 3.575
1.585 1.344 1.309 1.284 1.411
1.337 1.204 1.193 1.238 1.398
1.083 1.097 1.079 1.161 1.318
1.110 1.118 1.119 1.206 1.378
1.080 1.096 1.079 1.162 1.318
1.109 1.118 1.119 1.206 1.378
1.008 1.009 0.984 0.980 0.988
1.009 1.005 0.982 0.978 0.987
1.005 0.990 0.988 0.992 0.994
1.008 0.988 0.986 0.991 0.994
0.2 0.2 0.2 0.2 0.2
0 0.1 0.2 0.4 0.6
1.059 1.269 1.262 1.949 4.149
1.097 1.346 1.428 2.244 5.372
1.437 1.171 1.144 1.065 1.023
1.203 1.058 1.048 1.023 1.007
1.004 0.994 0.976 0.980 0.994
1.001 0.999 0.977 0.992 0.999
0.999 0.996 0.978 0.981 0.995
1.000 0.999 0.977 0.992 0.999
1.011 1.196 1.218 1.896 4.051
1.012 1.290 1.346 2.209 5.275
0.992 1.183 1.192 1.861 3.979
0.997 1.268 1.323 2.154 5.156
0.992 1.183 1.191 1.861 3.979
0.997 1.268 1.323 2.154 5.155
1.550 1.340 1.320 1.335 1.596
1.320 1.188 1.234 1.274 1.691
1.069 1.127 1.108 1.214 1.541
1.101 1.125 1.149 1.243 1.681
1.066 1.127 1.110 1.215 1.542
1.100 1.125 1.149 1.244 1.681
1.000 1.009 0.989 0.983 0.995
0.998 1.002 0.985 0.982 0.995
1.001 1.001 0.978 0.992 0.999
1.000 1.000 0.978 0.992 0.999
0.4 0.4 0.4 0.4
0 0.1 0.2 0.4
1.290 1.522 1.650 3.510
1.343 1.707 1.948 4.294
1.421 1.149 1.114 1.027
1.211 1.046 1.029 1.005
1.012 0.993 0.969 0.998
1.005 0.993 0.983 0.999
1.009 0.995 0.970 0.999
1.004 0.993 0.983 1.000
1.002 1.291 1.344 2.988
1.004 1.411 1.555 3.631
1.002 1.284 1.338 2.974
0.999 1.399 1.543 3.598
1.002 1.284 1.338 2.974
0.999 1.399 1.543 3.598
1.962 1.652 1.768 2.297
1.690 1.537 1.651 2.342
1.297 1.356 1.433 2.085
1.349 1.433 1.553 2.288
1.295 1.355 1.430 2.084
1.348 1.433 1.553 2.288
1.009 1.003 0.976 0.998
1.009 0.999 0.972 0.998
1.002 0.994 0.984 0.999
1.003 0.994 0.983 0.999
0.6 0 1.615 0.6 0.1 2.154 0.6 0.2 2.754
1.789 2.636 3.460
1.429 1.090 1.052
1.190 1.025 1.013
1.010 0.998 0.974
1.007 0.996 0.994
1.006 0.999 0.975
1.006 0.997 0.994
1.002 1.521 1.814
1.002 1.790 2.248
0.998 1.518 1.809
1.000 1.783 2.237
0.998 1.518 1.809
1.000 1.783 2.237
2.657 2.378 2.993
2.277 2.320 3.005
1.619 1.914 2.383
1.790 2.173 2.790
1.619 1.912 2.381
1.791 1.003 1.004 1.002 1.006 2.173 1.001 1.000 0.996 0.996 2.789 0.976 0.975 0.994 0.994
0.8 0 2.739 0.8 0.1 5.495
3.139 7.043
1.447 1.045
1.176 1.012
1.013 0.996
1.003 0.992
1.008 0.999
1.002 0.992
1.001 2.956
1.001 3.472
0.999 2.953
0.999 3.465
1.000 2.953
0.999 3.465
4.871 6.307
4.138 6.298
2.740 4.834
3.139 5.821
2.745 4.825
3.142 1.005 1.004 1.000 1.001 5.821 0.997 0.996 0.992 0.992
A Comparative Study of Pure and Pretest Estimators
23
4. CONCLUSION Our experiments show that the correct FGLS procedure is always the best, followed closely by the correct ML estimator. However, the researcher does not have perfect foresight on the true specification, whether it is two-way, one-way, or a pooled regression model with no time or individual effects. The pretest estimator proposed in this paper provides a viable alternative given that its MSE performance is a close second to correct FGLS for all type of misspecified models considered. The fixed effects estimator has the advantage of being robust to possible correlation between the regressors and the individual and time effects. It is clear from our experiments that the wrong fixed effects estimator, like its counterpart the wrong random effects estimator can lead to huge loss in MSE performance. We checked the sensitivity of our results to doubling the significance level for the pretest estimator from 5 to 10% as well as to specifying non-normal random effects. We found our pretest estimator to be robust to non-normality.
NOTE 1. The results of the experiments are invariant to the choice of the true values of the regression parameters ␥ and  and the remainder variance, see Breusch (1980).
ACKNOWLEDGMENTS The authors would like to thank Carter Hill for his helpful comments as well as the participants of the LSU econometrics conference in Baton Rouge, Louisiana, November 2–3, 2002. Baltagi would like to thank the Private Enterprise Research Center for its research support.
REFERENCES Baltagi, B. H. (2001). Econometric analysis of panel data. Chichester: Wiley. Baltagi, B. H., Chang, Y. J., & Li, Q. (1992). Monte Carlo results on several new and existing tests for the error component model. Journal of Econometrics, 54, 95–120. Baltagi, B. H., & Li, Q. (1997). Monte Carlo results on pure and pretest estimators of an error component ´ model with autocorrelated disturbances. Annales d’Economie et de Statistique, 48, 69–82. Breusch, T. S. (1980). Useful invariance results for generalized regression models. Journal of Econometrics, 13, 327–340. Breusch, T. S., & Pagan, A. R. (1980). The Lagrange multiplier test and its applications to model specification in econometrics. Review of Economic Studies, 47, 239–253.
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BADI H. BALTAGI, GEORGES BRESSON AND ALAIN PIROTTE
Giles, J. A., & Giles, D. E. A. (1993). Pre-test estimation and testing in econometrics: Recent developments. Journal of Economic Surveys, 7, 145–197. Gouri´eroux, C., Holly, A., & Monfort, A. (1982). Likelihood ratio test, Wald test, and Kuhn-Tucker test in linear models with inequality constraints on the regression parameters. Econometrica, 50, 63–80. Judge, G. G., & Bock, M. E. (1978). The statistical implications of pre-test and Stein-Rule estimators in econometrics. Amsterdam: North-Holland. Judge, G. G., & Bock, M. E. (1983). Biased estimation. In: Z. Griliches & M.D. Intrilligator (Eds), Handbook of Econometrics (Vol. 1, pp. 601–649). Amsterdam: North-Holland. Magnus, J. R. (1999). The traditional pretest estimator. Theory of Probability and Its Applications, 44, 293–308. White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50, 1–25. Ziemer, R. F., & Wetzstein, M. E. (1983). A Stein-rule method for pooling data. Economics Letters, 11, 137–143.
APPENDIX This appendix gives the SAS computer code behind the flow chart in Fig. shows how the pretest estimator was constructed. DO z = 1 TO rep; /* rep = number of replications /* Pretest estimator /* /* Gouri´ eroux, Holly and Monfort (GHM) test: /* H(0,A): sigma mu = 0 , sigma lambda = 0 /* res ols1 = SHAPE(res ols,N,T);/* OLS residuals */ bet ires = SHAPE(REPEAT(res ols1[,:],1,T),NT,1) bet tres = SHAPE(REPEAT(res ols1[:,],N,1),NT,1) u1 = SQRT (NT / (2*(T-1)) ) u2 = SQRT (NT / (2*(N-1)) ) A = u1 * (((T*bet ires‘*bet ires)/ (res ols‘*res os )) - 1) B = u2 * (((N*bet tres‘*bet tres)/ (res ols‘*res os )) - 1) IF A > 0 THEN DO; IF B > 0 THEN DO; GHM[z,1] = A**2 + B**2; END; IF B One-Way time FGLS */ IF BCL[z,1] 0) /* res 1 = SHAPE(res mli,N,T) ; /* One-way ind. ML residuals */ res bar = res mli[:] res 1 = SHAPE(res mli,N,T) bet ires = SHAPE(REPEAT(res 1[,:],1,T),NT,1) bet tres = SHAPE(REPEAT(res 1[:,],N,1),NT,1) bt res = bet tres - REPEAT(res bar,NT,1) wi res = res mli - bet ires mg res = REPEAT(res bar,NT,1) sigt2 = (t(bet ires)*bet ires ) / N sigtnu = (t(wi res)*wi res ) / (N*(T-1)) Dmu1 = t(mg res)*mg res Dmu2 = t(bt res)*bt res (N/2)* ((1/sigt2)*(dmu1/sigt2 - 1) D lamb = +((T-1)/sigtnu) *(Dmu2/((T-1)*sigtnu ) - 1)) num = SQRT(2.0)*sigt2*sigtnu den = SQRT (N*(N-1)* (sigtnu**2 + (T-1)*sigt2**2) ) BCL[z,3] = (num/den)*D lamb;
*/ */ */ */
; ; ; ; ; ; ; ; ; ; ; ; ; ; ;
/* Count for H(0,C) => One-Way individual FGLS */ IF BCL[z,3] 0.
H1 :
An alternative to model (1) is obtained by computing partial sums of the raw data z t = t + 
t2 + t 2
+ St ,
(2)
where z t = tj=1 y j and S t = tj=1 u j . For each of models (1) and (2), we consider t-tests that are not only robust to serial correlation in ut but are also robust to the possibility that ut has a unit root. These tests do not suffer from the usual over-rejection problem that is often present with HAC robust tests when serial correlation is highly persistent. For model (1), we use one of the tests proposed by Bunzel and Vogelsang (2003) and for model (2), we used one of the tests proposed by Vogelsang (1998). Both tests depend on a common adjustment factor that controls the over-rejection problem when ut has a unit root. It is convenient to first describe this adjustment factor before defining the test statistics. Consider model (1) augmented with additional regressors that are polynomials in time y t = + t +
9
␦i t i + u t .
(3)
i=2
Under the assumption that model (1) adequately captures the deterministic component of yt , the ␦i coefficients in regression (3) are all zero. Consider the following test statistic which is the standard Wald test (normalized by T−1 ) for testing the joint hypothesis that the ␦i coefficients are zero J=
RSS1 − RSS3 RSS3
(4)
where RSS1 is the OLS residual sum of squares from regression (1) and RSS3 is the OLS residual sum of squares from regression (3). The J statistic was originally proposed by Park and Choi (1988) and Park (1990) as a unit root test. When ut has a unit root, J has a well-defined asymptotic distribution while when ut is covariance stationary, J converges to zero. Bunzel and Vogelsang (2003) analyzed t-tests for  using OLS estimates from model (1) and recommended the following test. Let ˆ denote the OLS estimate
Tests of Common Deterministic Trend Slopes Applied
33
from model (1). Let uˆ t denote the OLS residuals from model (1) and define ˆ 2 = ␥ˆ 0 + 2 T
T −1 j=1
sin(j/M) ␥ˆ j , j/M
where ␥ˆ = t=j+1 uˆ t uˆ t−j and M = max(0.02T, 2). ˆ 2 is the Daniell kernel non-parametric zero frequency spectral density estimator. Define the t-test ˆ −  t DAN = exp(−bJ), ˆ se() ˆ = ˆ 2 ( (t − ¯t )2 )−1 with ¯t = T −1 T t and b is a pre-specified where se() t=1 constant (see Bunzel & Vogelsang, 2003) that depends on the significance level of the test. Bunzel and Vogelsang (2003) showed that t DAN has nearly optimal power when ut is covariance stationary. The adjustment factor, exp(−bJ), was proposed by Vogelsang (1998) as a method of solving the over-rejection problem that occurs when ut has a unit root. Without the adjustment factor, t DAN would over-reject quite substantially when ut has strong serial correlation or a unit root. ˜ We also consider a test proposed by Vogelsang (1998). Let t z = (˜ − )/se() ˜ ˜ where  is the OLS estimate of beta from model (2) and se() is the classic (non-HAC robust) OLS standard error. Define the statistic t − PS = T −1/2 t z exp(−bJ). While t − PS is not as powerful as tDAN when ut is covariance stationary, it can have higher average power when ut has strong (nearly integrated) serial correlation. As with tDAN , t − PS does not over-reject when ut has a unit root. We applied the t DAN and t − PS tests to each of the global temperature series in our data sets and report the results in Tables 1 and 2. Table 1 gives results for the JOBP data and Table 2 gives the results for the VGL data. We follow the global climate literature convention and define the seasons as follows: the winter quarter (season) consists of December, January, and February in the northern hemisphere and June, July, and August in the southern hemisphere. The spring quarter consists of March, April, and May in the northern hemisphere and September, October, and November in the southern hemisphere. The summer quarter consists of June, July, and August in the northern hemisphere and December, January, and February in the southern hemisphere. Finally, the autumn quarter consists of September, October, and November in the northern hemisphere and March, April, and May in the southern hemisphere. In the tables, we see that the null hypothesis of no global warming is strongly rejected in nearly all cases.2 The exception is the summer in the northern
34
Table 1. Global Warming Trends By Season; JOBP Data 1857–1998. Location
Season
ˆ
t − PS
J
5%
2.5%
1%
5%
2.5%
1%
4.861b
3.626a
2.851c
2.524b
4.639c
3.442b
2.013c
0.437∗∗∗ 0.630∗∗ 0.978 0.584∗∗
0.0045 0.0043 0.0038 0.0045
2.835c 5.974c
1.785 4.532b
North
DJF (Winter) MAM (Spring) JJA (Summer) SON (Autumn)
0.0049 0.0041 0.0026 0.0044
5.889c 4.055c 1.415 5.667c
South
DJF (Summer) MAM (Autumn) JJA (Winter) SON (Spring)
0.0041 0.0044 0.0048 0.0043
7.357c 6.358c 5.945c 5.658c
2.255 0.927 3.064a
0.0038 0.0036 0.0033 0.0042
1.634 3.479c
1.689 1.245 2.956b
2.023 1.227 0.758 2.200
5.277b 3.167b 0.892 4.637b
4.517a 2.234 0.464 3.490a
0.0041 0.0033 0.0019 0.0044
2.962c 1.690 0.649 3.349c
2.777b 1.462 0.495 2.975b
2.469 1.122 0.302 2.401
0.232∗∗∗ 0.521∗∗ 0.976 0.424∗∗∗
6.311b 5.123b 4.493b 4.256b
5.078a 3.772a 3.022a 2.843a
0.0034 0.0036 0.0041 0.0036
3.506c 2.599c 2.899c 2.356c
3.203b 2.288b 2.458b 1.993
2.719a 1.816 1.822 1.470
0.324∗∗∗ 0.457∗∗∗ 0.592∗∗ 0.602∗∗
Notes: DJF = Dec–Feb, MAM = Mar–May, JJA = Jun–Aug, SON = Sep–Nov. The superscripts a, b, and c denote rejection of the null hypothesis of no global warming at the 1, 2.5, and 5% levels of significance, respectively. The superscripts ∗∗∗ , ∗∗ , and ∗ denote rejection of a unit root in the errors at the 1, 2.5, and 5% levels of significance, respectively. The J is a left tail test with 1, 2.5, and 5% critical values of 0.488, 0.678 and 0.908, respectively. The pre-specified constants for the t DAN test are 1.322 (5%), 1.795 (2.5%) and 2.466 (1%). The pre-specified constants used for the t − PS test are 0.716 (5%), 0.995 (2.5%) and 1.501 (1%). Critical Values for Test Statistics % tDAN t − PS 0.950 1.710 1.720 0.975 2.052 2.152 0.990 2.462 2.647
THOMAS B. FOMBY AND TIMOTHY J. VOGELSANG
DJF MAM JJA SON
5.979c
Globe
˜
t DAN
Location
Season
ˆ
˜
t DAN 5%
2.5%
1%
t − PS
J
5%
2.5%
1%
Globe
DJF MAM JJA SON
0.0058 0.0064 0.0047 0.0041
6.263c 6.568c 6.117c 4.287c
5.379b 5.557b 5.131b 3.448b
4.336a 4.385a 4.000a 2.532a
0.0057 0.0061 0.0044 0.0042
3.893c 3.847c 3.744c 2.193c
3.559b 3.486b 3.375 1.929
3.024a 2.914a 2.797a 1.528
0.321∗∗∗ 0.353∗∗∗ 0.372∗∗∗ 0.460∗∗∗
North
DJF (Winter) MAM (Spring) JJA (Summer) SON (Autumn)
0.0075 0.0064 0.0029 0.0041
3.675c 5.147c 2.231c 1.829c
2.709b 4.280b 1.660 1.248
1.758 3.296a 1.091 0.726
0.0079 0.0060 0.0024 0.0046
2.044c 3.237c 1.368 1.042
1.708 2.903b 1.149 0.832
1.233 2.383 0.837 0.552
0.645∗∗ 0.389∗∗∗ 0.625∗∗ 0.808∗
South
DJF (Summer) MAM (Autumn) JJA (Winter) SON (Spring)
0.0039 0.0063 0.0063 0.0040
5.269c 8.162c 9.164c 7.157c
4.695b 7.272b 8.350b 6.784b
3.987a 6.174a 7.317a 6.289a
0.0034 0.0060 0.0064 0.0038
3.544c 4.430c 6.017c 6.504c
3.311b 4.139b 5.695b 6.302b
2.927a 3.658a 5.156a 5.952a
0.244∗∗∗ 0.244∗∗∗ 0.197∗∗∗ 0.113∗∗∗
Tests of Common Deterministic Trend Slopes Applied
Table 2. Global Warming Trends By Season; VGL Data 1881–1993.
Notes: DJF = Dec–Feb, MAM = Mar–May, JJA = Jun–Aug, SON = Sep–Nov. The superscripts a, b, and c denote rejection of the null hypothesis of no global warming at the 1, 2.5, and 5% levels of significance, respectively. The superscripts ∗∗∗ , ∗∗ , and ∗ denote rejection of a unit root in the errors at the 1, 2.5, and 5% levels of significance, respectively. The J is a left tail test with 1, 2.5, and 5% critical values of 0.488, 0.678 and 0.908, respectively. The pre-specified constants for the t DAN test are 1.322 (5%), 1.795 (2.5%) and 2.466 (1%). The pre-specified constants used for the t − PS test are 0.716 (5%), 0.995 (2.5%) and 1.501 (1%). Critical Values for Test Statistics % tDAN t − PS 0.950 1.710 1.720 0.975 2.052 2.152 0.990 2.462 2.647
35
36
THOMAS B. FOMBY AND TIMOTHY J. VOGELSANG
hemisphere using the JOBP data. In that case, the OLS point estimate of  is only 0.026 and the null hypothesis of no global warming cannot be rejected by either t-test. The corresponding OLS point estimate using the VGL data is also relatively small, 0.029, but in that case the null hypothesis of no global warming can be rejected at the 5% level using t DAN . Differences in the strength of the serial correlation in the JOBP series versus the VGL series, as indicated by the J statistic, could explain these conflicting results. Note that overall, the J statistic suggests that most time series in our data sets have covariance stationary errors. It is interesting to note that in general the VGL data leads to larger slope estimates than the JOBP data set. This may be a function of the years spanned by the data because unreported results for the JOBP data set using the years 1881–1993 have larger slope point estimates than reported in Table 1.
3. MULTIVARIATE ANALYSIS: SEASONAL PATTERNS IN GLOBAL WARMING NORTH/SOUTH COMPARISONS Because evidence of global warming is convincing in the data, we can then ask whether warming is occurring at the same rates across the seasons and hemispheres. The point estimates in Tables 1 and 2 suggest there are interesting differences and similarities. For the globe, there is relatively little difference in warming trends across the seasons in the JOBP data set, but some differences are evident in the VGL data where warming in the December–May months appears greater than in the June–November months. Within hemispheres, there are interesting warming patterns across the seasons. For both data sets and both hemispheres, summers appear to be warming the slowest and winters appear to be warming the fastest. Patterns across seasons seem similar in the north and south. Are they the same? Hypotheses such as these can be written as linear restrictions on the slope parameters across the series and we can apply the multivariate linear trend tests of Franses and Vogelsang (2002). Consider the multivariate trend model y1,t = 1 + 1 t + u1,t y2,t = 2 + 2 t + u2,t .. . yn,t = n + n t + un,t that can be compactly written as Y t = + t + U t
(5)
Tests of Common Deterministic Trend Slopes Applied
37
where Y t = ( y 1,t , y 2,t , . . . , y n,t ) , U t = (u 1,t , u 2,t , . . . , u n,t ) ,  = (1 , 2 , . . . , n ) and = (1 , 2 , . . . , n ) . For hypotheses about seasonal differences in the slopes within the globe or a hemisphere, n = 4. For hypothesis involving differences or similarities between the north and south, n = 8. We are interested in testing hypotheses of the form H0 :
R = r,
H1 :
R = r,
(6)
where R is a q × n matrix and r is a q × 1 vector of known constants. The linear hypotheses of (6) are quite general. They include linear hypotheses on slopes within given trend equations as well as joint trend hypotheses across equations. We apply two tests analyzed by Franses and Vogelsang (2002). Both statistics are functions of the following HAC variance covariance matrix estimator. Let ˆ i and ˆ i denote the single equation OLS estimates of i and i . Let uˆ i,t = y i,t − ˆ i − ˆ i t denote the equation by equation OLS residuals. Stack the OLS estimates and residuals into vectors ˆ = (ˆ 1 , ˆ 2 , . . . , ˆ n ) and Uˆ = (uˆ 1,t , uˆ 1,2 , . . . , uˆ 1,n ) . Define T −1 j ˆ ˆ
= 0 + k (ˆ j + ˆ j ), M where ˆ j = T
−1 T
j=1
ˆ ˆ t=j+1 Ut Ut−j .
ˆ The first, denoted by ˆ QS ˆ , follows Andrews We use two configurations of . M=M (1991) and uses the quadratic spectral kernel3 for k(x) and a data dependent ˆ based on the VAR(1) plug-in formula. See Andrews (1991) bandwidth, M, ˆ Bart for the formulas. The second configuration is denoted by M=T and uses the Bartlett kernel for k(x) and bandwidth M = T. As shown by Kiefer and Vogelsang ˆ Bart (2002), Kiefer, Vogelsang and Bunzel (2000) first proposed using M=T in stationary regressions, and Franses and Vogelsang (2002) extended their results to multivariate linear trend models. The statistics we use are defined as −1 −1 T QS ˆ M=M R (R ˆ − r), W = (R ˆ − r) R
(t − ¯t )2 t=1
−1 −1 T R ˆ − r ˆ Bart (t − ¯t )2
. F ∗ = (R ˆ − r) R M=T R q t=1
Note that the F∗ statistic was labeled F ∗2 by Franses and Vogelsang (2002). Under the assumption that the errors are covariance stationary, the distribution theory
38
THOMAS B. FOMBY AND TIMOTHY J. VOGELSANG
for W is standard and W converges in distribution to a 2q random variable. The asymptotic distribution theory for F∗ is non-standard and was developed by Franses and Vogelsang (2002) for the case where the errors are covariance stationary4 . Simulation evidence reported by Franses and Vogelsang (2002) suggests that the F∗ tests suffers much less from the over-rejection problem caused by strong positive serial correlation than W whereas power of F∗ is only slightly lower than power of W. Therefore, rejections obtained using F∗ are more robust to the overrejection problem. Finite sample simulation evidence in Franses and Vogelsang (2002) also suggested that the performance of both W and F∗ are improved when ˆ is computed using VAR(1) prewhitening (see Andrews & Monahan, 1992 for
details). Therefore, we use VAR(1) prewhitening in all cases. Note that the use of prewhitening has no effect on the asymptotic distributions of the tests. The hypotheses we are interested in testing vis-`a-vis the JOBP and VGL data sets are the following: (i) For the northern and southern hemispheres, separately, and for the globe as a whole, are all seasons, simultaneously, without trend? In the presence of trend, are the trends of all seasons in each hemisphere equal to each other? For the globe, are the trends of all seasons equal to each other? For each hemisphere and, separately, for the globe, are there pairs of seasons that have equal trends? We refer to these hypotheses as “Intra-seasonal” hypotheses because they relate to the comparison of trends of seasons within separate hemispheres or the globe itself (ii) Are the trends for the four seasons the same across hemispheres? Across the hemispheres do the winters have the same trend? the springs? the summers? the autumns? We refer to these hypotheses as “Inter-seasonal” hypotheses because they relate to the comparison of trends of seasons across hemispheres. Tables 3 and 4 report the results of the Intra-seasonal hypothesis tests. Some major conclusions that can be drawn are as follows: The null hypothesis that all of the seasons jointly have zero slopes is strongly rejected using both the W and F∗ statistics. The null hypothesis that the warming trends are the same across the seasons is rejected in all cases with the possible exception of the southern hemisphere in the JOBP data set. There the slope point estimates are similar and the F∗ statistic cannot reject the null hypothesis that they are equal. Pair-wise tests of the null of equal slopes across the seasons confirm what is apparent in the point estimates. Summers (June–August) in the northern hemisphere are warming slower than the winters (December–February) and to
Tests of Common Deterministic Trend Slopes Applied
39
Table 3. Intra-Seasonal Hypotheses; JOBP Data 1857–1998. Global
All seasons zero (q = 4) All seasons equal (q = 3) DJF = MAM (q = 1) DJF = JJA (q = 1) DJF = SON (q = 1) MAM = JJA (q = 1) MAM = SON (q = 1) JJA = SON (q = 1)
North
South
W
F∗
W
F∗
W
F∗
131.7a 20.96a 0.822 8.489a 0.080 5.754b 0.179 12.92a
364.6a 45.13c 6.260 42.40c 0.155 15.39 0.227 22.42
144.4a 40.15a 2.607 24.83a 0.838 15.90a 0.517 26.17a
248.0a 52.50c 28.08 145.9a 1.434 88.56a 0.692 27.96
106.0a 8.137c 4.397c 7.952a 1.563 3.397 0.410 4.286c
123.0a 16.98 10.81 45.88c 5.744 14.54 6.052 26.64
Notes: The superscripts a, b, and c denote rejection of a null hypothesis at the 1, 2.5, and 5% levels of significance, respectively. Critical Values for Test Statistics, q = 4 % W F∗ 0.950 9.488 43.83 0.975 11.14 54.59 0.990 13.28 69.29 Critical Values for Test Statistics, q = 3 % W F∗ 0.950 7.815 41.45 0.975 9.348 52.86 0.990 11.34 68.67 Critical Values for Test Statistics, q = 1 % W F∗ 0.950 3.841 41.53 0.975 5.024 58.56 0.990 6.635 83.95
a lesser extent slower than the spring or autumn (according to the slope point estimates). The same pattern, summers (December–February) warming slower than winters (June–August), occurs in the southern hemisphere as well and the formal statistical evidence is convincing in the case of the VGL data. The Inter-seasonal hypothesis tests are reported in Table 5. There appears to be some heterogeneity in seasonal patterns of warming between the northern and southern hemispheres. The joint hypothesis that winters are warming at the same rate in the northern and southern hemispheres, springs are warming at the same rate, etc. can be rejected for both series using the W statistic.
40
THOMAS B. FOMBY AND TIMOTHY J. VOGELSANG
Table 4. Intra-Seasonal Hypotheses; VGL Data 1881–1993. Global
All Seasons Zero (q = 4) All Seasons Equal (q = 3) DJF = MAM (q = 1) DJF = JJA (q = 1) DJF = SON (q = 1) MAM = JJA (q = 1) MAM = SON (q = 1) JJA = SON (q = 1)
North
W
F∗
82.78a 45.74a 2.286 9.312a 16.17a 22.88a 39.83a 3.007
137.0a 127.8a 12.04 82.94b 78.94b 95.53a 130.2a 6.014
W 86.56a 83.66a 3.359 43.81a 27.57a 56.01a 15.94a 3.728
South F∗
W
F∗
247.8a 321.3a 8.027 63.91b 280.8a 234.0a 21.74 3.421
142.7a 24.50a 13.97a 15.66a 0.001 0.042 20.76a 18.50a
238.3a 34.07a 23.12 27.40 0.002 0.162 63.92b 73.81b
Notes: The superscripts a, b, and c denote rejection of a null hypothesis at the 1, 2.5, and 5% levels of significance, respectively. Critical Values for Test Statistics, q = 4 % W F∗ 0.950 9.488 43.83 0.975 11.14 54.59 0.990 13.28 69.29 Critical Values for Test Statistics, q = 3 % W F∗ 0.950 7.815 41.45 0.975 9.348 52.86 0.990 11.34 68.67 Critical Values for Test Statistics, q = 1 % W F∗ 0.950 3.841 41.53 0.975 5.024 58.56 0.990 6.635 83.95
Using the more size robust F∗ test, this null hypothesis can also be rejected for the VGL data set but not for the JOBP data set. Pair-wise tests of equal warming for given seasons between the hemispheres suggest that warming patterns in the winter, spring, and autumn are equal in the northern and southern hemispheres for the JOBP data set with only a difference in the summers where warming has been greater in the southern hemisphere. For the VGL data set, winters and summers have been warming at the same rates in the northern and southern hemispheres with differences in warming rates in the spring and autumn. Overall, these tests suggest that there is substantial heterogeneity in seasonal warming trends across hemispheres.
Tests of Common Deterministic Trend Slopes Applied
41
Table 5. Inter-Hemisphere Hypotheses. JOBP Data
All seasons in north and south equal (q = 4) North winter = South winter (q = 1) North spring = South spring (q = 1) North summer = South summer (q = 1) North autumn = South autumn (q = 1)
VGL Data
W
F∗
W
F∗
18.91a 0.020 0.293 8.921a 0.000
29.10 0.157 1.600 25.90 0.000
49.79a 1.731 9.480a 1.872 6.043b
233.3a 5.186 46.75c 7.394 15.05
Notes: The superscripts a, b, and c denote rejection of a null hypothesis at the 1, 2.5, and 5% levels of significance, respectively. Critical Values for Test Statistics, q = 4 % W F∗ 0.950 9.488 43.83 0.975 11.14 54.59 0.990 13.28 69.29 Critical Values for Test Statistics, q = 1 % W F∗ 0.950 3.841 41.53 . 0.975 5.024 58.56 0.990 6.635 83.95
4. CONCLUSIONS The multivariate deterministic trends model of Franses and Vogelsang (2002) is used to examine some interesting hypotheses concerning global warming by season and hemisphere. Global warming appears to be present in each season in the global series of the JOBP and VGL data sets with the possible exception of the June–August season in the northern hemisphere for the JOBP data set. A more informative look at seasonal warming, however, can be obtained by looking at the hemispheric temperatures separately. For both hemispheres and both data sets, winters appear to be warming the fastest and summers the slowest with their trends being significantly different. This heterogeneity of warming occurs not only within hemispheres but also across hemispheres. Only the winter season appears to have the same warming trend across hemispheres. Coming to understand why differential warming trends exist by season and hemisphere would provide an interesting topic for future research.
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NOTES 1. The idea of using heteroskedasticity robust standard errors appeared earlier in the statistics literature in work by Eicker (1967). White (1980, p. 821) points out this fact and notes with surprise that the econometrics literature seemed unaware of these results for over ten years. 2. These results confirm for the quarterly frequency the statistical significance of warming reported by Fomby and Vogelsang (2002) for annual global temperature. 3. Franses and Vogelsang (2002) only considered the Bartlett kernel. 4. The assumption of covariance stationary errors is reasonable for the global warming data given the J statistic results reported in Tables 1 and 2.
REFERENCES Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59, 817–854. Andrews, D. W. K., & Monahan, J. C. (1992). An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica, 60, 953–966. Bunzel, H., & Vogelsang, T. J. (2003). Powerful trend function tests that are robust to strong serial correlation with an application to the Prebisch-Singer hypothesis, Working Paper, Department of Economics, Cornell University. den Haan, W. J., & Levin, A. (1997). A practictioner’s guide to robust covariance matrix estimation. In: G. Maddala & C. Rao (Eds), Handbook of Statistics: Robust Inference (Vol. 15, pp. 291–341). New York: Elsevier. Eicker, F. (1967). Limit theorems for regressions with unequal and dependent errors, Proceedings of the fifth Berkeley Symposium on mathematical statistics and probability, Vol. 1, University of California Press, 59–82. Fomby, T. B., & Vogelsang, T. J. (2002). The application of size robust trend analysis to global warming temperature series. Journal of Climate, 15, 117–123. Franses, P. H., & Vogelsang, T. J. (2002). Testing for common deterministic trend slopes, center for analytic economics, Working Paper 01–15, Cornell University. Gallant, A. (1987). Non-linear statistical models. New York: Wiley. Gallant, A., & White, H. (1988). A unified theory of estimation and inference for non-linear dynamic models. New York: Blackwell. Grenander, U., & Rosenblatt, M. (1957). Statistical analysis of stationary time series. New York: Wiley. Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica, 50, 1029–1054. Hansen, B. E. (1992). Consistent covariance matrix estimation for dependent heterogenous processes. Econometrica, 60, 967–972. Intergovernmental Panel on Climate Change (IPCC) Report: Climate change 2001: Synthesis report available at the URL www.ipcc.ch/pub/un/syreng/spm.pdf Jones, P. D., Parker, D. E., Osborn, T. J., & Briffa, K. J. (1999). Global monthly and annual temperature anomalies (degrees C), 1856–1998, Feb., Climatic Research Unit, School of Environmental Sciences, University of East Angelia, Norwich, United Kingdom. This data is available from
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the authors or a more recent version of the data (through 2000) is available from the Oak Ridge National Laboratory at http://cdiac.esd.ornl.gov/ftp/trends/temp/jonescru Kaufmann, R. K., & Stern, D. I. (2002). Cointegration analysis of hemispheric temperature relations. Journal of Geophysical Research, 107 (DX), 10.1029/2000JD000174, 2002. (On-line version.) Kiefer, N. M., & Vogelsang, T. J. (2002). Heteroskedasticity-autocorrelation robust standard errors using the Bartlett Kernel without truncation. Econometrica, 70, 2093–2095. Kiefer, N. M., Vogelsang, T. J., & Bunzel, H. (2000). Simple robust testing of regression hypotheses. Econometrica, 68, 695–714. Newey, W. K., & West, K. D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55, 703–708. Newey, W. K., & West, K. D. (1994). Automatic lag selection in covariance estimation. Review of Economic Studies, 61, 631–654. Park, J. Y. (1990). Testing for unit roots and cointegration by variable addition. In: T. Fomby & F. Rhodes (Eds), Advances in Econometrics: Cointegration, Spurious Regressions and Unit Roots (pp. 107–134). London: JAI Press. Park, J. Y., & Choi, I. (1988). A New Approach to Testing for a Unit Root, Center for Analytic Economics, Cornell University, Working Paper 88–23. Priestley, M. B. (1981). Spectral analysis and time series (Vol. 1). New York: Academic Press. Robinson, P. (1998). Inference-without smoothing in the presence of non-parametric autocorrelation. Econometrica, 66, 1163–1182. Vinnikov, K. Y., Groisman, P. Y., & Lugina, K. M. (1994). Global and hemispheric temperature anomalies from instrumental surface air temperature records. In: T. A. Boden, D. P. Kaiser, R. J. Sepanski & F. W. Stoss (Eds), Trends 1993: A compendium of data on global change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, 615–627. This data is available from the authors or from the Oak Ridge National Laboratory at http://cdiac.esd.ornl.gov/ftp/trends93/temp/global.618 http://cdiac.esd.ornl.gov/ftp/trends93/ temp/nhem.621 http://cdiac.esd.ornl.gov/ftp/trends93/temp/shem.624 Vogelsang, T. J. (1998). Trend function hypothesis testing in the presence of serial correlation correlation parameters. Econometrica, 65, 123–148. White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48, 817–838. White, H., & Domowitz, I. (1984). Non-linear regression with dependent observations. Econometrica, 52, 143–161.
THE SANDWICH ESTIMATE OF VARIANCE James W. Hardin ABSTRACT This article examines the history, development, and application of the sandwich estimate of variance. In describing this estimator, we pay attention to applications that have appeared in the literature and examine the nature of the problems for which this estimator is used. We describe various adjustments to the estimate for use with small samples, and illustrate the estimator’s construction for a variety of models. Finally, we discuss interpretation of results.
1. INTRODUCTION Section 2 examines the assumptions allowing construction of the sandwich estimate of variance. Section 3 illustrates the robustness property of the sandwich estimator in a simple univariate setting along with further developments and generalizations. We illustrate this construction with several detailed illustrations in the subsequent sections. Various applications are examined subject to the assumptions of the sandwich estimator including linear regression, two-stage models compared to the Murphy–Topel variance estimate, other two-stage models, multi-stage models, and generalized estimating equations. Section 8 then presents a summary.
Maximum Likelihood Estimation of Misspecified Models: Twenty Years Later Advances in Econometrics, Volume 17, 45–73 Copyright © 2003 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(03)17003-X
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2. HISTORICAL DEVELOPMENT The history of the sandwich estimate of variance focuses on the results presented by Huber (1967) wherein the author formally developed properties of maximum likelihood estimators (i) not assuming that the true distribution underlying observations belonged to the parametric family defining the estimator and (ii) the regularity conditions do not involve second and higher derivatives of the likelihood. Huber took advantage of the consistency results of Wald (1949) and was likely aware of hints and motivations for this particular problem in Fisher (1935). The Berkeley conference at which Huber presented his results was also the forum where Eicker (1967) discussed results for the particular case of linear regression. His work extended an earlier illustration of the asymptotic results for families of regressions in Eicker (1963). While there are several results provided in Huber’s paper, our focus is on the corollary (p. 231) which states that if the expected value of the estimating equation E((x, 0 )) has a nonsingular derivative A at 0 , then √ for the estimating function T n (x 1 , . . . , x n ), or simply T n , of 0 , we have that n(T n − 0 ) is asymptotically normal with mean zero and covariance matrix given by V S = A−1 BA−T . We call V S the sandwich estimator. Here, B is the covariance matrix of the estimating equation (x, 0 ) and A is the Hessian matrix for the estimating equation. The assumptions underlying the corollary include (1) x 1 , x 2 , . . . , x n are independent random variables with a common probability distribution. (2) The expected value of the estimating equation has a unique zero at 0 which is in the parameter space of . (3) The estimating equation is measurable and separable; this condition ensures that the estimating equation infima is measurable. (4) The expectation of the covariance of the estimating equation is finite (this is matrix B above). Other assumptions ensure that the estimating equation is smooth near 0 ; that the derivative of the estimating equation at 0 is finite. Huber’s paper is aimed at statistical theorists such that the applicability of the results was not immediately apparent to a wide audience of interested researchers. Independently, White (1980) derived the sandwich estimate of variance (and its properties) in the case of linear regression. While the presentation was still theoretical, the practical utility of the results was more easily discerned; White (1982b) subsequently placed even more emphasis on the robustness properties of the estimator. These papers brought the sandwich estimator to the forefront
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for many researchers and is the reason that the sandwich estimator is known as White’s estimator to so many scientists. White’s goal mirrors that of Huber’s in that both articles (Huber, 1967; White, 1980) investigate the assumptions that justify the construction of a valid variance estimate under misspecification. The end result is that test statistics and confidence intervals based on the sandwich estimate of variance are robust to model misspecification. In general, the construction of the sandwich estimator is straightforward. More sophisticated analysis is required to use the resulting estimator as a genesis for deriving test statistics for model misspecification. This application is emphasized in White (1982b). In Section 7, we address the interpretation of analyses using the sandwich estimator. While the title of Huber’s paper implies that the corollary is in terms of maximum likelihood models, the assumptions actually include the application ˆ of estimating equations of the form of this variance estimator for solutions, , (x, ) = 0. This specification does not require that the estimating equation is a derivative of a log-likelihood. In the usual application of the sandwich estimate of variance, a model is derived from a likelihood. The estimating equation is then the derivative of the log-likelihood with respect to the vector of parameters. Setting this equation to zero and solving for the parameters yields maximum likelihood estimates. However, we need not start from a likelihood specification. A simple example is quantile regression which directly specifies the estimating equation. A second example is generalized linear models (GLMs) or the class of GEE models that have a genesis in the estimating equation. The GEE models directly introduce second order variance components. The most general presentation of GLMs and GEE do not start from a likelihood. Rather, these models specify an estimating equation that implies a quasi-likelihood. The results presented here apply to this larger class of models based on estimating equations. Stefanski and Boos (2002) build on the results of Hampel (1974) by pointing out that we are constructing the variance estimate of an M-estimator such that the corollary holds for this entire class. Another powerful result is given in Binder and Patak (1994) where the authors show that the estimating equation may be partitioned. This result allows us to easily justify the application of the corollary to two (and higher) stage models. We highlight this result in the next section. Miller (1974) presents a review of the jackknife and infinitesimal jackknife wherein the equivalence of the variance estimate built from the infinitesimal jackknife and the variance matrix described in the corollary can be inferred. We use this result to highlight the derivation of the sandwich estimate of variance for the proportional hazards model in the next section; see Cox (1972) and Lin and Wei (1989).
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Over the last 35 years there have been many rediscoveries, studies, summaries, and extensions of this estimator. Royall and Cumberland (1978) look at the estimator in regard to survey sampling, and Royall and Cumberland (1985) investigate the coverage probabilities of confidence intervals constructed from the sandwich estimate of variance. Gail et al. (1988) investigate the usefulness of the sandwich estimator in biostatistical applications. Newey and West (1987) expand on the work of White (2000) to extend the sandwich estimator for the case of autocorrelation in linear regression. White’s text presents a general treatment for this case. The Newey–West contribution was to derive a formula for which the resulting estimator was always positive definite since White’s treatment was not guaranteed to produce results with that property; this focus by Newey and West is the genesis of the title of the their paper. The so-called Newey–West estimator replaces the usual B matrix with a weighted sum of matrices calculated using lagged residuals where the weights are taken from the Bartlett kernel. Following the Newey and West paper, several papers focused on alternatives to the Bartlett kernel weights. These papers include Gallant (1987) specifying weights from the Parzen (1957) kernel and Andrews (1991) specifying weights from the Tukey–Hanning kernel. Newey and West’s original description required a specification of the number of lags to consider in constructing the matrix in the middle of the sandwich. Later, Newey and West (1994) considered automated techniques, and Lumley and Heagerty (1999) considered adaptive, as opposed to kernel, methods for selecting weights. Wooldridge (1991) provides an excellent review that focuses on the interpretation of the sandwich estimator including the extensions of Newey and West. Another review which focuses on the properties of the sandwich estimate of variance is given in Kauermann and Carroll (2001). Application of the sandwich estimate of variance for panel data is discussed in Xie et al. (2000). Due to the specific circumstances and assumption sets under which various authors have derived this estimator, it is known by various names. These names include the sandwich estimator (used throughout this article), Huber’s estimator, White’s estimator, the survey design estimator, and the robust variance matrix. It is also called the empirical variance estimator since the B matrix is formed from the observations without regard to underlying parametric theory, but we dislike this practice since only the B matrix in the construction is empirical. Hardin and Hilbe (2001) present a more complete list with comments.
3. ROBUSTNESS This section presents a simple illustration of the robustness property of the sandwich estimator; see Royall (1986) for further illuminating examples. We will
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consider estimating the probability of success p from an assumed binomial model. In addition, we will examine both variance estimators (naive and sandwich) when the underlying true model is actually hypergeometric. Consider a sample of n observations y 1 , . . . , y n from the binomial (m, p) distribution. We want to estimate the variance of pˆ in order to develop valid asymptotic confidence intervals. The log-likelihood (where NF is the normalization factor) is given by L( p; y 1 , . . . , y n ) =
n
{y i ln p + (m − y i ) ln(1 − p) + NF}
i=1
where the estimating equation for p is then n n 1 1 ∂L yi − (m − y i ) = 0 = ( p) = ∂p p 1−p i=1
i=1
It is clear from the estimating equation that the maximum likelihood estimator is pˆ = y¯ /m. Further, we can easily derive the expected value of the inverse second derivative A = E( ( p)) as pq/mn where q = 1 − p. The empirical part of the sandwich estimator (the matrix B) is given by n yi m − yi 2 − p q i=1
Assuming that the Binomial model is true, the expected value of B is given by mn/pq, and the sandwich estimate of variance has expected value pq mn pq pq = mn pq mn mn Since the variance of pˆ is also pq/mn, we see that both the usual Hessian estimator and the sandwich estimator have the same (correct) value. Now, let’s assume that the true underlying model is hypergeometric (M, Y, m). Instead of considering this distribution in terms of the number of success in the population Y, we consider the distribution in terms of the proportion of successes in the population p = Y/M. In this case, the expected value of the empirical part of the sandwich estimator is pq(M − m)/[mn(M − 1)] and the sandwich estimate of variance then has expected value pq pq M − m pq pq(M − m) = (1) mn mn M − 1 mn mn(M − 1) while the expected value of the Hessian does not change. The variance of pˆ assuming the hypergeometric distribution is the same as given for the sandwich estimator (Eq. (1)) illustrating the robustness to misspecification
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of the underlying model. It is also important to note that the sandwich estimator is not necessarily larger than the information-based variance estimator. In this particular case, it is smaller on average. The validity of a confidence interval is affected by (i) the consistency of the √ estimator pˆ for p, (ii) the asymptotic normality (limiting distribution) of n(pˆ − p), and (iii) the consistency of the information-matrix-based estimate of the variance of the limiting distribution. The first two properties are quite robust – another focus of Huber (1967) is a formal investigation of these properties. The weakness in results (for model-based variance estimators) is in the last property where the (Hessian) variance estimate is defined by the assumed distribution of the outcome. Hessian-based variance estimators are not robust to misspecification of the distribution. In addition to this simple illustration, we can also find direct results for generalized linear models with misspecified variances. Here, assume a misspecified model where the true variance is V 2 (), but we fit the model assuming the variance is V 1 (). The estimating equation for the fitted model is given by n y i − i ∂ () = =0 a()V 1 () ∂ i=1
where g() = and g −1 () = are the pair of link/inverse link functions, and we assume that the variance is a function of the mean V(y) = a()V 1 (). For the constant only model (single parameter case = X = ), the maximum likelihood estimator is given by ˆ = g( y/n). Under the true model, we have E(y) = , V(y) = a()V 2 (), and, therefore, E(y 2 ) = a()V 2 () + 2 . Now, we can investigate the variance of the maximum likelihood estimator and compare that to the model-based and sandwich variance estimators. The sandwich estimator is robust to misspecification of the underlying model and therefore should be robust to misspecification of the variance function. The variance of the maximum likelihood estimator (fitted under the incorrect model) under the true model is given by y 1 1 ˆ V() = V g = g 2 2 nV(y) = g 2 a()V 2 () n n n 2 ∂ 1 = a()V 2 () n ∂
The Sandwich Estimate of Variance
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while the model based variance and the middle of the sandwich estimator (the empirical estimator) are given by 2 1 ∂ −1 A = a()V 1 () n ∂ n n ∂ 2 1 B= i ()2 = ( y − i )2 ∂ a()2 V 1 ()2 i i=1
i=1
Using the expected value of the empirical variance estimator to examine the expected value of the sandwich estimator results in 2 n 1 ∂ E(y 2i − 2y i i + 2i ) E(B) = ∂ a()2 V 1 ()2 =n
V 2 () a()V 1 ()2
E(V S ) = A
−1
E(B)A
−T
∂ ∂
2
i=1
2 1 ∂ = a()V 2 () n ∂
Compared to the variance of the estimator, we see that the model-based variance ˆ under misspecification of variance, while the estimator is incorrect A−1 = V() ˆ sandwich estimator is correct on average E(V S ) = V().
4. APPLICATIONS In the following subsections, we highlight the detailed construction of the sandwich estimate of variance for various models. This collection of examples allows us to emphasize the techniques that may be utilized. We include in our discussion the use of the so-called modified sandwich estimate of variance. See Kim and White (2003) for a detailed use of the sandwich variance estimate for quantile regression, and Vogelsang (2003) for an application under generalized methods of moments models.
4.1. Single Equation Maximum Likelihood Models The applicability and derivation of the sandwich variance estimate for single equation maximum likelihood models is straightforward. Models derived assuming
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independent and identically distributed outcomes clearly satisfy the assumptions of the corollary and, thus admit construction of the sandwich estimator. Derivation of the estimator involves writing the log-likelihood. We then take a derivative with respect to the parameter vector of interest. This derivative, called the score equation, is the estimating equation; it is the equation which when solved provides the estimates of the parameters. The contributions to the score equation, apart from the covariates, are residuals called scores. The sandwich estimate of variance for single equation maximum likelihood models V S = A−1 BA−T is constructed using n
∂() i ()i (T ) , B= A= − ∂ i=1
For example, consider the Poisson regression model for which the expected count is reparameterized via the log link to ensure non-negativity of the predicted counts. Our goal is to construct the sandwich estimate of variance for the regression parameters . The log-likelihood L is given by L(|X, y 1 , . . . , y n ) =
n
−exp(xi ) + y i xi  − ln (y i + 1)
i=1
and the estimating equation is n ∂L yi ∂ () = = −1 x ji = [0] ∂j i ∂ i i=1
Estimation of the A and B matrices assuming the log link xi  = ln(i ) is then Aˆ = −XT Diag()X, ˆ
ˆ = B
n
xi (y i − ˆ i )2 xTi
i=1
ˆ and xi is the ith row of the design matrix. This straightforwhere ˆ = exp(xi ) ward construction is valid for both FIML models, and LIML models including ancillary parameters. Consider an underlying true model where observations are grouped into panels. Assuming that the observations are correlated within panels, but that panels are uncorrelated, admits a further generalization of the usual sandwich estimate of variance. We believe that this generalization is first formally used in Rogers (1993), though the author, who was a student of Huber, claims that the extension was obvious to his advisor. The modified sandwich estimate of variance is a modification of the meat of the sandwich; the B matrix. Instead of summing the outer product of the individual
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53
score contributions, we instead sum the products of the panel-based sums of score contributions. The bread of the sandwich, the A matrix, is unchanged in this modification. We describe this procedure assuming that there are n panels where each panel has n i observations. Constructing the appropriate estimated matrix for Poisson regression, we have n n n i i T ˆ = B xit (y it − ˆ it ) (y it − ˆ it )xit i=1
t=1
t=1
See Section 7 for further notes on interpreting this estimator. Other modifications focus on the score contributions. Kauermann and Carroll (2001) point out that these contributions are biased and can be replaced with unbiased contributions. At least by these authors, the resulting estimator is then called the unbiased sandwich estimate of variance. MacKinnon and White (1985) also consider the unbiased sandwich variance estimator among other estimators for a study in linear regression, and we will look at their findings in more detail in Section 6.
4.2. The Infinitesimal Jackknife Approach While the focus of Cain and Lange (1984) is the development of influence statistics, we follow the same arguments for building the infinitesimal jackknife variance estimate in order to highlight the equivalence to the sandwich estimate of variance for the Cox proportional hazards model formally derived in Lin and Wei (1989). These arguments are also presented in Miller (1974) and Efron (1981). The partial likelihood L is given by exp(xi ) L= k∈R i exp(xk ) i∈D
The model is developed for survival analysis. The terminology of this area of study assumes that a group of patients are followed over time. Some of these patients will die. The likelihood is in terms of the death times D i in the set of all death times D. At each death time, there is a pool of patients R i who are at risk to die. Though the terminology is in terms of death and people, the model holds for studies in which we change death to failure and patients to items. The specification attempts to model survival times and so individual patients actually appear in many of the contributions to the likelihood. This violates the assumption of independence needed to construct the sandwich estimate of
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variance. What is needed is an equivalent representation of the likelihood in terms of the independent contributions to the likelihood for each patient. Equivalently, we seek the change in ˆ when we leave out an individual patient j. Initially, we consider a (patient) weighted analysis where all of the weights are one except for patient j. A first order Taylor expansion about the weight for patient j ˆ given by wj is approximately equal to the change in the fitted coefficients; ∂/∂w j is approximately equal to ˆ − ˆ (j) where ˆ (j) is the usual leave-out notation indicating the the fitted coefficients are obtained leaving out observations on patient j. The variance of this estimator then has the same form as the sandwich estimator. The derivative we seek is obtained by considering that the estimating equation is a function of both  and wj so that an application of the chain rule defines ˆ /∂w ˆ (∂/∂)(∂ j ) = 0. If we solve for the derivative of interest, we then have ˆ ˆ −1 ∂/∂wj . The variance is formed noting that the first term ∂/∂wj = −(∂/∂) in the resulting product is the A−1 in the sandwich estimator. To obtain the middle of the infinitesimal (sandwich) variance estimator ∂/∂wj , we must specify the manner in which the weights enter the partial likelihood wj exp(xi ) L= k∈R i exp(xk ) i∈D
Under this specification, the estimating equation is then given by ˆ w w x exp(x ) i k k k k∈R ˆ = w i xi − i ˆ k∈R i wk exp(xk ) i∈D and we differentiate this with respect to wj to obtain a form of the estimating equation that satisfies the independence assumption. The result is an estimating equation that is squared (in the outer matrix sense) to estimate the meat of the sandwich and is the same as given in Lin and Wei (1989). Binder (1992) illustrates the same variance estimate including sampling weights in the original design.
4.3. Two-Stage Maximum Likelihood Models Numerous models have been presented in the literature in which one model is embedded in another. Such models are broadly known as two-step estimation problems and are characterized as Model 1 :
E[y1 |x1 , 1 ]
Model 2 :
E[y2 |x2 , 2 , E[y1 |x1 , 1 ]]
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The overall model indicates that there are two parameter vectors to estimate. The first parameter vector 1 appears in both models, but the second parameter vector 2 appears only in the second model. Model 1 has n 1 observations and Model 2 has n 2 observations. In general, n 1 = n 2 and selection models, in particular, are characterized by n 1 > n 2 . There are two standard approaches to estimation. The first approach is a full information maximum likelihood, FIML, model in which we specify the joint distribution f(y1 , y2 |x1 , x2 , 1 , 2 ) and maximize the joint log-likelihood function. Alternatively, we can adopt a limited information maximum likelihood, LIML, two-step procedure. In this approach, we estimate the first model since it does not involve the second parameter vector. Subsequently, we estimate the second parameter vector conditional on the results of the first step estimation; we maximize the conditional log-likelihood L given by L=
n
f [y 2i |x2i , 2 , (x1i , ˆ 1 )]
i=1
Here, and throughout this section, we assume that there are n observations, x1i is the ith row of the X1 design matrix, x2i is the ith row of the X 2 design matrix, and ˆ 1 is the maximum likelihood estimate obtained from the estimation of Model 1. 4.3.1. A Comparison to the Murphy–Topel Estimator We present a comparison of the sandwich estimate of variance and the eponymous Murphy–Topel variance estimator for two-stage models. Greene (2000) gives a concise presentation of one of the results in Murphy and Topel (1985). The presentation, summarized from Hardin (2002), describes a general formula of a valid variance estimator for 2 in a two-stage maximum likelihood estimation model. Interested readers can also consult Lee et al. (1980) for illustration of specific two-stage models. This LIML estimation fits one model which is then used to generate covariates for a second model of primary interest. Calculation of a variance estimate for the regressors 2 in the primary model of interest must address the fact that one or more of the regressors have been generated via (x1 , ˆ 1 ). In order to highlight the derivation and comparison to the sandwich estimate of variance, we assume that 1 is a q × 1 vector of unknown parameters associated with an n × q matrix of covariates X. In addition, 2 is a p × 1 vector of unknown parameters associated with an n × p matrix of covariates W . Following Greene (2000), the formula for the Murphy–Topel variance estimator for 2 is given by V 2 + V 2 [CV 1 C T − RV 1 C T − CV 1 R T ]V 2
(2)
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where V 1 = (q × q) Asymptotic variance matrix of ˆ 1 based on L1 (1 ) V 2 = (p × p) Asymptotic variance matrix of ˆ 2 based on L2 (2 |1 ) ∂L2 ∂L2 C = (p × q) matrix given by ∂2 ∂T 1 ∂L2 ∂L1 R = (p × q) matrix given by T ∂2 ∂T1
(3)
It should be emphasized that this estimator is only one of several results presented in the original paper. Most textbooks, however, refer only to this specific formula. The component matrices of the Murphy–Topel estimator are estimated by evaluating the formulas at the maximum likelihood estimates ˆ 1 and ˆ 2 . The presentation assumes the existence of a log-likelihood for the first model L1 (1 ) and a conditional log-likelihood for the second (primary) model of interest L2 (2 |1 ). Examining the formula, we see that there is a penalty term added to the naive variance for the estimation that is performed in the first stage. While this aspect of the formula strikes us as a necessity, the origin of the penalty term is not intuitively obvious. The illustration of the sandwich estimate of variance in this instance serves two purposes. First, we highlight the construction of the estimator when the estimating equation is partitioned. Second, we illustrate that the derivation of the sandwich estimate of variance admits an intuitive feel for the penalty term in the Murphy–Topel variance estimate. Our goal is the construction of the sandwich estimate of variance for = (1 , 2 ) partitioning the overall estimating equation as [()] =
1 (1 ) 2 (2 |1 )
= [0]
It is obvious that if our goal is the covariance matrix for 2 instead of the covariance matrix for the complete vector (1 , 2 ), we need only look at the lower ˆ −1 . Likewise, the sandwich estimate of ˆ −1 B ˆA right p × p partition of Vˆ S = A variance for 1 is given by the upper left q × q partition of Vˆ S .
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Using log-likelihood notation L, we calculate the sandwich estimator working separately with the individual terms of the two model log-likelihoods to get 2 2 ∂ L1 ∂ L1 ∂2 L1 0 ∂ ∂T ∂ ∂T ∂ ∂T 1 2 1 1 1 A= 2 1 = ∂ L2 ∂2 L2 ∂2 L2 ∂2 L2 ∂2 ∂T ∂2 ∂T2 ∂2 ∂T1 ∂2 ∂T2 1 ∂L1 ∂L1 ∂L1 ∂L2 ∂1 ∂1 ∂T1 ∂T2 B = ∂L2 ∂L2 ∂L1 ∂L2 ∂2 ∂2 ∂T1 ∂T2
(4)
The upper right matrix entry of A is zero since 2 does not enter into L1 . That a particular partition is zero is either clear from the two models or is assumed, as in the case of the population averaged generalized estimating equations of Liang (1987). Inserting the component matrices for the Murphy–Topel estimator, we have −V −1 V ∗−1 0 RT 1 1 A= , B= (5) −C ∗ −V −1 R V ∗−1 2 2 The use of asterisks (∗) as superscripts distinguishes similar matrix components where the superscript appears when the component in the sandwich estimator differs from the corresponding component in the Murphy–Topel estimator. For example, the C matrix in the Murphy–Topel estimator is the outer product of the gradients, Eq. (3), while the C ∗ matrix in the sandwich estimator is the inverse matrix of second derivatives; see the lower left matrix of Eq. (4). Carrying out the matrix multiplication A−1 BA−T , we have the partitioned sandwich estimate of variance given by V S (1 ) = V 1 V ∗−1 1 V 1 = V S1
∗T ∗T T CovS (1 , 2 ) = V 1 R T V 2 − V 1 V ∗−1 1 V 1 C V 2 = V 1 R V 2 − V S1 C V 2 ∗T ∗T V S (2 ) = V 2 C ∗ V 1 V ∗−1 1 V 1 C V 2 − V 2 RV 1 C V 2
−V 2 C ∗ V 1 R T V 2 + V 2 V ∗−1 2 V2
∗−1 ∗ ∗T = V 2 V ∗−1 − RV 1 C ∗T − C ∗ V 1 R T ]V 2 2 V 2 + V 2 [C V 1 V 1 V 1 C
= V S2 + V 2 [C ∗ V S1 C ∗T − RV 1 C ∗T − C ∗ V 1 R T ]V 2
(6) As anticipated, the sandwich estimate of variance for 1 is the usual sandwich estimate of variance which considers only the first model. This is the anticipated
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result since the first model does not involve the second parameter vector 2 . The sandwich estimate of variance for 2 given in Eq. (6) is similar in form to the Murphy–Topel variance estimate in Eq. (2). The two key differences are the use of the sandwich estimators from the separate estimating equation partitions, V S1 and V S2 from the individual models, and the specification of the matrix of second derivatives estimator C ∗ over the outer product of the gradient estimator C. Application of the usual sandwich estimator V S2 considering only the second model provides an estimator that is asymptotically correct when the model is correctly specified. However, V S2 does not have the usual robustness properties if the model is misspecified since the penalty term (the second summand in Eq. (6) is missing). The two-stage Heckman selection, or heckit, model described in Heckman (1976), is an example for which the previous derivation applies. The maximum likelihood Heckman selection model is a single equation maximum likelihood specification that admits a sandwich estimate of variance as described in Section 4.1. 4.3.2. Two-Stage Linear Regression In the two-stage linear regression case (identity link and Gaussian variance) referred to as two stage least squares, the derivation greatly simplifies. This simplification is shown in detail in White (1982a). Here, we present the results of the simplification and summarize the implications for construction of the sandwich variance estimate. The naive (model-based) covariance matrix takes advantage of the fact that V(Y − X W ) = V{Y i − X Wi }I = 2 I where 2 is the mean square ˆ Thus, the variance of the regression coefficients is estiof Y i − X Wi . ˆ = ˆ 2 (XT X P )−1 , where the projection matrix given by ˆ mated by V () P T XP = XS (X S X S )−1 X TS X W is a matrix of the predicted values from using the instruments XS . Therefore, the correct asymptotic variance can be obtained simply by performing a standard linear regression of Y on X P . The sandwich estimate of variance is then clearly given by (XTP X P )−1 X TP (Y − X W )(Y − X W )T X P (X TP X P )−1 such that the usual sandwich estimate from the (second stage) linear regression of Y on X P is correct. Thus, for the standard linear regression case, we may obtain both a modelbased and a sandwich estimate of variance by considering only the second stage regression – making it very easy to obtain results in software that includes the sandwich estimate of variance for linear regression. These simplifications are not true in the general case of a GLM as illustrated in the previous subsection.
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4.4. Multi-Stage Models We continue development of the sandwich estimate of variance where there are more than 2 models. Our specific example is a presentation in terms of generalized linear models where some of the covariates are measured with error. While there are many techniques for addressing measurement error, our application assumes access to unbiased instruments. The construction would be easy were it not so notationally vexing, but we provide all of the details to highlight the application. We introduce a notation altered from the usual notation for measurement error models. The usual notation involves naming individual matrices: Z for covariates measured without error, W for covariates measured with error, S for the instruments of W , and R for the augmented matrix of exogenous variables [ZS]. To avoid confusion with the measurement error notation and the usual notation associated with generalized linear models (the W weight matrix in the iteratively reweighted least squares, IRLS, algorithm), we demote the usual measurement error notational conventions as subscripts of X. We begin with an n × p matrix of covariates measured without error given by the augmented matrix X = (X1 X2 ), and consider the case for which X 1 = XZ , X2 is unobserved, and X W = X2 plus measurement error. X Z is an n × p z matrix of covariates measured without error (possibly including a constant), and X W is an n × p x (p z + p x = p) matrix of covariates with classical measurement error that estimates X2 . We wish to employ an n × p s (where p s ≥ p x ) matrix of instruments XS for X W . Greene (2000) discusses instrumental variables and provides a clear presentation to supplement the following concise description. The method of instrumental variables assumes that some subset X W of the independent variables are correlated with the error term in the model. In addition, we have a matrix X S of independent variables which are correlated with XW . Given X, Y and X W are uncorrelated. Using these relationships, we can construct an approximately consistent estimator which may be succinctly described. One performs a regression for each of the independent variables (each column) of XW on the instruments and the independent variables not correlated with the error term (X Z X S ). Predicted values are obtained from each regression and substituted for the associated column of XW in the analysis of the GLM of interest. This construction provides an approximately consistent estimator of the coefficients in the GLM (it is consistent in the linear case). If we have access to the complete matrix of covariates measured without error (if we know X 2 instead of just X W ), we denote the linear predictor p = j=1 [X 1 X 2 ]j j , and the associated derivative as ∂/∂j = [X1 X 2 ]j . The
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estimating equation for  is then
n y i − i ∂ [X 1 X 2 ]ji V(i ) ∂ i i=1
However, since we do not know X 2 , we use X R = (XZ X S ) to denote the augmented matrix of exogenous variables which combines the covariates measured without error and the instruments. We regress each of the p x components (each of the j columns) of XW on X R to obtain an estimated (p z + p s ) × 1 coefficient vector ␥j for j = 1, . . . , p x . The complete coefficient vector ␥ = (␥T1 , ␥T2 , . . . , ␥Tp x )T for these IV regressions is described by the estimating equation T XR (X W1 − X R ␥1 ) T XR (X W2 − X R ␥2 ) 2 = . (7) .. XTR (X Wpx − X R ␥px ) ˆ 2 = [XR ␥ˆ 1 , XR ␥ˆ 2 , . . . , X R ␥ˆ p ] of We may then form an n × p x matrix X x predicted values from the instrumental variables regressions to estimate X2 . Combining the (predicted value) regressors with the independent variables measured without error, we may write the estimating equation of the GLM as n y i − i ∂ [XZ X R ␥]ji (8) 1 = V(i ) ∂ i i=1
where
[XZ X R ␥]ji =
(XZ )ji (X R ␥j−pz )i
if 1 ≤ j ≤ pz if pz < j ≤ pz + px
Operationally, we obtain a two-stage estimate ˆ by first replacing each unknown covariate X W i for i = 1, . . . , p x with the fitted values of the regression of X Wi on (XZ X S ). We call the resulting n × p x matrix of fitted values Xˆ 2 . We then perform a (second stage) usual GLM fit of Y on (X ZXˆ 2 ). This GLM fit provides an estimate of . The construction of the sandwich estimate of variance then follows as in the previous section. The variance matrix obtained from the (second stage) GLM fit described in the previous section assumes that Xˆ 2 = X2 . This is clearly unacceptable. We derive an appropriate sandwich estimate of variance that takes into account the estimation of X2 .
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The two-stage derivation resulting in an estimate for  involves estimating the combined parameter vector given by = (T ␥T )T . These results are from the estimating equations given in Eqs (7) and (8). While we are ultimately interested in , we must consider all of the parameters in forming the associated variance matrix. Although there are several partitions of the total estimating equation, we specify only two partitions. The partitions devoted to specifying the instrumental variables regression are block diagonal and grouped into a single block for discussion. Again, our goal is construction of the sandwich estimate of variance given by V S = A−1 BA−T . We form the variance matrix, A, for by obtaining the necessary derivatives. The variance matrix A (information matrix) may be calculated numerically, but the analytic derivatives are not difficult. The detailed presentation of these derivatives is
∂1 − ∂ (p +p )×(pz +px ) z x A= ∂2 − ∂ (px (ps +pz ))×(pz +px )
−1 ∂1 − ∂␥ (pz +px )×(px (pz +ps )) ∂2 − ∂␥ (px (pz +ps ))×(px (pz +ps ))
where the derivatives of the first partition are given by 2 n ∂1 1 ∂ =− ∂k V(i ) ∂ i i=1 2 2 ∂ 1 1 ∂V(i ) ∂ − (i − y i ) − V(i ) ∂2 i V(i )2 ∂ i ∂ × [XZ X R ␥]ji [X Z XR ␥]ki
(9)
j = 1, . . . , p z + p x ; k = 1, . . . , p z + p x yields a matrix of size (p z + p x ) × (p z + p x ). 2 n 1 ∂1 ∂ =− V(i ) ∂ i ∂␥k i=1 2 2 ∂V(i ) ∂ 1 1 ∂ − (i − y i ) − 2 ∂ i ∂ V(i ) ∂2 i V(i ) × [XZ X R ␥]ji X Rki +p z
(10)
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j = 1, . . . , p z + p x ; k = 1, . . . , p z + p s ; = 1, . . . , p x yields a matrix of size (p z + p x ) × (p x (p z + p s )) and the derivatives of the second partition are given by ∂2 =0 ∂k
(11)
j = 1, . . . , p z + p s ; k = 1, . . . , p z + p x ; = 1, . . . , p x yields a matrix of size (p x (p z + p s )) × (p z + p x ). ∂2 =− X Rji X Rki ∂␥k n
(12)
i=1
j = 1, . . . , p z + p s ; k = 1, . . . , p z + p s ; = 1, . . . , p x yields a block diagonal matrix of size(p x (p z + p s )) × (p x (p z + p s )) where each block matrix is of size (p z + p s ) × (p z + p s ). The elements of the variance matrix are formed from the definitions above. Mapping these equations is accomplished by defining the matrix A using [XZ X R ␥]ji =
XZji
if 1 ≤ j ≤ pz
(X R ␥(j−pz ) )i
if pz < j ≤ pz + px
in which we apply the notation XZji = ith observation of the jth column of XZ X Rji = ith observation of the jth column of XR ␥j−pz = IV coefficient vector from regressing X W(j−pz ) on XR +pz = ( + pz )th coefficient of  (X R ␥(j−pz ) )i = ith observation of (the predicted values from) X R ␥j−pz ␥k = kth coefficient of the th IV coefficient vector Equation (9) defines the (j, k) elements of A for j, k = 1, . . . , p z + p x . Equation (10) defines the (j, k) elements of A for j = 1, . . . , p z + p x and k = p z + p x + , where = 1, . . . , p x (p z + p s ). This notation addresses the cross derivatives for all of the (p z + p s ) × 1 coefficient vectors ␥m for m = 1, . . . , p x . Equation (11) calculates the (j, k) elements of A for j = p z + p x + , where = 1, . . . , p x (p z + p s ) and k = 1, . . . , p z + p x . Equation (12) defines the (j, k) elements of A for j, k = p z + p x + , where = 1, . . . , p x (p z + p s ). These are the covariances of all of the (p z + p s ) × 1 coefficient vectors ␥m for m = 1, . . . , p x .
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Given that i = [1i 2i ], the middle of the sandwich estimate of variance is then B = ni=1 i Ti . A suitable estimate may be calculated using j=1,...,(p z +p x ) y i − i ∂ [X Z XR ␥] ˆ ji V(i ) ∂ i (p z +p x )×1 j=1,...,(p +p ) [(X W1 − X R ␥ˆ 1j )X Rji ](pz +ps )×1z s [(X − X ␥ˆ )X ]j=1,...,(pz +ps ) W2 R 2j Rji (pz +ps )×1 ˆ 2i = .. . j=1,...,(pz +ps ) [(X Wpx − X R ␥ˆ pxj )X Rji ](pz +ps )×1
ˆ 1i =
(p x (p z +p s ))×1
The sandwich estimate of variance for  is then the upper (p z + p x ) × (p z + p x ) matrix of V S constructed from the illustrated matrices A and B.
5. GENERALIZED ESTIMATING EQUATION MODELS Liang and Zeger (1986), hereafter LZ, present methods for extending the class of generalized linear models to the analysis of longitudinal data. The estimating equation derived by the authors is not specified from joint distributions of the subject’s observations. Instead, they use familiar quasi-likelihood arguments to introduce a score equation for multivariate Gaussian outcomes. Our previous examples and discussion focused on the robustness of the sandwich estimator when the underlying parametric model was misspecified. In this application, we see that the sandwich estimator is robust even when the underlying model is left completely unspecified. See Gourieroux et al. (1984) for further justification of the use and properties of GEE-based estimators for Poisson models. The data of interest are characterized by an outcome variable y it and a p × 1 vector of covariates x it where multiple observations t = 1, . . . , n i are collected from each of the subjects (panels) i = 1, . . . , n. The total number of observations is given by N = ni=1 n i . If we have a single observation per panel (n i = 1 for all i = 1, . . . , n), then we can apply the generalized linear modeling techniques of McCullagh and Nelder (1989) for a wide variety of continuous and discrete models. Repeated observations per panel, however, introduce an added level of correlation among the observations that must be addressed. The approach of LZ focuses on a marginal distribution of the outcome variable in which the repeated observations per panel are addressed through the inclusion
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of a working correlation matrix. The correlation matrix is usually parameterized subject to some form, and the associated parameter estimation is treated as ancillary to the generalized linear model (GLM) of interest. Notation is introduced by describing the marginal density of y it given as a member of the exponential family of distributions f(y it ) = exp[y it it − a(it ) + b(y it )] where it = g(it ), it = x it . In this notation, g(·) is a link function relating the linear predictor it to the expected value of y it . The exponential family of distributions also admits a scalar component of the variance . We know that the mean is given by E(y it ) = a (it ) = it , and the variance by V(y it ) = a (it )/. An estimate of  is obtained through n theTscore equation (from the likelihood analysis) is given by solution to i=1 X i i S i = 0 where X i is the n i × p matrix of observations for panel i, i = Diag(∂it /∂it ) is an n i × n i matrix, and S i = Y i − a (i ) is an n i × 1 vector. In the definition of i , and throughout the remainder of this paper, we write Diag(·) to denote a diagonal matrix. Following the arguments of LZ, one may introduce a working correlation matrix in the calculation of the variance by defining Ai as the n i × n i matrix 1/2 1/2 Diag(a (it )). The variance matrix is V i = (1/)Ai R(␣)Ai where we write R(␣) to indicate the parameterization of the correlation matrix by ␣. Note that if R(␣) is the identity matrix, the definition of the variance matrix is unchanged from the definition in the usual GLM approach. The generalized estimating equation for  is given by ni=1 D Ti V −1 i Si = 0 where D i = ∂i /∂ = Ai i X i . Similarly, there is an estimating equation for the structural parameters ␣ of the working correlation matrix. The partitioned GEE is given by n ∂ T −1 yi − i xji Diag [V(i )] () = ∂ a() i=1
(␣) =
n i=1
∂i ∂␣
T
H −1 i (W i − i )
where W i = (ri1 ri2 , ri1 ri3 , . . . , rini −1 rini )T H i = Diag(V (W ij )) i = E(W i ) and r ij is the Pearson residual for the jth observation of the ith panel. To construct the sandwich estimate of variance for the class of GEE models described in LZ, note that an explicit assumption of the model is that  and ␣ are uncorrelated. Therefore, the cross covariances are zero as we saw in the
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illustration comparing the sandwich estimate of variance to the Murphy–Topel variance estimate. The result of this assumption is that the sandwich estimate of variance may be formed exactly as illustrated for a single equation model. The remaining issue is that the single estimating equation of interest fails to satisfy the independence assumption; we must consider the sums of score contributions per panels (the panels are independent). The modified sandwich estimate of variance is the correct choice for this specification. Pan (2001) suggested an alternative to the usual modified sandwich estimate of variance. As usual, the alternative derives from modifying the meat of the sandwich (cf. the Newey–West extension). The author first points out that the empirical correction factor may be written n ∂i T ∂i B= Diag V(i )−T Cov(yi )V(i )−1 Diag ∂ ∂ i=1
This presentation emphasizes that the covariance of the outcome is estimated using information from individual panels Cov(yi ) = S i S Ti ,
S i = yi − i
The author argues that a better estimate of the covariance would consider information across panels, and he altered the correction factor to n −1/2 ∂i 1/2 1 1/2 T −1/2 Cov(yi ) = Ai Ai , Ai S i S i Ai Ai = Diag n ∂ i=1
We emphasize that one could easily add instrumental variables into the LZ population-averaged GEE model. This addition to the model only incurs further partitioning of the estimating equation. Other than complicating the derivatives and notation, these additional pieces of the overall estimating equation are addressed in the same manner illustrated in earlier examples. Hardin and Hilbe (2002) provides the details on this more complicated example.
6. SMALL SAMPLE ADJUSTMENTS The price we pay for obtaining consistency for misspecified models is that the sandwich variance estimator is more variable than a comparable naive variance estimator. Sometimes, it is far more variable; Kauermann and Carroll (2001) provide specific examples illustrating this property. Additional references that highlight the increased variability of the sandwich variance estimator include
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Efron (1986) discussing results from Wu (1986) and Breslow (1990) for overdispersed Poisson regression. More recently, Long and Ervin (2000) investigate small sample adjustments for linear regressions, and Hill et al. (2003) provide results and analysis from a comparison of several variance estimators for the heckit model. Finally, Rothenberg (1988) investigates the power of tests of linear regression coefficients based on sandwich estimators. Due to increased interest in the sandwich estimator and its use in construction of hypothesis tests and confidence intervals, there has been a corresponding interest in improving performance in small samples. These adjustments are ad hoc; they are constructed for specific applications. While the sandwich estimator plays an important role in the construction of confidence intervals and tests that are robust to model misspecification, the estimator is not without criticism. Drum and McCullagh (1993) cite the simple case of the comparison of means from two independent samples. The usual and sandwich estimates of variance correspond to the alternate forms of the t-test; separate and pooled variances, respectively. The authors point out that when one or both of the two samples is small, there is a substantial loss of power using the pooled (sandwich) variance estimator. Bias in small samples is further addressed in Chesher and Jewitt (1987). There are two approaches we could take to address the small sample behavior of the sandwich estimator. The first approach is to apply a scalar adjustment (possibly based on the number of independent score contributions), and the second approach is a distributional change in the construction of confidence intervals and tests to use a distribution with heavier tails than the normal. MacKinnon and White (1985) investigate both approaches for small samples in linear regression. Their investigation includes the unadjusted sandwich estimator (labeled hc0), scalar adjustments to the sandwich estimator (labeled hc1), the unbiased sandwich estimator (labeled hc2), and an alternative that is equivalent to an asymptotic jackknife estimator (labeled hc3). Hill et al. (2003) include similar estimators in their investigation of the heckit estimator. For linear regression treating 2 as an ancillary parameter, we have the estimating equation for  given by () =
1 T X (Y − X) 2
yielding Aˆ = ˆ 2 (X T X)−1 ˆ 2X ˆ = 1 X T Diag(Y i − xi ) B ˆ 4
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Thus, the unadjusted sandwich estimate of variance is given by ˆ 2 X(X T X)−1 hc0 = (XT X)−1 X T Diag(Y i − xi ) Focusing on the meat of the sandwich estimator B, the authors noted first that a reasonable scalar adjustment is given by n/(n − p). This adjustment inflates the usual sandwich estimator. n ˆ 2 X(X T X)−1 hc1 = (X T X)−1 X T Diag(Y i − xi ) n−p It should be noted that the use of this particular scalar adjustment is usually limited to linear regression models. Other regression models include the common small sample scalar adjustment n/(n − 1). The modified sandwich variance estimator is similarly adjusted using the scale factor g/(g − 1) where g is the number of panels or groups. The inflation is unimportant for large samples, but for small samples can overcome the underestimation of the variance estimate. A second adjustment considers that the B matrix is biased for the variance. E(B −1 ) = ˆ 4 [X T (I − X(X T X)−1 XT )X]−1 To overcome this bias, the middle of the sandwich estimator can instead be estimated by ˆ Diag[1 − xi (X T X)−1 xi ]−1 (Y − X ) ˆ TX ˆ U = 1 X T (Y − X ) B ˆ 4 so that
−1
hc2 = (X X) T
Y i − xi ˆ X Diag √ hii T
2 X(X T X)−1
where h ii is the ith diagonal of the hat matrix I − X(X T X)−1 X. The final adjusted sandwich estimator in MacKinnon and White (1985) is 2 Y i − xi ˆ T −1 T X(X T X)−1 hc3 = (X X) X Diag h ii which is shown to be equivalent to an asymptotic jackknife estimator in Wu (1986). These approaches serve as starting points for investigating small sample adjustments in other types of models. Hill et al. (2003) present findings of a simulation study of the properties of variance estimators for the heckit model. Interestingly, they use the wrong sandwich estimator, but the results indicate that the small sample adjustment
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included in their estimator overcomes the underestimation in ignoring the extra term of the correct sandwich estimator. Still, a better estimator is likely found by including a small sample adjustment to the correct sandwich estimator. Confidence intervals are constructed based on the normal distribution as Huber and White both derive the asymptotic normality of the coefficients of the estimating equation. Kauermann and Carroll (2001) show that the sandwich estimator is more variable than the Hessian. Accordingly, they argue that the Student’s t distribution should be favored in small samples. The choice of degrees of freedom still remains an issue. For linear regression, Kauermann and Carroll (2001) recommend 3 degrees of freedom. Others have recommended n − p degrees of freedom where n is the number of independent score contributions (number of observations or the number of panels) and p is the number of estimated coefficients. Kent (1982) recommends using F(1, n − p) citing consistency arguments.
7. INTERPRETATION OF RESULTS Lin (1989) points out that the derivation in White (1980) has the idea that the Fisher information matrix can be consistently estimated by either the score derivative matrix A or the squared score matrix B. When these two estimates differ, it is an indication that the assumed model is incorrect. Binder (1983), in a discussion of complex surveys, showed that the information matrix A could be replaced by the observed matrix of second derivatives A = ∂(x, )/∂. To see this subtle distinction imagine the case of a maximum likelihood estimator such that the estimating equation is in fact the derivative of the log-likelihood, L, given by (x, ) = ∂L/∂. Huber’s result specified E(∂2 L/∂2 ) = (∂L/∂)2 whereas Binder specified (−∂2 L/∂2 )−1 . Hardin and Hilbe (2001) show in detail that these two approaches are numerically equal for the class of generalized linear models when the canonical link function is used, and Efron and Hinkley (1978) investigate the small sample properties of variance estimators based on the two approaches. Interpretation of results starts with noting that the construction of the variance for a model that is correctly specified involves specification of A and B that are asymptotically equal. The end result of this is that the sandwich estimate of variance is asymptotically equal to the inverse matrix of negative second derivatives (Hessian) of a likelihood-based model; it is the inverse matrix of derivatives of the estimating equation. This is the usual variance estimate such that if the model is correctly specified. Ignoring small sample discrepancies, this means that there is no difference in the two approaches. Since there are advantages to the sandwich estimate of variance, there is a clear motivation to using it in construction of tests and confidence intervals. However, we
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must point out that the common use of the modifier robust leads many researchers to the incorrect conclusion that the sandwich estimate of variance is more conservative than the usual specification of the inverse matrix of second derivatives. This is not true. The sandwich estimate of variance can provide standard errors that are smaller than Hessian-based standard errors. Kauermann and Carroll (2001) present a lucid review of these issues in terms of linear and logistic regression showing that the size of the standard errors is a function of the kurtosis of the design values. The use of the modified sandwich estimate of variance is more complicated. The estimator is asymptotically justified and so requires a large number of panels. Further, the rank of the resulting matrix can not exceed the number of panels. For example, a data set with five panels each containing 100,000 observations would not be of sufficient size for testing a model with six coefficients. The bread of the sandwich estimator is the expected Hessian, Huber (1967), or the observed Hessian, Binder (1983). While these estimators are asymptotically equivalent, in the case of generalized linear models using noncanonical links, they are numerically different. In constructing the sandwich estimator, the use of the expected Hessian is sometimes called semi-robust while the use of the observed Hessian is called robust. The distinction is irrelevant in the case of the canonical link since the expected and observed Hessians are the same. Using noncanonical links, the distinction refers to results which, while still robust to variance misspecification, are not robust to misspecification of the link function. Knowledge that the sandwich estimate of variance is constructed assuming misspecification, has led some researchers to misinterpret analysis results. Specifically, consider the modified sandwich estimate of variance. This estimator is said to be robust to any within-panel correlation. That it is robust to any within-panel correlation comes from the fact that no specific structure was assumed in the construction of the variance estimate. We merely added the correlated score contributions together, and considered the collection of uncorrelated sums. Lurking underneath this statement is that we have misidentified the underlying true model; we have estimated an observation-level model on panel data. It would therefore be inconsistent, and incorrect, to interpret likelihood-based tests of our fitted model. Further, it is incorrect to interpret individual coefficients as if they were panel-based since we did not fit a panel model.
8. SUMMARY The sandwich estimate of variance may be applied in a number of situations. We illustrated this application for single-equation models, estimating equation models, multi-stage models, estimating equation models, generalized estimating equation
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models, and proportional hazards models which failed to satisfy the independence assumption. Utilizing results that allow partitioning of the estimating equation, summation of the score contributions, and the infinitesimal jackknife, we can derive formulas for the sandwich estimate of variance for a wide variety of applications. For the two-stage models, we provided intuition for the (naive) Murphy–Topel estimator by illustrating the derivation of the appropriate sandwich estimate of variance. The power of the estimator is not without its limits. While we enjoy many benefits of robustness for associated confidence intervals, we must take care in the interpretation of the model and tests of coefficients. The popularity of the (asymptotically justified) variance estimator has also led to research into improving properties in small samples. This research introduces many ad hoc small sample adjustments, but there is no universally accepted adjustment.
ACKNOWLEDGMENTS The author would like to thank Carter Hill and Tom Fomby for organizing such a successful conference. Several conference participants, including Hal White and Tim Vogelsang, pointed out additional references that have been incorporated in this paper. Hill and Fomby also provided several useful suggestions to improve the final presentation.
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TEST STATISTICS AND CRITICAL VALUES IN SELECTIVITY MODELS R. Carter Hill, Lee C. Adkins and Keith A. Bender ABSTRACT The Heckman two-step estimator (Heckit) for the selectivity model is widely applied in Economics and other social sciences. In this model a non-zero outcome variable is observed only if a latent variable is positive. The asymptotic covariance matrix for a two-step estimation procedure must account for the estimation error introduced in the first stage. We examine the finite sample size of tests based on alternative covariance matrix estimators. We do so by using Monte Carlo experiments to evaluate bootstrap generated critical values and critical values based on asymptotic theory.
1. INTRODUCTION Many researchers use Heckman’s (1979) two-step estimation procedure (Heckit) to deal with selectivity in the linear regression model. Selection bias results when the regression dependent variable is observed only when a “latent” selection variable is positive. While the two-step estimation procedure is easy to implement, i.e. a probit estimation of the selection equation followed by least squares estimation of an augmented regression, the applied literature reveals that researchers (and econometric software vendors) take a variety of approaches when computing standard errors. This is important since the standard errors are the basis for t-statistics that are used in significance tests. Maximum Likelihood Estimation of Misspecified Models: Twenty Years Later Advances in Econometrics, Volume 17, 75–105 Copyright © 2003 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(03)17004-1
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The literature on sample selection bias is huge, and we will not attempt a survey. Two articles of note are by Vella (1998) and Puhani (2000). In this paper we focus on two issues. First, how do the alternative versions of asymptotic variance-covariance matrices used in selectivity models capture the finite sample variability of the Heckit twostep estimator? Second, is it possible to use bootstrapping to improve finite sample inference? Three aspects of this question are: ◦ Do bootstrap standard errors match finite sample variability better than nominal standard errors computed from asymptotic covariance matrices? ◦ Do critical values of test statistics generated by bootstrapping pivotal t-statistics lead to better test size (and power?) than those based on usual asymptotic theory? ◦ Does modern software make obtaining bootstrap standard errors and critical values feasible for empirical researchers? Our plan is to develop the model and Heckman’s two-step estimator in Sections 2 and 3, respectively, and then describe the alternative covariance matrix estimators we consider in Section 4. In Section 5, we comment on the practices used in empirical research and some choices available in commercial software. Section 6 contains the design of the Monte Carlo experiment as well as discussion of the output measures we recover. Section 7 presents the results of the Monte Carlo experiment, followed by conclusions in Section 8.
2. THE SELECTIVITY MODEL Following Greene (1997, pp. 974–981) consider a model consisting of two equations. The first equation is the “selection equation,” which is defined as z ∗i = wi ␥ + u i ,
i = 1, . . . , N
(2.1)
where z ∗i is a latent variable, ␥ is a K × 1 vector of parameters, wi is a 1 × K row vector of observations on K exogenous variables and ui is a random disturbance. The latent variable is unobservable, but we do observe the dichotomous variable 1 z∗i > 0 zi = (2.2) 0 otherwise The second equation is the linear model of interest. It is y i = x i  + e i ,
i = 1, . . . , n,
N>n
(2.3)
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where yi is an observable random variable,  is an M × 1 vector of parameters, x i is a 1 × M vector of exogenous variables and ei is a random disturbance. We assume that the random disturbances are jointly distributed as 1 0 ui ∼N , (2.4) 0 ei 2e A “selectivity problem” arises when yi is observed only when z i = 1, and if = 0. In such a situation, the ordinary least squares estimator of  in (2.3) is biased and inconsistent. A consistent estimator is the two-step procedure suggested by Heckman (1979) and clarified by Greene (1981). It is known as “Heckman’s twostep estimator,” or more simply as “Heckit.” The basis for this estimation procedure is the conditional regression function E[y i |z i > 0] = E[y i |u i > −wi ␥] = x i  + E[e i |u i > −wi ␥] = x i  + ( · e )i (2.5) where i =
(wi ␥) (wi ␥)
(2.6)
is the “inverse Mill’s ratio,” (·) is the standard normal probability density function evaluated at the argument, and (·) is the cumulative distribution function for a standard normal random variable evaluated at the argument. In a regression equation format, y i = E[y i |z i > 0] + vi = x i  + ( · e )i + vi = x i  +  i + vi
(2.7)
where the random disturbance vi has conditional mean and variance given by E[vi |z i > 0] = 0,
var(vi |z i > 0) = 2e (1 − 2 ␦i )
(2.8)
with ␦i = i (i + wi ␥)
(2.9)
Thus, the regression error vi in (2.7) is heteroskedastic. If i were known and non-stochastic, then the selectivity corrected model (2.7) could be estimated by generalized least squares. Alternatively, the heteroskedastic model (2.7) could be estimated by ordinary least squares, and the heteroskedasticity consistent covariance matrix estimator (HCE) of Huber (1967), Eicker (1963) and White (1980) used for hypothesis testing and construction of confidence intervals. Unfortunately i is not known and must estimated, introducing variability not accounted for by a heteroskedasticity correction.
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3. HECKIT: THE HECKMAN TWO-STEP ESTIMATOR The widespread use of Heckit is no doubt in part due to the ease with which it is computed. In the first step of this two-stage estimation process, the method of maximum likelihood is used to estimate the probit model parameters ␥, based upon the observable random variable zi . Denote the MLE as ␥ˆ and its asymptotic covariance matrix as V. We will compute V as the negative of the inverse Hessian evaluated at the MLE. Specifically, if f i1 is the (Bernoulli) probability function of zi , then ln f i1 = ln ([ (wi ␥)]z i [1 − (wi ␥)]1−z i ) (3.1) N and the log-likelihood function for the probit model is L 1 = i=1 ln f 1i . For convenience let i = (wi ␥) and i = (wi ␥). Then, for future reference, the gradient vector of the probit model log-likelihood function is N
N
i=1
i=1
(w ␥) (wi ␥) ∂L 1 i = zi (1 − z i ) wi − wi ∂␥ (wi ␥) 1 − (wi ␥)
(3.2)
and the Hessian is
N i + (w ␥i ) i i − (w ␥i )(1 − i ) ∂2 L 1 wi wi = − i z i + (1 − z i ) 2 2 ∂␥∂␥ ) (1 − i i i=1 (3.3) Then, using the negative of the inverse Hessian evaluated at the MLE, we compute the asymptotic covariance matrix V of the probit estimator as 2 −1 ∂ L1 V=− (3.4) ∂␥∂␥ ␥), Given the MLE ␥, ˆ we compute ˆ i = (wi ␥)/ (w ˆ i ˆ i = 1, . . . , n. The variable ˆi is used as a regressor in the second step equation,
y i = x i  +  ˆ i + vi +  (i − ˆ i ) = x i  +  ˆ i + vi ˜  ˆ ∗ + vi = xi  + vi = [x i i ]  ˜ ˜ ˜
(3.5)
Stacking the n complete observations in (3.5) into matrices, the Heckit estimator of the M + 1 parameters ∗ is the ordinary least squares estimator applied to the augmented regression in (3.5), that is b ∗ = (X X)−1 X y
˜ ˜
˜
(3.6)
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¯ˆ 2 , where vˆ = y − Xb ∗ , b is the least An estimator of 2e is ˆ 2e = (ˆv vˆ /n) + ␦b ˜ ˜ ˜ ˜ ˆ squares estimator of  , ␦i is the plug-in estimator of ␦i in (2.9), and ␦ˆ¯ = n ˆ ˆ 2 = b 2 /ˆ 2e . i=1 ␦i /n. An estimator of is
4. HECKIT: ALTERNATIVE COVARIANCE MATRIX ESTIMATORS CONSIDERED IN THE MONTE CARLO The focus in this paper is the computation of the covariance matrix estimate. There are a number of alternatives used in the literature, some of which are appropriate and some of which are not. (1) One alternative is to use the standard estimator for the covariance matrix of the OLS estimator, V OLS = ˆ 2v (X X)−1 ,
ˆ 2v =
˜ ˜
vˆ vˆ ˜˜ n − (M + 1)
(4.1)
This estimator is incorrect for two reasons: it ignores the heteroskedasticity in (2.7), even if we assume i is known, and it ignores that in (3.5) λˆ i is stochastic. (2) A second possibility is to account for the heteroskedasticity in (2.7), using the estimator ˆ ](X X)−1 V HET = ˆ 2e (X X)−1 [X (1− 2 )X
˜ ˜
˜
˜ ˜ ˜
(4.2)
ˆ = diag(␦ˆ i ) is a diagonal matrix with non-zero elements ␦ˆ i . This where estimator is the covariance matrix of the ordinary least squares estimator in the presence of heteroskedasticity of the form in (2.8). This estimator does not account for the fact that λˆ i in (3.5) is stochastic. The error term vi in (3.5) is for each observation a function of the probit MLE ␥. ˆ This means˜that cov(vi , vj ) = 0, which (4.2) does not account for. ˜ same ˜ (3) In this spirit, one might use the White (1980) heteroskedasticity consistent estimator (HCE) V HCE = (X X)−1 X D p X(X X)−1
˜ ˜
˜
˜ ˜ ˜
(4.3)
where Dp is a diagonal matrix. For the basic heteroskedasticity consistent estimator V HC0 the squared least squares residuals are on the diagonal, D 0 = diag(ˆv2i ). This estimator suffers the same flaw as VHET , but is easier to compute since˜V HC0 is readily available in most regression software. We also consider a modification of the standard heteroskedasticity consistent estimator V HC0 . This modification is denoted V HC3 by Davidson
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and MacKinnon (1993, p. 554) and has been studied recently by Long and Ervin (2000). To obtain V HC3 we replace the diagonal matrix D0 with D 3 = diag(ˆv2i /(1 − xi (X X)−1 xi )2 ). Davidson and MacKinnon note (1993, ˜ ˜ ˜model with homoskedastic errors the expected p. 70) that in˜ a linear˜regression value of these weighted residuals would equal the error variance. Justification for using V HC0 is found in Lee (1982). Amemiya (1984, p. 33) summarizes the argument by noting that the Heckman estimator is consistent under assumptions less restrictive than the usual bivariate normality of the error terms. Under the less restrictive assumptions White’s HCE provides a consistent estimator of the asymptotic covariance matrix. The specific form we use is Lee (1982, p. 365) equation (51), which is V Lee = (X X)−1 (V HC0 − X GVG X)(X X)−1
˜ ˜
˜
˜ ˜ ˜
where G is the matrix with i’th row λ ∂λi /∂␥ . See (4.8) below for the derivative. (4) The asymptotic covariance matrix for b ∗ was obtained by Heckman (1979, p. 159) and refined by Greene (1981, pp. 795–798). It is, following Greene (1997, p. 981), ˆ V HECK = ˆ 2e (X X)−1 [X (1 − 2 )X + Q](X X)−1
˜ ˜
˜
˜
˜ ˜
(4.4)
ˆ with V the asymptotic covariance matrix ˜ W where Q = ˆ 2 FVF and F = X of the probit estimator, the negative of the inverse Hessian, and with W being the regressor matrix from the probit estimation in the first stage. (5) An asymptotically equivalent, but seldom if ever used in selectivity models, asymptotic covariance matrix estimator is based on the Murphy-Topel (1985) general result. We are including the MT estimator because of its use in an increasingly wide variety of contexts, and its availability as an automated command in LIMDEP 8.0. In Greene’s (1997, p. 467) notation y i = h(x i , ∗ , wi , ␥) + vi = x i + i + vi = xi ∗ + vi
˜
(4.5)
Let cov(b ∗ |ˆ i ) = V HET = 2e V b (see Eq. (4.2)). Then the unconditional covariance matrix of the least squares estimator is V MT = 2e V b + V b [CVC − RVC − CVR ]V b
(4.6)
where V is the asymptotic covariance matrix of the probit estimator. The matrix C is given by C=
x 0i
∂h , ∂␥
x 0i =
∂h = x˜ i , ∂∗
∂i ∂h =  ∂␥ ∂␥
(4.7)
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where (wi ␥) ∂i = − [ (wi ␥) · (wi ␥) + (wi ␥)]wi = ␥i ∂␥ [ (wi ␥]2
(4.8)
The matrix R is R=
∂ln f i1 ˜ ˜ ∂␥
xi vˆ i
(4.9)
For i = 1, . . ., n the selectivity indicator z i = 1 in the probit model so that for these observations the i’th term of the log-likelihood function L1 is ln f i1 = (wi ␥) and ∂ln f i1 /∂␥ = φ(wi ␥)/ (wi ␥)wi = i wi . (6) Hardin (2002, 2003) has introduced a robust variance estimator for two-stage estimators similar in construction to the Murphy-Topel estimator. In it, which we call VRMT , the matrix C in (4.6) is replaced by C∗ = −
∂2 L 2 ∂1 ∂2
(4.10)
where L2 is the log-likelihood function of the second stage estimator, conditional on the parameters in the first stage. In the context of Heckit the first stage estimator is probit, and 1 = ␥. The second stage is least squares estimation of the augmented model (4.5), so 2 = ∗ = (  ). To compute the matrix ∗ C we write the log-likelihood function for the second stage regression as n 1 1 (y i − x i  − i  )2 1 2 L2 = − ln 2 − ln − 2 2 2 2 i=1 n n 1 1 v2i 1 2 − ln 2 − ln − = ln f i2 (4.11) = 2 2 2 2 i=1
i=1
Then, 1 ∂ln f i2 ∗ = 2 ∂ and
(yi − xi  − i  )xi
(yi − xi  − i  )i
(4.12a)
∂i  − xi − 2 xi ∂2 ln f i2 1 ∂␥ = ␥i = xi ␥i = ∗ 2 v  ∂ ∂ i ∂ ∂␥ i i − 2 i vi − i  2 i ∂␥ ∂␥ (4.12b)
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Finally, n C ∗ = − xi ␥i = −X ␥
(4.13)
i=1
where X is a matrix whose rows are xi and γ is a matrix with rows λγi which is given in (4.8). The second modification of the Murphy-Topel estimator Hardin introduces is to replace the probit covariance matrix V by the “sandwich” estimator V s = V · V −1 opg · V, where V −1 opg =
∂L 1 ∂L 1 ∂γ ∂γ
where L1 is the probit log-likelihood function, and Vopg is the “outer product of the gradient” estimator of the probit covariance matrix. (7) Our concern is not just the asymptotic reliability of covariance matrix estimators. Selectivity corrections are often applied in samples that are not “large.” It is known that standard errors computed from asymptotically valid covariance matrices can seriously understate true estimator variability in finite samples (Horowitz, 1997). In the context of models with limited dependent variables this has been illustrated by Griffiths, Hill and Pope (1987). We use the bootstrap approach (Freedman & Peters, 1984 a, b; Jeong & Maddala, 1993) to compute measures of finite sample variability. As suggested by Jeong and Maddala (1993, p. 577) we resample, with replacement, from the rows of the data matrix to create a large number of “bootstrap samples.” LIMDEP and STATA employ this scheme for automated bootstrapping. Using each bootstrap sample we re-estimate the model, storing the parameter estimates as we go. The bootstrap estimate of the estimator standard error is the sample standard deviation of all the estimates. Horowitz (1997) argues that while the bootstrap can be used to approximate standard errors of estimation, it is preferable to use the bootstrap to obtain critical values for t-statistics that are used as a basis for hypothesis testing. The crux of the argument is that bootstrap standard errors converge to the true standard errors as the sample size gets larger, but that bootstrapped critical values of test statistics converge to the true critical values at an even faster rate. Thus, instead of computing bootstrap standard errors and using these as a basis for a new t-statistic, it is perhaps better to bypass the computation of the standard error and just compute the critical value of the commonly used asymptotic t-statistic.
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Using standard econometric practice, to test the null hypothesis H 0 : k = 0k against the alternative H 0 : k = 0k we use the t-statistic t = (b k − 0k )/se(b k ), where se (b k ) is a valid asymptotic standard error for the consistent estimator b k . For example, se (b k ) might be the Heckit standard error from VHECK , or the standard error computed from the Murphy-Topel asymptotic covariance matrix VMT . The null hypothesis is rejected if |t| ≥ t c , where tc is the critical value from the standard normal or t-distribution. The problem is that using the standard critical values with t-statistics derived from asymptotic theory leads to tests with incorrect size, or probability of type I error. Horowitz suggests that we can obtain improved critical values using the bootstrap. In each bootstrap sample we obtain the t-statistic value for the (true in the sample) hypothesis H 0 : k = b k . That is, we compute for each bootstrap sample t b = (ˆ b − b k )/se b , where ˆ b is an estimate from a consistent estimator in the b’th bootstrap sample, and seb is the value of the asymptotic standard error of the estimator ˆ b . In the bootstrap world it is b k , the estimate based on the original and full sample, that plays the role of the true parameter value. Hence the statistic tb is centered at the “true in the sample” parameter value, and it is asymptotically pivotal. The absolute values of these t-statistics are sorted by magnitude and the positive t-critical value tc is chosen to be the upper ␣-percentile value. We will report, from the Monte Carlo experiment that we outline in the next section, both the standard errors computed by the bootstrap, and the bootstrap t-critical values.
5. APPROACHES IN THE EMPIRICAL LITERATURE: SOME COMMENTS At least four different strategies are being used to obtain standard errors in selectivity models appearing in the applied literature.
5.1. Heckman’s Asymptotic Covariance Matrix Many authors cite the analytic results of Heckman (1979) or Greene (1981), who derive expressions for asymptotic covariance matrices in selectivity models, as the source of their standard errors. We may include in this group the authors who indicate they use software packages LIMDEP or STATA which will compute the Heckman asymptotic covariance matrix. It is our view that authors of empirical papers should clearly state which software is being used, which version, and
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the commands for options actually employed. Not doing so hinders the work of other researchers in the area and obscures calculations that may be incorrect. For example, early versions of STATA 6.0 contained an error when computing the Heckman asymptotic covariance matrix, a reminder that we should all keep our software updated.
5.2. Maximum Likelihood Estimator Less seldom used in applications is the method of maximum likelihood. The most recent versions of LIMDEP and STATA include both two-step estimator and maximum likelihood options. In addition, both packages offer a Huber/White/sandwich estimator for the asymptotic covariance matrix when maximum likelihood estimation is chosen. Examples in the applied literature are Hunter (2000) and Wu and Kwok (2002). Nawata (1994) notes that standard software routines for finding the MLE may not converge, or may converge to a local rather than a global maximum. This may explain why the two-step estimator is widely used instead of maximum likelihood. Nawata (1994) offers an alternative to standard optimization routines for maximizing the log-likelihood function. Nawata and Nagase (1996) compare the finite sample properties of the MLE and Heckit. They conclude, confirming the result in Nawata (1993), that a key indicator to the likely performance of Heckit relative to the MLE is the collinearity between the systematic portion of the selection equation and the regressors in the equation of interest. If the selection (probit) equation and the equation of interest have a substantial number of variables in common then the Heckit estimator is not a good choice relative to the MLE. Nawata and McAleer (2001) compare the t-test, likelihood ratio test and Lagrange multiplier test of the hypothesis that the errors in the selection equation and equation of interest are uncorrelated using maximum likelihood. They find that even if there is no collinearity between the regressors in the two equations the t-test based on maximum estimates performs poorly due to poor variance estimates.
5.3. White’s Heteroskedasticity Consistent Estimator We discovered a large group of authors who rely upon White’s HCE in selectivity models. Puhani (2000, p. 55) states: “In order to obtain a simple and consistent estimator of the asymptotic variance-covariance matrix, Lee (1982, p. 364f.) suggests to use White’s (1980) method.” We are extremely curious about this
Test Statistics and Critical Values in Selectivity Models
85
point. We note first that blind use of the HCE does not account for the fact that the parameters of the first stage probit model are estimated. Second, Lee (1982, p. 365) indicates that he suggests HCE only for part of his two-step estimator’s (which is a generalization of the standard Heckit estimator) variance-covariance matrix. We are unsure whether applied researchers have relied on Lee’s (1982, p. 364) sentence “A relatively simpler approach which avoids the above complication is to adopt the method in White (1980),” assuming it means that a simple heteroskedasticity correction is adequate asymptotically in the two-step estimation process. Amemiya (1984, p. 33) summarizes the argument by noting that the Heckman estimator is consistent under assumptions less restrictive than the usual bivariate normality of the error terms. Under the less restrictive assumptions White’s HCE provides a consistent estimator of the asymptotic covariance matrix. Simpson (1986, p. 801), in the context of a model somewhat more involved than the usual selectivity model, acknowledges that Ordinary least squares estimates . . . will be consistent, but the resulting standard errors will be incorrect and the estimation of correct standard errors will be complex. Therefore, the estimates of the standard errors reported . . . use White’s (1980) procedure to adjust for heteroskedasticity induced by sample selection. This procedure does not correct for the fact that γ is unknown and must be estimated.
Similarly, Ermisch and Wright (1993, p. 123) obtain t-statistics using OLS standard errors, but then say, When the t-statistics are computed from standard errors corrected for heteroskedasticity using White’s (1980) method, conclusions about statistical significance are the same. These t-statistics are not, however, the appropriate ones . . . While they correct for heteroskedasticity, they do not allow for the fact that the λ regressor is estimated. The correct asymptotic standard errors for the dichotomous selection case are derived in Heckman (1979, pp. 158–159).
5.4. Ordinary Least Squares Another strategy is to use least squares regression. The ill-conditioning in the Heckit regression model, which includes the inverse Mills ratio, becomes severe when the variables in the probit selection equation are highly correlated with the variables in the regression equation. The ill-conditioning in the augmented regression occurs in this case because the inverse Mills ratio, despite the fact that it is a non-linear function the selection equation variables, is well approximated by a linear function over broad ranges. In such cases Monte Carlo evidence exists (summarized in Puhani, 2000) supporting the small sample estimation efficiency of OLS relative to Heckit and Heckit alternatives.
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5.5. Recent Developments in Covariance Matrix Estimation In recent years the standard approaches to computing standard errors have been expanded in two directions. First, the use of bootstrapping has become more refined. Originally used as an empirical method for computing standard errors, bootstrapping is now seen as a basis for computing empirically relevant critical values for hypothesis tests (Deis & Hill, 1998; Horowitz, 1997; Horowitz & Savin, 2000). The advantage of this approach is that it leads to the use of a test procedure having Type I error specified by the researcher in finite samples. As Horowitz and Savin (2000) point out, this feature facilitates power comparisons among alternative tests. Newer versions of the software STATA and LIMDEP have automated bootstrap commands. These can be used to obtain bootstrap standard errors for any estimation procedure via resampling the data. With the appropriate modifications the pivotal statistics can be resampled, making bootstrap critical values easy to obtain. Some sample programs are provided in the appendix to this paper showing the commands required. “Sandwich” covariance matrix estimators are being used in a wide variety of contexts. The heteroskedasticity-consistent covariance matrix estimator (HCE) introduced by Huber (1967), Eicker (1967) and White (1980) is ubiquitous. The software STATA offers robust covariance estimation as an option for virtually all models that can be estimated by maximum likelihood (STATA 7.0 User’s Guide, p. 254). These robust covariance estimators are used to obtain “consistent estimates of the covariance matrix under misspecified working covariances as well as under heteroscedastic errors” (Kauermann & Carroll, 2001, p. 1387). Recently Hardin (2002) has proposed a sandwich estimator for two-step models, a class of models that includes Heckit. He develops a sandwich covariance matrix estimator similar in construction to the Murphy-Topel (1985) estimator for two-step estimators, and he provides instructions for using STATA to implement the estimator. LIMDEP 8.0 offers a general procedure for two-step estimators with the Murphy-Topel covariance matrices automated for some special cases. It is our conjecture that bootstrapping a pivotal t-statistic, using a standard error from any consistent asymptotic covariance matrix estimator for the Heckit model (or any other), should yield critical values such that hypothesis tests are the proper size. Then the question is “Do any of the alternative covariance matrix estimators lead to power advantages?” We have not yet investigated this question.
6. THE MONTE CARLO DESIGN The Monte Carlo Design we employ is that of Zuehlke and Zeman (1991) with a modification used by Nawata and Nagase (1996). The performance of each
Test Statistics and Critical Values in Selectivity Models
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covariance estimator is examined under various circumstances likely to affect their performances. The sample size, severity of censoring, degree of selection bias and the correlation between the independent variables in the selection and regression equations are all varied within the Monte Carlo experiment. The specific model we employ consists of the selection equation z ∗i = ␥1 + ␥2 wi + u i = ␥1 + 1wi + u i ,
i = 1, . . . , N
(6.1)
The sample size N = 100 and 400. The value of the parameter ␥1 controls the degree of censoring. Following Zuehlke and Zeman (1991) we specify ␥1 = −0.96, 0 or 0.96, which correspond to expected sub-samples of size n equaling 25, 50 and 75% of N, given that u i ∼ nid(0, 1) The regression equation of interest is y i = 1 + 2 x i + e i = 100 + 1x i + e i ,
i = 1, . . . , n,
N>n
(6.2)
We assume the regression error e i ∼ nid(0, e = 1). The selection bias is controlled through the error correlation = [0, 0.5 or 1] in (2.4). The remaining element of the experiment is the correlation between the regressor w in the selection equation and the regressor x in the regression equation. Following Nawata and Nagase (1996) we specify this correlation to be xw = [0.90, 0.95 or 1]. Varying ␥1 , and xw as describes gives us 27 Monte Carlo design points for each sample size N. For each design point and sample size we generate 500 Monte Carlo samples. We use 200 bootstrap samples. The Monte Carlo results we report are designed to measure (i) The finite sample accuracy of nominal standard errors of the Heckit estimator based on the alternative covariance matrix estimators described in Section 4. The accuracy is evaluated by comparing the average nominal standard errors to the Monte Carlo estimates’ standard deviations. (ii) The size of t-tests of significance based on each standard error method for the slope parameter in the regression equation, based on the usual asymptotic critical values ±1.645 for tests of nominal 10% size and ±1.96 for nominal 5% size tests. That is, we compute the t-statistic tm =
ˆ m −  se m
(6.3)
where ˆ m and  are Heckit estimates for parameters in the augmented regression equation from the Monte Carlo sample and the true parameter value, respectively, and se m is a standard error computed from the Monte Carlo sample. The true null hypothesis is rejected if |t m | ≥ t c , with t c = 1.645 or 1.96. The test size is the percentage of times in 500 Monte Carlo samples
88
R. CARTER HILL, LEE C. ADKINS AND KEITH A. BENDER
we reject the null hypothesis. Clearly we prefer tests of an assumed nominal size of 0.05 or 0.10 to have actual size close to those values. (iii) The size of t-tests of significance based on bootstrapping critical values of the distribution of the “pivotal” statistic associated with each standard error method. Specifically, in each Monte Carlo sample m we resample, with replacement, from the rows of the data matrix [y z x w] to obtain a bootstrap sample b of size N. A pivotal statistic is obtained for each of 200 bootstrap samples by computing tb =
ˆ b − ˆ m se b
(6.4)
where ˆ b and ˆ m are Heckit estimates for parameters in the augmented regression equation from the bootstrap and Monte Carlo samples respectively, and se b is a standard error computed from the bootstrap sample. Recall that for the m’th Monte Carlo sample, the estimate ˆ m based on the full sample is the “true in the sample” value and thus using it to center the t-statistic makes it asymptotically pivotal. The bootstrap critical values are obtained by taking the 0.90 and 0.95 percentiles of empirical distribution of |t b | , for the nominal 10 and 5% size tests, respectively. The size of a two-tailed test using the bootstrap critical values is computed by calculating the percentage of the time the t-statistic in (6.3) falls above or below the computed percentiles. That is, a nominal 5% test is obtained by rejecting the (true) null hypothesis if |t m | ≥ t c , with t c being the bootstrap critical value, which is the 95th percentile of the sorted |t b | values. The actual test size using the bootstrap critical values is the percentage of times in 500 Monte Carlo samples we reject the null hypothesis.
7. THE MONTE CARLO RESULTS When viewing the tabled results recall that the interpretation of the parameters controlling the experimental design: ␥ controls the degree of censoring in the sample. When ␥ = −0.96, 0.0, 0.96 1 1 the sizes of the selected sub-samples of size n are approximately 25, 50 and 75%, respectively, of the original sample of N observations. is the correlation between the errors in the regression and selection equation. It takes the values 0.0 (under which OLS on the sub-sample is BLUE), 0.5 and 1.0. As the value of increases the magnitude of the selection problem, as measured by λ = · e , increases.
Test Statistics and Critical Values in Selectivity Models
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is the correlation between the explanatory variables in the selection equaxw tion and the regression model of interest. The parameter xw takes the values 0.90, 0.95 and 1.0. The latter case represents the undesirable situation in which the explanatory variables in the selection equation are identical to those in the regression equation. The columns labeled OLS are results that use the standard errors from V OLS in (4.1). Columns labeled HCE0 contain results based on the covariance matrix estimator V HC0 , and so on. Table 1a reports for N = 100 the average of the nominal (asymptotic) standard errors relative to the true estimator standard error as measured by the sampling variation of the Heckit estimator of the parameter 2 . The column labeled MCSE is the standard error of the estimates in the Monte Carlo simulation. This is the true finite sample variability. Several cases are of interest: When ␥ = −0.96 and = 1 the Monte Carlo standard errors become signifi1
xw
cantly larger than in the other cases. This is an indication of the general difficulties encountered when the independent variables used in each of the two-steps are the same, especially if the sample for the regression is heavily censored. When = 0, the OLS standard error should accurately reflect the sampling variation of the Heckit estimator since it is BLUE. For the extreme design point ␥1 = −0.96 or 0.0, and xw = 1, the OLS estimator understates the Heckit estimator variability. The other estimators HCE3, HECK (and its near image, MT) and BOOT all overestimate the variability in this case. Bootstrap standard errors measure finite sample variation quite well when = 0 except for the case when xw = 1. Given = 0 and xw = 1, the performance of HCE3, HECK and BOOT improve as the degree of censoring diminishes. Of the two White HCE estimators, the usual HCE0 tends to understate the finite sample variation of Heckit and HCE3 tends to overstate it, but by a relatively small margin. Both estimators improve as censoring is reduced. When the degree of censoring is large the Murphy-Topel and Robust MurphyTopel estimators are not reliable for this sample size. For less censored cases, in this example, the usual MT estimator performs better than the robust version, RMT. Indeed, the MT estimator mirrors the results of HECK even in this small sample. The LEE estimator is appropriate when errors are non-normal, but in this case it severely understates the true sample variation of Heckit when censoring is severe or moderate. In our experiment we did not consider non-normal errors or the robustness of alternative standard error estimators. Comparing HECK and BOOT, we see that HECK does not fare well when censoring is large, but it becomes progressively better as the degree of censoring
␥1
xw
MCSE
OLS
HCE0
HCE3
HECK
MT
RMT
LEE
90
Table 1a. Nominal Standard Errors Relative to the Monte Carlo Standard Errors, N = 100 Coefficient: 2 . BOOT
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.3879 0.5654 1.4435 0.3796 0.7730 1.3504 0.2679 0.4568 1.9669
0.9830 0.9868 0.8550 1.0066 1.0203 0.8554 0.9476 0.9137 0.8628
0.8865 0.8852 0.6240 0.9107 0.8775 0.6831 0.8611 0.8295 0.7620
1.0694 1.0988 1.2476 1.0793 1.1497 1.0929 1.0579 1.0870 1.0860
0.9277 0.9571 1.2703 0.9557 0.9063 1.1962 0.9103 0.9117 1.3601
0.9270 0.9564 1.2555 0.9556 0.9050 1.1827 0.9101 0.9111 1.3675
0.9971 1.0808 1.2485 0.9779 1.0484 1.1726 0.9738 0.9903 1.3373
0.8700 0.8837 0.9763 0.9054 0.8593 0.8989 0.8758 0.8414 0.9200
0.9923 1.0284 1.4686 1.0346 1.0210 1.2761 1.0019 1.0435 1.1929
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.3131 0.3901 0.7672 0.2966 0.2904 0.6089 0.2121 0.2728 0.5708
1.0177 1.0063 0.9378 0.9811 0.9389 0.8743 0.9640 0.9088 0.7867
0.9655 0.9362 0.8461 0.9370 0.7811 0.7671 0.9029 0.9199 0.7927
1.0705 1.0654 1.0414 1.0348 1.1728 1.0205 1.0333 1.0548 0.9392
0.9886 1.0259 1.1127 0.9503 1.0037 1.0587 0.9611 0.9937 0.9993
0.9877 1.0222 1.0966 0.9494 0.9982 1.0387 0.9558 0.9853 0.9769
1.1285 1.2529 1.0906 1.0354 1.3274 1.0435 1.0926 1.2130 0.9738
0.9635 0.9381 0.9610 0.9276 0.8900 0.8958 0.9356 0.9139 0.8800
1.0152 1.0186 1.0871 0.9869 1.1165 1.0490 0.9735 0.9897 0.9681
0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.1996 0.2625 0.3075 0.1996 0.2585 0.2967 0.1888 0.1975 0.2725
1.0309 0.9917 1.0136 0.9957 1.0170 0.9830 1.0125 0.9735 0.9297
0.9808 0.9514 0.9637 0.9610 0.9638 0.9018 0.9813 0.9575 0.8682
1.0662 1.0307 1.0443 1.0288 1.0644 1.0746 1.0606 1.0591 1.0424
1.0213 0.9841 1.0560 0.9909 1.0111 1.0675 1.0228 1.0353 1.0678
1.0173 0.9792 1.0372 0.9881 1.0062 1.0553 1.0165 1.0165 1.0451
1.7107 1.2908 1.0508 1.4615 1.4529 1.1216 1.3934 1.6213 1.1670
0.9769 0.9456 1.0208 0.9650 0.9726 0.9842 0.9865 0.9888 0.9735
1.0060 0.9646 1.0291 0.9834 0.9961 1.0610 1.0029 0.9845 1.0310
R. CARTER HILL, LEE C. ADKINS AND KEITH A. BENDER
−0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96
Test Statistics and Critical Values in Selectivity Models
91
declines. When censoring is severe (␥1 = −0.96) the bootstrap estimator of sampling variation is strongly preferred, except for xw = 1. If censoring is moderate (␥1 = 0.0) BOOT is preferred when = 0. For other cases of moderate censoring HECK and BOOT are similar, except in the unusual case = 0.5 and xw = 9.5. In the case with mild censoring (␥1 = 0.96) HECK and BOOT perform similarly. Table 1b reports for N = 400 the same information as Table 1a. Having more data is better, and all estimators more accurately capture finite sampling variation of the Heckit estimator. Furthermore, the same relative relationships exist between them. When the degree of censoring is large, HECK does not do as well as BOOT or HCE3. The bootstrap standard error BOOT also provides a close measure of finite sample variation in other designs. The exceptions are the designs ␥1 = 0.0, = 0.5 and xw = 9.0 and ␥1 = 0.96, = 0 and xw = 1 In these anomalous cases all standard errors understate the true variation. Tables 2a and 2b report for N = 100 and N = 400, respectively, the ␣ = 0.05 critical values for asymptotic tests. These critical values were computed by sorting the absolute values of the difference between the Monte Carlo Heckit estimates and the true parameter value divided by the relevant asymptotic standard error. Given the evidence in Table 1a it is not surprising that these critical values are not ±1.96, as asymptotic theory would predict. When N = 100 and ␥ = −0.96 (censoring is severe), the empirically deter1 mined critical values are quite different from ±1.96 for all estimators in almost all cases. This is especially true for xw = 1. It is worth noting, however, that even when N = 100, for ␥ = 0.0 or 0.96 (less 1 severe censoring) and = 0.5 or 1.0 (moderate or severe selection bias) with xw = 0.90 or xw = 0.95, then the usual Heckit standard error, HCE3 and BOOT are in a range consistent with the usual 1.96. For N = 100, using the bootstrap standard error, BOOT, produces nominal critical values closer to ±1.96 than HECK when censoring is severe, but the results illustrate that simply using bootstrap standard errors does not solve the problem of testing in small samples. When N = 100 using the bootstrap standard error yields a t-statistic whose critical values are close to ±1.96 even in the severe censoring case. Compare, for example, the critical values based on BOOT with those from HECK (and the others) when censoring is severe and xw = 1. Blindly comparing a Heckit tstatistic to ±1.96 is a less troubling when using the bootstrap standard error than the alternatives. In this context HCE3 also provides a definite improvement over HECK.
␥1
92
Table 1b. Nominal Standard Errors Relative to the Monte Carlo Standard Errors, N = 400 Coefficient: 2 . xw
MCSE
OLS
HCE0
HCE3
HECK
MT
RMT
LEE
BOOT
−0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.2388 0.3108 0.8865 0.2047 0.3032 0.8982 0.1440 0.1933 0.5204
0.9658 1.0394 1.0087 0.9403 0.9442 0.9134 0.8999 1.0029 0.8198
0.9419 0.9988 0.9258 0.9186 0.9164 0.8637 0.9024 0.9542 0.8906
0.9905 1.0544 1.0422 0.9631 0.9666 0.9848 0.9508 1.0033 1.0814
0.9517 1.0250 1.1326 0.9302 0.9307 1.0635 0.9319 0.9718 1.1485
0.9516 1.0248 1.1263 0.9300 0.9302 1.0575 0.9295 0.9693 1.1340
0.9662 1.0603 1.1234 0.9466 0.9552 1.0532 0.9361 0.9833 1.1334
0.9230 0.9913 0.9715 0.9184 0.9197 0.9010 0.8939 0.9513 0.9338
0.9695 1.0274 1.0391 0.9404 0.9478 0.9758 0.9311 0.9786 1.0567
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.1398 0.1853 0.2964 0.1405 0.1795 0.3006 0.1063 0.1412 0.2503
1.0055 1.0034 1.0171 0.9445 1.0264 1.0294 1.0008 0.8934 0.7969
0.9805 0.9890 0.9411 0.9344 1.0090 1.0116 1.0015 0.9371 0.8545
1.0144 1.0190 1.0613 0.9601 1.0423 1.0753 1.0265 0.9723 0.9801
1.0027 1.0005 1.0565 0.9472 1.0298 1.0810 1.0159 0.9739 1.0071
1.0026 1.0000 1.0538 0.9461 1.0284 1.0742 1.0118 0.9634 0.9893
1.0727 1.0517 1.0552 0.9871 1.1029 1.0740 1.0206 1.0001 0.9970
0.9813 0.9860 0.9676 0.9378 1.0018 1.0142 1.0004 0.9400 0.8980
0.9980 1.0030 1.0519 0.9497 1.0260 1.0578 1.0200 0.9625 0.9633
0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.1111 0.1138 0.1769 0.0997 0.1376 0.1643 0.0966 0.1079 0.1409
0.9841 1.0576 0.9365 0.9909 0.9956 1.0011 0.9825 0.8835 0.8950
0.9701 1.0454 0.9182 0.9797 0.9874 0.9894 0.9757 0.9273 0.9288
0.9894 1.0714 0.9487 1.0010 1.0095 1.0332 0.9941 0.9545 0.9600
0.9818 1.0580 0.9533 0.9955 1.0003 1.0421 0.9963 0.9596 0.9861
0.9817 1.0574 0.9502 0.9941 0.9975 1.0354 0.9887 0.9468 0.9672
1.1602 1.2216 0.9554 1.1672 1.1762 1.0474 1.0744 1.0662 0.9778
0.9661 1.0505 0.9369 0.9802 0.9831 1.0014 0.9811 0.9410 0.9635
0.9759 1.0575 0.9431 0.9893 1.0013 1.0290 0.9890 0.9567 0.9733
R. CARTER HILL, LEE C. ADKINS AND KEITH A. BENDER
␥1
xw
OLS
HCE0
HCE3
HECK
MT
RMT
LEE
BOOT
−0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
2.1564 1.9765 2.1854 2.0495 1.9881 2.1838 2.1766 2.1372 2.3388
2.4997 2.4784 3.3944 2.4514 2.5483 2.9830 2.5664 2.5000 2.8207
2.0921 2.0300 2.2879 2.0473 2.0967 2.1977 2.1281 1.9635 2.1199
2.2959 1.9034 1.0708 2.1605 2.4759 1.2258 2.2015 2.0984 1.2288
2.2960 1.9148 1.0896 2.1606 2.4722 1.2676 2.2045 2.0981 1.2198
2.2913 1.8836 1.0890 2.1153 2.4870 1.2799 1.9579 1.9684 1.2635
2.4987 2.4783 2.4424 2.4514 2.5119 2.2716 2.5658 2.4867 2.2081
2.2475 2.1307 1.4372 2.0689 2.1011 1.5520 2.1364 2.0471 1.6392
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
1.9518 2.1152 2.0411 2.0201 2.0782 2.3039 2.0709 2.1359 2.3178
2.0702 2.2941 2.3298 2.1913 2.8374 2.7945 2.3460 2.2299 2.7172
1.8953 2.0492 1.8919 2.0182 2.1033 2.3425 2.0931 1.9406 2.3249
1.9681 1.8647 1.2812 2.0889 1.9094 1.5838 2.0664 1.9634 2.0401
1.9701 1.8697 1.2866 2.0950 1.9186 1.6462 2.0850 1.9823 2.0613
1.8695 1.7915 1.3344 1.9332 1.6178 1.6463 2.0038 1.6567 1.9249
2.0635 2.2503 1.9916 2.1881 2.7738 2.4435 2.3231 2.2175 2.3276
2.0093 2.0836 1.6607 2.1184 1.9887 2.0065 2.1118 2.0637 2.1305
0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
1.9096 1.9942 1.9253 2.1139 1.9368 2.0130 2.0607 1.9905 2.1119
2.0825 2.1298 2.1346 2.2656 1.9385 2.2224 2.1647 2.0239 2.3250
1.9320 1.9958 1.9263 2.1211 1.7726 1.9966 2.0214 1.8937 2.0976
1.8630 1.9569 1.6599 2.1102 1.9358 1.7260 2.0333 1.8877 1.8567
1.8879 1.9955 1.7813 2.1126 1.9411 1.7470 2.0370 1.9386 1.8631
1.6744 1.8357 1.7885 1.7838 1.6993 1.6652 1.5831 1.4887 1.7598
2.0453 2.0626 1.9637 2.2339 1.9153 2.0930 2.1646 1.9670 2.1743
2.0021 2.1470 1.9374 2.2155 1.9506 1.8365 2.0948 1.9886 2.0130
93
Test Statistics and Critical Values in Selectivity Models
Table 2a. Nominal 0.05 Critical Values for Asymptotic Tests, N = 100 Coefficient: 2 .
␥1
94
Table 2b. Nominal 0.05 Critical Values for Asymptotic Tests, N = 400 Coefficient: 2 . xw
OLS
HCE0
HCE3
HECK
MT
RMT
LEE
BOOT
−0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
2.0500 1.8944 1.8752 2.1030 2.1649 2.0592 2.2114 2.0727 2.3443
2.1451 2.0885 2.2606 2.1336 2.1965 2.2756 2.2487 2.1301 2.4222
2.0523 1.9700 2.0678 2.0331 2.0492 2.0304 2.1376 2.0216 2.0593
2.0785 1.9001 1.3247 2.1268 2.1724 1.4440 2.1462 2.1494 1.8965
2.0795 1.8986 1.3388 2.1280 2.1748 1.4656 2.1491 2.1520 1.8980
2.0447 1.8787 1.3454 2.0869 2.1238 1.4657 2.1304 2.1324 1.8969
2.1435 2.0874 2.1320 2.1313 2.1959 2.0921 2.2427 2.1224 2.2716
2.1276 2.0033 1.9484 2.0827 2.1395 1.9184 2.2424 2.1424 1.9422
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
1.9659 1.8788 1.9069 2.0941 1.9059 1.9118 1.9238 2.1775 2.3878
2.0334 1.9846 2.1461 2.1415 1.9633 1.9348 1.8755 2.2601 2.5730
1.9766 1.9231 2.0072 2.0938 1.9035 1.8304 1.8320 2.1856 2.2710
1.9674 1.8731 1.6711 2.0622 1.9073 1.7409 1.9160 2.0703 1.9941
1.9665 1.8791 1.6868 2.0679 1.9237 1.7682 1.9227 2.0854 2.0103
1.9243 1.8339 1.6884 1.9699 1.7781 1.7681 1.8550 1.9107 1.9815
2.0330 1.9735 2.1198 2.1296 1.9625 1.9340 1.8742 2.2142 2.4375
1.9672 1.9741 1.9156 2.1316 1.9302 1.8751 1.8836 2.1287 2.1631
0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
1.9685 1.9170 2.1018 1.9689 1.9287 1.9138 1.9865 2.1878 2.1758
2.0711 1.8881 2.2210 2.0432 2.0100 2.0252 1.8974 2.0817 2.2229
2.0390 1.8575 2.1754 2.0042 1.9722 1.9444 1.8620 2.0182 2.1715
1.9733 1.8954 1.9464 1.9731 1.9223 1.7525 1.9589 2.0688 2.0610
1.9735 1.9034 1.9670 1.9788 1.9223 1.7698 1.9754 2.0788 2.0626
1.8247 1.6659 1.9777 1.7784 1.8189 1.7754 1.8189 1.8654 2.0304
2.0570 1.8753 2.1772 2.0251 2.0015 1.9883 1.8956 2.0299 2.1527
2.0502 1.8722 2.0706 2.0590 2.0391 1.9547 1.9626 2.0396 2.1981
R. CARTER HILL, LEE C. ADKINS AND KEITH A. BENDER
Test Statistics and Critical Values in Selectivity Models
95
When ␥ = 0.0 or 0.96 (less severe censoring) and = 1, BOOT, HCE3 and 1 xw other estimators exhibit critical values that are larger than 1.96. In Tables 3a and 3b , we report for N = 100 and N = 400, respectively, the sizes of the nominal ␣ = 0.05 asymptotic tests. These sizes are computed by calculating the percentage of true null hypotheses rejected in the Monte Carlo experiment, when the t-statistics based on alternative standard error estimators are compared to ±1.96. When N = 100, HECK does not produce tests of predicted size when censoring is severe (␥1 = −0.96) or if xw = 1. Using the bootstrap standard error in these cases is certainly better, except when xw = 1. In other cases using the bootstrap standard error is a good choice; comparing the resulting t-statistic to ±1.96 leads to tests of approximately the correct size. The exception to this rule is when censoring is moderate (␥1 = 0.0), = 0 (so that OLS is BLUE) and xw = 1. When N = 100, HCE3 results in less size distortion than HCE0, which rejects too frequently, a finding consistent with its “too small” standard errors noted in Table 1a. If the usual critical values ±1.96 are employed, then using standard error HCE3 by far the best alternative in our simulation when censoring is severe and xw = 1. In fact using HCE3 seems a good choice overall, with its only real problem occurring when censoring is moderate and the error correlation is = 0.5 and xw = 1. When N = 400 using t-statistics based on the bootstrap standard error yield sizes closer to the nominal ␣ = 0.05 value than using HECK, especially for xw = 1. MT and HECK perform similarly. RMT under-rejects in all cases when censoring is not severe. HCE3 is again preferable to HCE0 over all degrees of censoring and in virtually all cases. Tables 4a and 4b contain the sizes of the nominal ␣ = 0.05 tests based on bootstrap critical values using the pivotal statistic in Eq. (6.4). The usefulness of bootstrapping critical values is immediately obvious. When N = 100, using bootstrap critical values for the t-statistic based on HECK provides tests of close to proper size in all cases other than xw = 1. The MurphyTopel (MT) results are very similar to HECK and the robust version RMT is close, though with a few more extreme values. HCE3 seems to perform relatively well with the bootstrap critical values, even when xw = 1 and censoring is severe (␥1 = −0.96) or not (␥1 = 0.96). If N = 400, using the bootstrap critical values with HCE3 again seems a good alternative across all designs. For smaller amounts of censoring (␥1 = 0.96)
␥1
96
Table 3a. Sizes of the Nominal 0.05 Asymptotic Tests, N = 100 Coefficient: 2 . xw
OLS
HCE0
HCE3
HECK
MT
RMT
LEE
BOOT
−0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.0780 0.0540 0.0720 0.0680 0.0560 0.0740 0.0660 0.0760 0.0940
0.1180 0.0940 0.1940 0.1020 0.1000 0.1500 0.1000 0.1300 0.1640
0.0580 0.0600 0.0740 0.0580 0.0500 0.0700 0.0680 0.0500 0.0620
0.0940 0.0380 0.0000 0.0760 0.1040 0.0020 0.0680 0.0680 0.0000
0.0940 0.0380 0.0000 0.0760 0.1040 0.0020 0.0680 0.0680 0.0020
0.0900 0.0380 0.0000 0.0700 0.1040 0.0020 0.0480 0.0500 0.0020
0.1180 0.0940 0.1360 0.1020 0.0980 0.0900 0.1000 0.1300 0.0940
0.0920 0.0680 0.0040 0.0640 0.0640 0.0180 0.0660 0.0540 0.0200
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.0480 0.0660 0.0540 0.0580 0.0680 0.0780 0.0700 0.0720 0.1140
0.0660 0.0880 0.0900 0.0740 0.1620 0.1580 0.1060 0.0820 0.1360
0.0460 0.0580 0.0440 0.0500 0.0700 0.1000 0.0620 0.0420 0.0880
0.0500 0.0400 0.0000 0.0600 0.0400 0.0100 0.0720 0.0500 0.0600
0.0500 0.0420 0.0000 0.0600 0.0460 0.0120 0.0720 0.0540 0.0620
0.0420 0.0320 0.0000 0.0480 0.0200 0.0120 0.0580 0.0280 0.0440
0.0660 0.0820 0.0520 0.0740 0.1540 0.1160 0.1000 0.0780 0.1020
0.0600 0.0620 0.0100 0.0660 0.0500 0.0520 0.0760 0.0540 0.0620
0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.0420 0.0520 0.0420 0.0680 0.0480 0.0540 0.0620 0.0560 0.0680
0.0640 0.0760 0.0660 0.0860 0.0460 0.0920 0.0680 0.0640 0.1140
0.0440 0.0500 0.0460 0.0680 0.0380 0.0520 0.0540 0.0420 0.0620
0.0440 0.0460 0.0140 0.0640 0.0440 0.0140 0.0640 0.0460 0.0360
0.0440 0.0540 0.0180 0.0640 0.0460 0.0160 0.0640 0.0460 0.0360
0.0240 0.0380 0.0260 0.0340 0.0300 0.0080 0.0180 0.0060 0.0200
0.0580 0.0620 0.0500 0.0820 0.0420 0.0700 0.0660 0.0520 0.0860
0.0640 0.0700 0.0460 0.0800 0.0480 0.0360 0.0600 0.0520 0.0500
R. CARTER HILL, LEE C. ADKINS AND KEITH A. BENDER
␥1
xw
OLS
HCE0
HCE3
HECK
MT
RMT
LEE
BOOT
−0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.0580 0.0460 0.0460 0.0740 0.0620 0.0560 0.0780 0.0600 0.1140
0.0740 0.0680 0.0880 0.0760 0.0780 0.0860 0.0880 0.0700 0.1020
0.0560 0.0500 0.0580 0.0540 0.0600 0.0520 0.0780 0.0540 0.0600
0.0620 0.0460 0.0000 0.0740 0.0680 0.0000 0.0660 0.0620 0.0400
0.0620 0.0460 0.0000 0.0740 0.0680 0.0020 0.0660 0.0640 0.0400
0.0580 0.0420 0.0000 0.0680 0.0680 0.0020 0.0600 0.0640 0.0380
0.0740 0.0680 0.0720 0.0760 0.0780 0.0680 0.0880 0.0700 0.0800
0.0700 0.0520 0.0480 0.0680 0.0720 0.0460 0.0800 0.0620 0.0460
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.0500 0.0420 0.0420 0.0640 0.0480 0.0400 0.0420 0.0800 0.1140
0.0620 0.0540 0.0660 0.0720 0.0500 0.0480 0.0400 0.0780 0.1260
0.0540 0.0420 0.0500 0.0680 0.0460 0.0400 0.0360 0.0700 0.0980
0.0520 0.0380 0.0120 0.0600 0.0460 0.0160 0.0360 0.0540 0.0540
0.0520 0.0380 0.0160 0.0620 0.0460 0.0200 0.0380 0.0560 0.0560
0.0420 0.0360 0.0180 0.0520 0.0320 0.0200 0.0300 0.0440 0.0520
0.0620 0.0520 0.0640 0.0720 0.0500 0.0460 0.0400 0.0720 0.1080
0.0500 0.0500 0.0460 0.0760 0.0480 0.0340 0.0420 0.0640 0.0800
0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.0520 0.0440 0.0600 0.0500 0.0460 0.0400 0.0540 0.0920 0.0820
0.0640 0.0460 0.0720 0.0580 0.0540 0.0620 0.0480 0.0800 0.0800
0.0580 0.0420 0.0640 0.0540 0.0520 0.0480 0.0480 0.0640 0.0720
0.0520 0.0440 0.0480 0.0500 0.0420 0.0260 0.0480 0.0700 0.0560
0.0520 0.0440 0.0500 0.0500 0.0440 0.0260 0.0500 0.0740 0.0600
0.0320 0.0260 0.0500 0.0240 0.0380 0.0240 0.0400 0.0300 0.0560
0.0640 0.0460 0.0620 0.0560 0.0540 0.0520 0.0480 0.0620 0.0680
0.0660 0.0380 0.0580 0.0600 0.0540 0.0460 0.0500 0.0640 0.0700
97
Test Statistics and Critical Values in Selectivity Models
Table 3b. Sizes of the Nominal 0.05 Asymptotic Tests, N = 400 Coefficient: 2 .
␥1
98
Table 4a. Sizes of the Nominal 0.05 Bootstrap Tests, N = 100 Coefficient: 2 . xw
OLS
HCE0
HCE3
HECK
MT
RMT
LEE
−0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.0740 0.0520 0.0400 0.0540 0.0440 0.0380 0.0540 0.0380 0.0320
0.0620 0.0400 0.0600 0.0560 0.0560 0.0540 0.0620 0.0380 0.0400
0.0620 0.0460 0.0420 0.0580 0.0580 0.0440 0.0640 0.0520 0.0400
0.0720 0.0480 0.0100 0.0520 0.0640 0.0220 0.0620 0.0640 0.0200
0.0720 0.0480 0.0120 0.0520 0.0640 0.0260 0.0620 0.0620 0.0220
0.0860 0.0780 0.0120 0.0560 0.1000 0.0380 0.0580 0.0680 0.0260
0.0660 0.0480 0.0680 0.0560 0.0560 0.0460 0.0620 0.0400 0.0460
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.0540 0.0600 0.0300 0.0600 0.0860 0.0700 0.0680 0.0560 0.0620
0.0420 0.0460 0.0220 0.0480 0.1060 0.0780 0.0680 0.0440 0.0560
0.0420 0.0440 0.0200 0.0500 0.0800 0.0720 0.0720 0.0440 0.0560
0.0500 0.0640 0.0020 0.0540 0.0860 0.0500 0.0680 0.0540 0.1060
0.0480 0.0640 0.0040 0.0520 0.0860 0.0540 0.0660 0.0540 0.0980
0.0520 0.0900 0.0060 0.0560 0.0800 0.0560 0.0800 0.0560 0.1020
0.0440 0.0500 0.0180 0.0460 0.1180 0.0820 0.0700 0.0480 0.0700
0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.0560 0.0660 0.0340 0.0700 0.0400 0.0520 0.0520 0.0380 0.0660
0.0460 0.0600 0.0340 0.0620 0.0360 0.0440 0.0500 0.0360 0.0560
0.0480 0.0620 0.0320 0.0620 0.0340 0.0460 0.0480 0.0380 0.0560
0.0560 0.0640 0.0220 0.0680 0.0480 0.0580 0.0560 0.0420 0.0820
0.0520 0.0680 0.0280 0.0680 0.0480 0.0620 0.0560 0.0420 0.0800
0.0740 0.0820 0.0520 0.0820 0.0560 0.0860 0.0480 0.0580 0.0760
0.0500 0.0580 0.0240 0.0640 0.0400 0.0600 0.0500 0.0380 0.0760
R. CARTER HILL, LEE C. ADKINS AND KEITH A. BENDER
xw
OLS
HCE0
HCE3
HECK
MT
RMT
LEE
−0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96 −0.96
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.0660 0.0480 0.0560 0.0700 0.0760 0.0560 0.0820 0.0660 0.0640
0.0660 0.0500 0.0480 0.0560 0.0720 0.0500 0.0640 0.0600 0.0560
0.0640 0.0500 0.0420 0.0560 0.0680 0.0460 0.0640 0.0600 0.0520
0.0640 0.0460 0.0280 0.0700 0.0780 0.0340 0.0760 0.0640 0.1260
0.0640 0.0460 0.0280 0.0700 0.0760 0.0340 0.0760 0.0660 0.1200
0.0680 0.0500 0.0280 0.0640 0.0760 0.0340 0.0840 0.0640 0.1140
0.0660 0.0500 0.0620 0.0560 0.0720 0.0580 0.0620 0.0600 0.0760
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0
0.90 0.95 1.00 0.90 0.95 1.00 0.90 0.95 1.00
0.0480 0.0400 0.0560 0.0740 0.0480 0.0380 0.0340 0.0620 0.0960
0.0500 0.0320 0.0520 0.0760 0.0420 0.0300 0.0260 0.0580 0.0840
0.0520 0.0320 0.0520 0.0760 0.0420 0.0260 0.0260 0.0580 0.0840
0.0540 0.0480 0.0540 0.0740 0.0500 0.0620 0.0380 0.0680 0.0900
0.0540 0.0460 0.0540 0.0740 0.0500 0.0640 0.0380 0.0660 0.0880
0.0620 0.0520 0.0560 0.0680 0.0540 0.0620 0.0440 0.0740 0.0920
0.0500 0.0320 0.0560 0.0760 0.0420 0.0340 0.0260 0.0580 0.0920
0.96 0.96 0.96
0.0 0.0 0.0
0.90 0.95 1.00
0.0620 0.0380 0.0600
0.0600 0.0360 0.0520
0.0600 0.0360 0.0540
0.0620 0.0420 0.0700
0.0620 0.0400 0.0660
0.0740 0.0420 0.0680
0.0580 0.0360 0.0580
␥1
Test Statistics and Critical Values in Selectivity Models
Table 4b. Sizes of the Nominal 0.05 Bootstrap Tests, N = 400 Coefficient: 2 .
99
100
R. CARTER HILL, LEE C. ADKINS AND KEITH A. BENDER
there is very little difference across the empirical sizes no matter which standard error is used. If N = 400, using HECK with bootstrap critical values is a good choice when censoring is not severe.
8. SUMMARY AND CONCLUSIONS Our research goals for this work were: First, how do the alternative versions of asymptotic variance-covariance matrices used in selectivity models capture the finite sample variability of the Heckit twostep estimator? The answer depends on the degree of censoring and on whether the explanatory variables in the selection and regression equation differ or not. With severe censoring and if the explanatory variables in the two equations are identical, then none of the asymptotic standard error formulations is reliable in small samples. In larger samples the bootstrap does a good job in reflecting estimator variability, as does the White HCE in which the diagonal matrix of squared residuals is weighted, element-wise, by the squared reciprocals of the diagonal of an identity less the “hat” matrix, which we have called V HC3 , following Davidson and MacKinnon (1993). These are both better choices than the usual Heckit asymptotic covariance matrix in the sense of coming closer to the finite sample variation or being slightly conservative. If selection and regression equations have different variables, and if the sample is small, the bootstrap does relatively well if censoring is strong or moderate. Second, is it possible to use bootstrapping to improve finite sample inference? Three aspects of this question are: ◦ Do bootstrap standard errors match finite sample variability better than nominal standard errors computed from asymptotic covariance matrices? As noted in the previous summary point, the answer to this is yes, unless censoring is severe, and the explanatory variables in the selection and regression equations are the same. ◦ Do critical values of test statistics generated by bootstrapping pivotal t-statistics lead to better test size than those based on usual asymptotic theory? The answer is yes, and this is the most interesting and useful finding in the current research. ◦ If censoring is severe, or when the sample is small, we recommend using HCE3 with bootstrapped critical values when carrying out tests about the slope parameters, or when constructing interval estimates. ◦ If the sample is large and censoring is moderate or less severe, then using the standard Heckit standard errors with bootstrapped critical values is satisfactory.
Test Statistics and Critical Values in Selectivity Models
101
◦ Does modern software make obtaining bootstrap standard errors and critical values feasible for empirical researchers? Yes. Sample programs in the appendix show that it is simple to obtain bootstrap standard errors, and almost as simple to obtain bootstrap critical values with Heckit or HCE3 standard errors.
ACKNOWLEDGMENTS The authors would like to thank Bill Greene for responding to our queries about LIMDEP and making an adjustment in the program. The authors benefited from discussion by participants at the LSU Advances in Econometrics Conference, November 2002, Tom Fomby and Viera Chmelarova. Any errors are ours.
REFERENCES Amemiya, T. (1984). Tobit models: A survey. Journal of Econometric, 24, 3–61. Davidson, R., & MacKinnon, J. (1993). Estimation and inference in econometrics. New York: Oxford University Press. Deis, D., & Hill, R. C. (1998). An application of the bootstrap method to the simultaneous equations model of the demand and supply of audit services. Contemporary Accounting Research, 15, 83–99. Eicker, F. (1963). Asymptotic normality and consistency of the least squares estimators for families of linear regressions. Annals of Mathematical Statistics, 34, 447–456. Ermisch, J. F., & Wright, R. E. (1993). Wage offers and full-time and part-time employment by British women. The Journal of Human Resources, 28, 111–133. Freedman, D., & Peters, S. (1984a). Bootstrapping and econometric model: Some empirical results. Journal of Business & Economic Statistics, 2, 150–158. Freedman, D., & Peters, S. (1984b). Bootstrapping a regression equation: Some empirical results. Journal of the American Statistical Association, 79, 97–106. Greene, W. H. (1981). Sample selection bias as a specification error: Comment. Econometrica, 49, 795–798. Greene, W. H. (1997). Econometric analysis (3rd ed.). Upper Saddle River, NJ: Prentice-Hall. Griffiths, W. E., Hill, R. C., & Pope, P. (1987). Small sample properties of probit estimators. Journal of the American Statistical Association, 73, 191–193. Hardin, J. W. (2002). The robust variance estimator for two-stage models. The Stata Journal, 2, 253–266. Hardin, J. W. (2003). The Sandwich estimate of variance. In: T. B. Fomby & R. C. Hill (Eds), Advances in Econometrics, 17, Maximum Likelihood Estimation of Misspecified Models: Twenty Years Later (pp. 45–73). Elsevier. Heckman, J. J. (1979). Sample Selection Bias as a Specification Error. Econometrica, 47, 153–161. Horowitz, J. L. (1997). Bootstrap methods in econometrics: Theory and numerical performance. In: D. M. Kreps & K. F. Wallis (Eds), Advances in Economics and Econometrics: Theory and Applications, Seventh World Congress (Vol. 3, pp. 188–222). Cambridge: Cambridge University Press.
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Horowitz, J. L., & Savin, N. E. (2000). Empirically relevant critical values for hypothesis tests: A bootstrap approach. Journal of Econometrics, 95, 375–389. Huber, P. J. (1967). The behavior of maximum likelihood estimation under non-standard conditions. In: L. M. LeCam & J. Neyman (Eds), Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (Vol. 1, pp. 221–233). Berkeley: University of California Press. Hunter, L. W. (2000). What determines job quality in nursing homes? Industrial and Labor Relations Review, 53, 463–481. Jeong, J., & Maddala, G. S. (1993). A perspective on application of bootstrap methods in econometrics. In: G. S. Maddala, C. R. Rao & H. D. Vinod (Eds), Handbook of Statistics (Vol. 11, pp. 573–610). New York: North-Holland. Kauermann, G., & Carroll, R. J. (2001). A note on the efficiency of Sandwich covariance matrix estimation. Journal of the American Statistical Association, 96, 1387–1396. Lee, L.-F. (1982). Some approaches to the correction of selectivity bias. Review of Economic Studies, 49, 355–372. Long, J. S., & Ervin, L. H. (2000). Correcting for heteroscedasticity with heteroscedasticity consistent standard errors in the linear regression model: Small sample considerations. American Statistician, 54, 217–224. Mroz, T. A. (1987). The sensitivity of an empirical model of married women’s hours of work to economic and statistical assumptions. Econometrica, 55, 765–799. Murphy, K. M., & Topel, R. H. (1985). Estimation and inference in two-step econometric models. Journal of Business & Economic Statistics, 34, 370–379. Nawata, K. (1993). A note on the estimation of models with sample selection biases. Economics Letters, 42, 15–24. Nawata, K. (1994). Estimation of sample selection bias models by the maximum likelihood estimator and Heckman’s two-step estimator. Economics Letters, 45, 33–40. Nawata, K., & McAleer, M. (2001). Size characteristics of tests for sample selection bias: A Monte Carlo comparison and empirical example. Econometric Reviews, 20, 105–112. Nawata, K., & Nagase, N. (1996). Estimation of sample selection bias models. Econometric Reviews, 15, 387–400. Puhani, P. A. (2000). The Heckman correction for sample selection and its critique. Journal of Economic Surveys, 14, 53–68. Simpson, W. (1986). Analysis of part-time pay in Canada. Canadian Journal of Economics, 19, 798– 807. Vella, F. (1998). Estimating models with sample selection bias: A survey. Journal of Human Resources, 33, 127–172. White, H. J. (1980). A Heteroskedasticity-consistent covariance matrix estimator and a direct test for Heteroskedasticity. Econometrica, 48, 817–838. Wu, C., & Kwok, C. C. Y. (2002). Why do U.S. firms choose global equity offerings? Finanacial Management, Summer, 47–65. Zuehlke, T. W., & Zeman, A. R. (1991). A comparison of two-stage estimators of censored regression models. Review of Economics and Statistics, 73, 185–188.
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APPENDIX A In this appendix we provide bare bones code in Stata 8.0 and LIMDEP 8.0 to carry out bootstrapping in the Heckit model. The data, code files and output are available at http://www.bus.lsu.edu/economics/faculty/chill/personal/Heckit.htm
The Example The data are from Thomas Mroz’s widely cited example (1987, “The Sensitivity of an Empirical Model of Married Women’s Hours of Work to Economic and Statistical Assumptions,” Econometrica, 55, 765–799) and consist of 753 observations from the 1987 Panel Study of Income Dynamics. The linear model of interest has the logarithm of wage dependent on education, experience and age: ln(wage) = 1 + 2 ed + 3 ex + 4 ex2 + 5 age + e
(A.1)
However, many in the sample are observed with zero wages, suggesting that there may be sample selection bias; omitted factors in the wage equation may be correlated with a decision to work. The selection equation considered is: z ∗ = ␥1 + ␥2 ed + ␥3 ex + ␥4 ex2 + ␥5 age + ␥6 klt6 + v
(A.2)
where klt6 is the number of children less than 6 years old present. The latent variable, z∗ , is only observed in one of two states; either one works for wages (z = 1) or does not (z = 0). The correlation between v and e is .
APPENDIX B STATA 8.0: THE .DO FILE clear #delimit; use c:\stata\mroz; heckman lwage educ exper expersq age, select(educ exper expersq age kidslt6) twostep; scalar b1= b[exper]; scalar se1= se[exper];
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di “Coefficient b1” b1 “Standard Error” se1; set seed 1431241; bootstrap “heckman lwage educ exper expersq age, select (educ exper expersq age kidslt6) twostep” “ b[exper] se[exper] e(lambda) e(rho),” reps(400) level(95) saving(bsstboot) replace; drop all; use bsstboot; gen ttest1 = ( bs 1 − b1)/ bs 2; gen abt=abs(ttest1); pctile abt, p(95); di “The 97.5% critical value is” r(r1); summarize ttest1 abt, detail; histogram ttest1, bin(20) normal; program notes: If the estimated ρ is outside [−1,1], Stata truncates the estimate and makes the error variance computation consistent. To obtain results comparable to LIMDEP, use the rhotrunc option. We are not advocating one approach or the other.
APPENDIX C LIMDEP 8.0 COMMAND FILE reset read;file=c:\mroz.raw;nobs=753;nvar=22;names=inlf,hours,kidslt6,kidsge6, age,educ,wage,repwage,hushrs,husage,huseduc,huswage,faminc,mtr,motheduc, fatheduc,unem,city,exper,nwifeinc,lwage,expersq$ ? M defines coefficient of interest. NSAM is number of bootstraps ? In this example different variables are in selection and regression equations namelist; w=one,educ,exper,expersq,age,kidslt6 $ namelist; x=one,educ,exper,expersq,age $ calc; m=3; nsam=400; k = col(x)+1$ matrix; boott = init(nsam,k,0); bootest = init(nsam,k,0) $
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probit; lhs=inlf; rhs=w; hold $ select; lhs=lwage; rhs=x$ matrix; truebeta = b $ calc; iter=0 $ proc calc; iter=iter+1 $ draw;n=753;replace$ probit; lhs=inlf; rhs=w; hold $ select; lhs=lwage; rhs=x$ ? tvec = absolute values of t-stats centered at sample estimates matrix; bhat = b; se = vecd(varb); sevec = esqr(se); rse = diri(sevec); dvec = bhat-truebeta; tvec = dirp(dvec,rse); tvec = dirp(tvec,tvec); tvec = esqr(tvec); boott(iter,∗) = tvec’; bootest(iter,∗) = bhat’ $ endproc exec; n=nsam; silent $ matrix; tm=part(boott,1,nsam,m,m); bm=part(bootest,1,nsam,m,m) $ sample; 1-400 $ create; boot bm=bm$ dstat; rhs=boot bm$ ? To sort t-statistics must create variable create; boot tm=tm$ sort; lhs = boot tm $ calc; nval = .95∗ nsam+1 $ matrix; list; tc = part(boot tm,nval,nval,1,1) $ stop $ program notes: If the estimated ρ is outside [−1,1], LIMDEP truncates the estimate. Reminder–in Project Settings specify enough memory for sample size of 753 observations.
ESTIMATION, INFERENCE, AND SPECIFICATION TESTING FOR POSSIBLY MISSPECIFIED QUANTILE REGRESSION Tae-Hwan Kim and Halbert White ABSTRACT To date, the literature on quantile regression and least absolute deviation regression has assumed either explicitly or implicitly that the conditional quantile regression model is correctly specified. When the model is misspecified, confidence intervals and hypothesis tests based on the conventional covariance matrix are invalid. Although misspecification is a generic phenomenon and correct specification is rare in reality, there has to date been no theory proposed for inference when a conditional quantile model may be misspecified. In this paper, we allow for possible misspecification of a linear conditional quantile regression model. We obtain consistency of the quantile estimator for certain “pseudo-true” parameter values and asymptotic normality of the quantile estimator when the model is misspecified. In this case, the asymptotic covariance matrix has a novel form, not seen in earlier work, and we provide a consistent estimator of the asymptotic covariance matrix. We also propose a quick and simple test for conditional quantile misspecification based on the quantile residuals.
Maximum Likelihood Estimation of Misspecified Models: Twenty Years Later Advances in Econometrics, Volume 17, 107–132 Copyright © 2003 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(03)17005-3
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1. INTRODUCTION Since the seminal work of Koenker and Bassett (1978) and Bassett and Koenker (1978), the literature on quantile regression and least absolute deviation (LAD) regression has grown rapidly in many interesting directions, such as simultaneous equation and two stage estimation (Amemiya, 1982; Powell, 1983), censored regression (Buchinsky & Hahn, 1998; Powell, 1984, 1986), serial correlation and GLS estimation (Weiss, 1990), bootstrap methods (Hahn, 1995; Horowitz, 1998), structural break testing (Bai, 1995), ARCH models (Koenker & Zhao, 1996), and unit root testing (Herce, 1996).1 All these papers, however, assume explicitly or implicitly that the conditional quantile regression model is correctly specified. When the model is misspecified, confidence intervals and hypothesis tests based on the conventional covariance matrix are, as we show, invalid. Even though misspecification is a generic phenomenon and correct specification is rare in reality, there has to date been no theory proposed for inference when a conditional quantile model may be misspecified. In this paper, we allow for possible misspecification of a linear conditional quantile regression model. We obtain consistency of the quantile estimator for certain “pseudo-true” parameter values and asymptotic normality of the quantile estimator when the model is misspecified. In this case, the asymptotic covariance matrix has a novel form, not seen in earlier work, and we provide a consistent estimator of the asymptotic covariance matrix. Of course, one can estimate the conditional quantile model without assuming correct specification using various non-parametric methods such as kernel estimation (Sheather & Marron, 1990), nearest-neighbor estimation (Bhattacharya & Gangopadhyay, 1990), or using artificial neural networks (White, 1992). Our results thus provide a convenient parametric alternative to non-parametric methods when researchers are not sure about correct specification or when they want to keep a parametric model for reasons of parsimony or interpretability even though it may not pass a specification test such as the non-parametric kernel based test proposed by Zheng (1998).
2. BASIC ASSUMPTIONS AND MODEL Consider a random series (Y t , X t ) where t = 1, 2, . . . , T, Y t is a scalar, Xt is a k × 1 vector, and k ≡ k˜ + 1. The first element in Xt is one for all t. First, we specify the data generating process. Assumption 1. The sequence {Y t , X t } is independent and identically distributed (iid).
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The iid assumption is made for clarity and simplicity. It can be straightforwardly relaxed. We denote the conditional distribution of Yt given X t = x by F Y|X (·|x). As is now standard in the quantile regression literature, we define the “check” function : R → R + for given ∈ (0, 1) as (z) ≡ z (z), where (z) ≡ − 1[z≤0] . We define the th conditional quantile of Yt given Xt as q (Y t |X t ) ≡ inf{y : F Y|X (y|X t ) ≥ }. Next, we impose the following assumption on the joint density of (Y t , X t ). Assumption 2. The random vector (Y t , X t2 , X t3 , . . . , X tk ) is continuously distributed with joint probability density f Y,X (·, ·) and conditional probability density f Y|X (·|x) for Yt given X t = x. Under Assumption 2, the conditional quantile q (Y t |X t ) satisfies f Y|X (y|X t )dy − = 0, which is equivalent to
q (Yt |Xt ) −∞
E[ (Y t − q (Y t |X t ))|X t ] = 0.
(2.1)
The condition (2.1) can be used to check if a given function of Xt is the th conditional quantile of Yt given Xt or not. It is well known that q (Y t |X t ) ∈ arg min E[ (Y t − f(X t ))], f
(2.2)
where f belongs to a space of measurable functions defined as F ≡ {g : R k → R such that g is measurable and E|g(X t )| < ∞}. Here, we focus only on the affine space A(⊂ F) ≡ {g : R k → R such that for some k × 1 vector , g(X t ) = X t  and E|g(X t )| < ∞}. For the objective function in (2.2) to be well-defined, we impose some moment conditions on (Y t , X t ). The following conditions are sufficient. Assumption 3. E|Y t | < ∞, and E||X t || < ∞, where ||X t || ≡ (X t X t )1/2 . Now we give a definition of correct model specification. Definition 1. We say a conditional quantile model {h(·, ) ∈ F,  ∈ R k } is correctly specified for q (Y t |X t ) if and only if there exists a vector 0 ∈ R k such that h(X t , 0 ) = q (Y t |X t ) almost surely, i.e. q θ ∈ F. We impose the following quantile version of the orthogonality condition. Assumption 4. There exists ∗ such that E(X t (Y t − X t ∗ )) = 0.
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Given the “pseudo-true parameters” ∗ of Assumption 4, we can define the “error” t ≡ Y t − X t ∗ . Assuming that E( (t )|X t ) = 0, which is stronger than Assumption 4, is equivalent to assuming that the conditional quantile model is correctly specified. This can be easily checked using (2.1). Thus, Assumption 4 permits the conditional quantile model to be misspecified. Let B denote a subset in Rk large enough to contain ∗ . Then, under our conditions Assumption 4 implicitly defines the parameter of interest ∗ as the solution to the minimization problem: min E[ (Y t − X t )] ∈B
which is well-defined by Assumption 3. As discussed by White (1994, pp. 74–75), this optimization problem corresponds to maximizing the expected log-likelihood for a particular density function. The regression quantile parameter estimator ˆ T , obtained by minimizing the sample analog min S T () ≡ T ∈B
−1
T
(Y t − X t ),
(2.3)
t=1
can therefore be viewed as a quasi-maximum likelihood estimator (QMLE).
3. CONSISTENCY AND ASYMPTOTIC NORMALITY First, we establish the consistency of the quantile estimator ˆ T for ∗ using Lemmas 2.2 and 2.3 in White (1980a). The consistency result is the first step in deriving the asymptotic normality of the quantile estimator. The following additional assumptions suffice for the proof of consistency. Assumption 3 . There exists ␦ > 1 such that E(||X t ||␦ ) < ∞. Assumption 5. ∗ ∈ B, where B is a compact subset of Rk . Assumption 6. For all x, f |X (0|x) > 0 where f |X (·|x) is the conditional density of t ≡ Y t − X t ∗ given X t = x. Lemma 1. Suppose that Assumptions 1, 2, 3 , 4, 5, and 6 hold. Then ˆ T − ∗ = o p (1). All proofs are provided in the Mathematical Appendix. There are several techniques available in the literature to derive asymptotic normality for the quantile and the LAD estimators. Among these are the linear programming method (Bassett & Koenker, 1978; Koenker & Bassett, 1978); the
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smoothing method (Bloomfield & Steiger, 1983); the convexity method (Pollard, 1991); and the generalized Taylor expansion method (Phillips, 1991). Here, we follow the method used by Huber (1967), Ruppert and Carroll (1980), and later extended by Pollard (1985). Huber (1967) gave sufficient conditions that deliver asymptotic normality for any sequence ˆ T satisfying T −1/2
T
(Z t , ˆ T ) = o p (1).
(3.1)
t=1
In our case, (Z t ,ˆ T ) = X t (Y t − X t ˆ T ). The condition (3.1) can be viewed as the first order condition for the quantile estimator ˆ T because the left hand term is essentially the vector of left partial derivatives of the objective function in (2.3) evaluated at ˆ T . Because we define the quantile estimator ˆ T using (2.3) rather than (3.1), we must establish (3.1). For this, we use the following assumption. Assumption 3 . There exists ␦ > 2 such that E(||X t ||␦ ) < ∞. Lemma 2. Suppose that Assumptions 2 and 3 hold. Then T −1/2 (Y t − X t ˆ T ) = o p (1).
T
t=1 X t
Next, we define () ≡ E[X t (Y t − X t )]. Since () can be shown to be continuously differentiable in , we have by the mean value theorem () = (∗ ) − Q ∗ ( − ∗ ),
(3.2)
≡ E[f |X (∗t |X t )X t X t ] and ∗t is between 0 and X t ( − ∗ ). Note that 0 by the definition of ∗ . We use the expression for () in (3.2) to obtain
Q∗
where (∗ ) = a variant of the Taylor expansion of (), which will be the key step obtaining the asymptotic distribution of the quantile estimator. We impose the following conditions. Assumption 3 . E(||X t ||3 ) < ∞.
Assumption 7. The conditional density f |X (|x) of t given X t = x is Lipschitz continuous: i.e. |f |X (1 |x) − f |X (2 |x)| ≤ L 0 |1 − 2 | for some constant 0 < L 0 < ∞ and for all x. Lemma 3. Suppose that Assumption 3 and 7 hold. Then ||() − (∗ ) + Q 0 ( − ∗ )|| = o(|| − ∗ ||), where Q 0 ≡ E[f |X (0|X t )X t X t ]. The final step in obtaining asymptotic normality of the quantile estimator ˆ T is to show that T 1/2 (ˆ T ) converges to some random variable in distribution. In fact, using Theorem 3 in Huber (1967), −T 1/2 (ˆ T ) and T −1/2 Tt=1 X t (t ) turn
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out to be asymptotically equivalent. To show this, we impose some additional assumptions. Assumption 8. ∗ is an interior point of B. Assumption 9. Q0 is positive definite. Assumption 10. There exists a constant f1 such that f |X (|x) ≤ f 1 for all and x. Lemma Assumptions 1, 2, 3 , 4−6 and 8–10 hold. Then T4. Suppose that1/2 −1/2 (ˆ T ) = o p (1). T t=1 X t (t ) + T Combining Lemma 3 and Lemma 4 and adding one more assumption permit us to state our main theorem. We impose Assumption 11. V ≡ E( (t )2 X t X t ) is positive definite. Theorem 1. Suppose that Assumptions 1, 2, 3 , 4–11 hold. Then d −1 T 1/2 (ˆ T − ∗ ) → N(0, C) where C ≡ Q −1 0 VQ 0 is positive definite. The asymptotic distribution in Theorem 1 includes all previously obtained results as special cases. Suppose that the linear conditional quantile model is correctly specified. Then it can be shown that V = (1 − )Q where Q ≡ E(X t X t ). This case thus corresponds to Powell (1984) who obtains d −1 T 1/2 (ˆ T − ∗ ) → N(0, (1 − )Q −1 0 QQ 0 ). Consider the more restricted case where not only is the linear conditional quantile model correctly specified but also there is no conditional heterogeneity in the density f at the origin (no “heteroaltitudinality”), that is f |X (0|X t ) = f(0). Then one can show that Q 0 = f(0)Q as well as V = (1 − )Q. This corresponds to the standard case of Koenker and d Bassett (1978), who obtain T 1/2 (ˆ T − ∗ ) → N(0, ((1 − )/f(0)2 )Q −1 ). Now consider testing a hypothesis about ∗ : H 0 : R∗ = r H a : R∗ = r, where R is a finite q × k matrix of full row rank and r is a finite q × 1 vector. Then Theorem 1 implies that, for example, d 2 ˆ −1 R ]−1 (R ˆ T − r)→ T(R ˆ T − r) [R C q T
(3.3)
p ˆT → under the null hypothesis, where C C is a covariance matrix estimator, consistent for C despite the possible misspecification. To implement such tests ˆ T . This is the focus of our next section. we require a consistent estimator C
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4. CONSISTENT COVARIANCE MATRIX ESTIMATION In this section, we provide an estimator for the asymptotic covariance matrix C that is consistent despite possible misspecification. The asymptotic covariance matrix consists of two components: Q0 and V. Powell (1984) suggested the following estimator for Q0 without formally proving its consistency: ˆ 0T ≡ (2ˆcT T)−1 Q
T
1[−ˆcT ≤ˆt ≤ˆcT ] X t X t ,
(4.1)
t=1
where cˆ T may be a function of the data and ˆ t ≡ Y t − X t ˆ T . We impose the following conditions on the sequence {ˆcT }. Assumption 12. There is a stochastic sequence {ˆcT } and a non-stochastic sequence {cT } such that p
(i) ccˆ TT → 1 (ii) c T = o(1)√ (iii) c −1 T = o( T ). ˆ 0T . We can now rigorously establish the consistency of Powell’s estimator Q Lemma 5. Suppose that T 1/2 ||ˆ T − ∗ || = O p (1) and Assumptions 1, 3 , 7, p ˆ 0T → Q0. 10 and 12 hold. Then Q Next, we use the plug-in principle to propose a consistent estimator of V: Vˆ T ≡ T −1
T uˆ 2t X t X t , t=1
where uˆ t ≡ (ˆt ). The estimator Vˆ T is completely analogous to White’s (1980b) estimator, for which uˆ t is the OLS residual. Lemma 6. Suppose that T 1/2 ||ˆ T − ∗ || = O p (1) and Assumptions 1, 3 , and p 10 hold. Then Vˆ T → V. We now define our estimator for the asymptotic covariance matrix C as ˆ −1 . ˆ −1 Vˆ T Q ˆT ≡ Q C 0T 0T
(4.2)
p ˆT → Together, Lemmas 5 and 6 imply C C, ensuring the consistency of the covariance estimator. This allows us to obtain a computable Wald statistic and its asymptotic distribution as follows.
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Theorem 2. Suppose that Assumptions 1, 2, 3 , 4–12 hold. Let the null hypothesis be given by H 0 : R∗ = r where R is a given finite q × k matrix of full row rank and r is a given finite q × 1 vector. Then under H0 , d
ˆ −1 R ]−1 (R ˆ T − r)→2q , T(R ˆ T − r) [R C T ˆ T is defined in (4.2). where C Although we do not pursue the issue here, we note that just as MacKinnon and White (1985) found modifications of Vˆ T that afforded improvements in finite sample properties, so also may there be analogous modifications of Vˆ T here. We leave this to subsequent research.
5. A TEST FOR CORRECT QUANTILE SPECIFICATION As we have seen in Section 3, if the conditional quantile model is correctly specified, then we have the quantile version of information matrix equality: V = (1 − )Q. We formally state this in the following lemma. Lemma 7. Suppose that Assumptions 2 and 3 hold and let u t ≡ (t ). If the conditional quantile model is correctly specified (i.e. E(u t |X t ) = 0), then we have V = (1 − )Q: that is, E(u 2t X t X t ) = E(u 2t )E(X t X t ). Equivalently, this can be written E((u 2t − 2u )X t X t ) = 0, where 2u ≡ E(u 2t ) = (1 − ). It is interesting to note that in order to have the same equality as in Lemma 7 in the context of OLS regression, we require two conditions: (1) the conditional expectation model is correctly specified; and (2) that there is no conditional heteroskedasticity in ut . See White (1980b) for details. However, for quantile regression, correct specification is the only sufficient condition due to the special structure of ut . According to Lemma 7, any misspecification in the conditional quantile is a form of conditional heteroskedasticity in ut . In such situations, inference based on the information equality is invalid. Zheng (1998) has developed a consistent test for conditional quantile misspecification. That test is based on a non-parametric kernel estimation and may accordingly be somewhat cumbersome to implement. We now propose a very easy to use specification testing procedure that exploits the quantile version of the information matrix equality in Lemma 7. Under the null hypothesis of correct specification of the conditional quantile function, we have E[h(X t )(u 2t − 2u )] = 0,
(5.1)
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where h(·) is a measurable s × 1 vector function. The information matrix test obtains when h(Xt ) selects some elements from the matrix X t X t . Using the fact that u 2t = 2u − (1 − 2) (t ) the expression in (5.1) can be shown to be equivalent to E[h(X t ) (t )] = 0.
(5.2)
A scaled sample version of the expectation in (5.2), which is our proposed test statistic for the null of correct specification, is given by T −1/2 m(ˆ T ),
where m(ˆ T ) = T −1 Tt=1 h(X t ) (Y t − X t ˆ T ). Note that to avoid degeneracies h(Xt ) cannot have elements equal to any element of Xt , as Lemma 2 ensures that T −1/2 Tt=1 X t (Y t − X t ˆ T ) = o p (1). We further restrict the space to which the function h(·) belongs using the following moment condition. Assumption 13. There exists ␦ > 2 such that E(||h(X t )||␦ ) < ∞. The following lemma is useful in deriving the asymptotic distribution of the proposed statistic. and 13 hold. Then Lemma 8. Suppose that Assumptions T 1, 3 ∗ ∗ 1/2 −1/2 sup ∈BT |T [m() − m( )] − T t=1 h(X t )[F t () − F t ( )]| = o p (1), ∗ 1/2 where B T ≡ { ∈ B : T || −  || ≤ M T and M T = O(1)}, and F t () ≡ F Y|X (X t |X t ).
The proof easily follows from Andrews (1989) once we show that T 1/2 [m T () − m T (∗ )] − T −1/2
T
h(X t )[F t () − F t (∗ )] = T −1/2
t=1
T
H(Z t ,),
t=1
where H(Z t ,) ≡ h(X t ){1[Y t ≤X t ] − 1[Y t ≤X t ∗ ] − [F t () − F t (∗ )]}. As the indicator functions 1[Y t ≤X t ] , 1[Y t ≤X t ∗ ] and the cumulative density functions Ft (), Ft (∗ ) are functions of bounded variation, the iid assumption (Assump tion 1) and the moment T conditions (Assumptions 3 and 13) are sufficient −1/2 to show that T t=1 H(Z t ,) is stochastically equicontinuous; that is, sup ∈BT |T −1/2 Tt=1 H(Z t , )| ≤ o p (1) using Theorems II.2 and II.3 in Andrews (1989). Theorem 3. Suppose that Assumptions 1, 2, 3 , 4–13 hold. (i) Suppose that the conditional quantile model is correctly specid −1/2 fied. Then 0 T 1/2 m(ˆ T ) → N(0, I s×s ) provided 0 ≡ (1 − )
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−1 −1 −1 (A 0 Q −1 0 QQ 0 A 0 − A 0 Q 0 A − A Q 0 A 0 + 0) is non-singular, where A 0 ≡ E[f |X (0|X t )X t h(X t ) ], A ≡ E(X t h(X t ) ) and D ≡ E(h(X t )h(X t ) ). (ii) Suppose that the conditional quantile model is correctly specified and that there is no conditional heteroaltitudinality in f Then d −1/2 T 1/2 m(ˆ T ) → N(0, I s×s ) provided ≡ (1 − )(D − A QA) is non-singular.
It is straightforward to derive consistent estimators for A, A0 and D using the plug-in principle. For example, ˆ T ≡ T −1 A
T
X t h(X t )
t=1
ˆ 0T ≡ (2ˆcT T)−1 A
T
1[−ˆcT ≤ˆt ≤ˆcT ] X t h(X t )
t=1
ˆ T ≡ T −1 D
T
h(X t )h(X t ) ,
t=1
ˆ 0T in (4.1). It can be easily where cˆ T and ˆ t are the same as in the definition of Q p p p ˆ T → A, A ˆ 0T → A 0 and D ˆT → shown using Lemma 5 that A D. Therefore, the null of correct specification of the conditional quantile function can be tested using the fact that under the null d 2 ˆ −1 m(ˆ T )→ Tm(ˆ T ) s
ˆ is a consistent estimator for either 0 or as desired, and which can be where ˆ 0T , Q ˆ T, A ˆ 0T , A ˆ T and D ˆ T. constructed using Q In particular applications, it may be of interest to test for conditional heteroaltiˆ 0T − fˆ T (0)Q ˆ T ), where fˆ T (0) is tudinality. Such a test can be based on vec h(Q an appropriately chosen consistent estimator of f(0) Because of the complexity introduced by the presence of kernel-type estimators in this difference, the analysis of the asymptotic distribution is somewhat involved, and we leave this to further work.
6. MONTE CARLO SIMULATIONS We conduct simulation experiments to investigate the finite sample properties of our new covariance matrix estimator and to compare it with conventional
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covariance matrix estimators. For later reference we label the various estimators as follows: ˆ −1 , (1 − )fˆ (0)−2 Q −1 ˆ Q ˆQ ˆ −1 , Q-SE2 : (1 − )Q 0 0 −1 ˆ ˆ −1 ˆ Q-SE2 : Q V Q .
Q-SE1 :
0
0
We compare the performance of these alternative estimators in three different set-ups: Case 1 The linear conditional quantile model is correctly specified and there is no conditional heteroaltitudinality in the density f, Case 2 The linear conditional quantile model is correctly specified but there is conditional heteroaltitudinality in the density f, Case 3 The linear conditional quantile model is misspecified and there is conditional heteroaltitudinality in the density f. We expect that Q-SE1 and Q-SE2 will achieve the best performance in Case 1 and Case 2 respectively, but Q-SE3 will be the winner in the most general case (Case 3). Bootstrapping the covariance matrix for quantile regressions has also gradually gained popularity. Hence, we also include the bootstrap covariance estimator in our simulation study. We use the design matrix bootstrap covariance estimator used in Buchinsky (1995), defined as B
1 ˜ ˆ (i () − ())( ˜ i () − ˆ i ()) B i=1
ˆ where () is the -quantile estimator, ˜ i () is the ith bootstrap estimator obtained by resampling the pair (y t , X t ) and B is the number of bootstrap samples. We set B to 500. When estimating Q-SE2 and Q-SE3, we need to make a choice for the bandwidth parameter cT . In our simulations we use three different methods to choose the bandwidth parameter, all of which are derived and discussed in Silverman (1986): two parametric choices based on the standard deviation (2 ) and the interquartile range (R ) of the underlying density f (the expressions are given by 1.06 T −1/5 and 0.79R T −1/5 respectively) and one non-parametric choice based on leastsquares cross-validation. Let (·) be the standard normal cumulative density function. We specify the data generating processes for our cases as follows:
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Case 1 y t = 1 + X t1 + X t2 + t , t ∼iid(−−1 (), 1) Case 2 y t = 1 + X t1 + X t2 + t , t = (X t )t , (X t ) = 1 + X 2t1 + X 2t2 , t ∼iid(−−1 (), 1) Case 3 y t = 1 + X t1 + X t2 + X t3 + 3i=1 3j=1 X ti X tj + t with t specified as in Case 2. In all cases the Xti s are drawn from the standard normal distribution with cross correlation 0.5; the value for the quantile () is set to 0.7. The number of observations and the number of replications are 50 and 1,000 respectively. For each replication we fit following quantile regression: y t = ˆ 0 () + ˆ 1 ()X t1 + ˆ 2 ()X t2 + e t (). The standard error for each coefficient is calculated using the various methods explained above. We report simulation means of the coefficient estimates and of the standard errors in Table 1. The results for Q-SE2 and Q-SE3 in Table 1 are based on the bandwidth choice using the interquartile range, that is cˆ T = 0.79R T −1/5 . We obtained qualitatively similar results using other bandwidth choices, so these are omitted. When the quantile regression model is correctly specified and there is no conditional heteroaltitudinality in the density f (Case 1), all methods to compute standard errors behave similarly except for the bootstrap method, which yields standard errors slightly larger than the other methods. In the last column under the heading “True Std. Dev.” we report the simulated standard deviations of the Table 1. Simulation Means of 0.7-Quantile Estimates and Standard Errors. Quantile Estimates
Q-SE1
Q-SE2
Q-SE3
Bootstrap Std. Errors
True Std. Dev.
Case 1 Xt 1 Xt 2 Constant
1 1 0.99
0.19 0.19 0.19
0.21 0.21 0.20
0.21 0.20 0.20
0.23 0.23 0.22
0.20 0.19 0.19
Case 2 Xt 1 Xt 2 Constant
1 0.99 0.99
0.30 0.30 0.29
0.42 0.41 0.32
0.41 0.40 0.31
0.44 0.43 0.34
0.40 0.37 0.28
Case 3 Xt 1 Xt 2 Constant
1.36 1.32 5.25
1.04 1.04 0.89
1.24 1.23 1.16
1.37 1.36 1.18
1.59 1.60 1.40
1.35 1.37 1.19
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quantile estimates, which closely approximate the true standard deviation of the sampling distribution for the quantile estimator. Not surprisingly, the correct covariance estimator Q-SE1 in Case 1 is smallest and closest to the true standard deviation. It is interesting to note that the efficiency loss caused by using Q-SE2 and Q-SE3 is practically negligible. In Case 2 where Q-SE2 is the valid covariance estimator, both Q-SE2 and Q-SE3 are reasonably close to the truth. The bootstrap standard errors are giving noticeable over-estimates and Q-SE1 is giving noticeable under-estimates. Finally, we turn to Case 3, where Q-SE3 is the only valid covariance estimator. It is clear from the table that only Q-SE3 is close to the truth. Again, the bootstrap standard errors provide over-estimates. While the other two covariance estimators (Q-SE1 and Q-SE2) both provide under-estimates, Q-SE1 is much worse than Q-SE2. The implication is that, at least in the case considered here, any null hypothesis is more likely to be rejected than it should be when the test is based on Q-SE1 or Q-SE2. Our simulation study illustrates that: (i) when the linear conditional quantile model is misspecified and there is conditional heteroaltitudinality in the density f, our new covariance estimator can allow researchers to conduct valid hypothesis tests; (ii) even when these conditions are not satisfied, using our covariance estimator is not likely to cause a serious efficiency loss; and (iii) the bootstrap method to calculate standard errors for quantile regressions should be used with care.
7. APPLICATION TO SHARPE STYLE ANALYSIS In this section, we apply our results to Sharpe style analysis. Following Sharpe’s (1988, 1992) seminal work, the Sharpe style regression has become a popular tool to analyze the style of an investment fund. The Sharpe style regression is carried out by regressing fund returns (over a period of time from a fund manager) on various factors mimicking relevant indices. By analyzing the coefficients of the factors, one can understand the style of a fund manager (e.g. style composition, style sensitivity, or style change over time). As Bassett and Chen (2001) point out, all previous papers have used the method of least squares and hence have concentrated on the relationship between the fund returns and the style factors focused on the mean of the return distribution conditional on the factors. Manager style, however, can be different in different parts of the conditional distribution. For example, a fund manager can change her style when the fund’s performance is very good or bad, which corresponds to high quantiles or low quantiles of the conditional distribution. Bassett and Chen (2001) have proposed using the quantile regression method to analyze the style
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of a fund manager over the entire conditional distribution. They used the returns on the Fidelity Magellan Fund (Rt ) over a 5 year sample period (January, 1992 – December, 1997) with 60 monthly observations. The Russell indices are the factors, which can be classified as follows:
Growth (G) Value (V)
Large (L)
Small (S)
Russell 1000 growth (X LG t ) LV Russell 1000 value (X t )
Russell 2000 growth (X SG t ) SG Russell 2000 value (X t )
The Sharpe style quantile regression equation is then given by LV SG SV R t = ␣() + LG ()X LG t + LV ()X t + SG ()X t + SV ()X t + t . (7.1)
On the grounds that for the equity-only funds (e.g. the Magellan Fund or S & P 500 Index), unconstrained and constrained (non-negativity and summing-to-one) cases are usually similar, Bassett and Chen estimate the equation in (7.1) without the non-negativity and summing-to-one constraints. Their findings can be summarized as follows: (i) for the conditional mean, the Magellan fund has an important LargeValue tilt (coefficient 0.69) and otherwise is equally divided between Large-Growth (0.14) and Small-Growth (0.20); (ii) the tilt to Large-Value appears at the other quantiles with an exception being the Quantile = 0.1, where the coefficient for Large-Growth is the largest; and (iii) in all the quantile regressions, most coefficients are not significant due to large standard errors. We use the same data set with a longer sample period (January, 1979 to December, 1997), yielding 228 monthly observations. Figure 1 shows a time-series plot of the Magellan Fund. Our belief is that the lack of significance encountered by Bassett and Chen is due to the relatively small number of observations. Since we are not sure about the correctness of the linear conditional quantile specification in (7.1), but would like to keep the linear specification, we calculate standard errors using the various methods explained in Section 6. The results for the least squares and quantile regressions are reported in Table 2. We take representative values of 0.1, 0.3, 0.5, 0.7 and 0.9 for in our quantile regressions. For the conditional mean of the distribution (that is, from the least squares regression), the Magellan fund appears to be heavily oriented toward Large-Growth (0.40) and also has an important Large-Value tilt (0.30). The remaining share is equally divided between Small-Growth (0.18) and Small-Value (0.21). In contrast to the findings of Bassett and Chen (2001), the Large-Growth component clearly stands out. This is, however, consistent with their finding that Large-Value orientation is an important component of the style of the Magellan
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Fig. 1. Time-Series Plot of Fidelity Magellan Fund Monthly Returns.
Fund. Further, it is obvious from Fig. 1 that the stock market crash in 1987 generated a huge outlier in the returns series. Considering that the least squares estimator is sensitive to outliers, one might like to see how robust the results are given this circumstance. The least absolute deviations (LAD) estimator is a potentially less sensitive alternative (but see Sakata & White, 1995, 1998). The LAD results are reported in the middle of Table 2 ( = 0.5). The coefficient for Large-Growth and Small-Growth are almost unchanged, but the coefficient for Large-Value changes from 0.30 to 0.38 while the coefficient for Small-Value has been reduced by half. As we change the value of , the style weight for Large-Growth (ˆ LG ()) is also gradually decreasing with while the style weight for Large-Value (ˆ LV ()) becomes more important as increases except at = 0.9 where there is a sudden drop. The style pattern for Small-Growth (ˆ SG ()) is also noticeably changing with . The tilt to Small-Growth is substantially increasing with , indicating that the fund tends to invest heavily in Small-Growth stocks when the fund’s performance is good, but reduces its share to a statistically insignificant point (when = 0.1, ˆ SG () is not significant) when the fund’s performance is poor. The allocation to Small-Value (ˆ SV ()) is decreasing with except at = 0.9 where it is sharply increasing. In order to see the change in the style against in detail, we examine a grid of values for (from 0.1 to 0.9 with 0.1 increment) and plot each quantile ˆ estimate () with its 95% confidence interval (constructed using Q-SE3) against . This plot is displayed in Fig. 2. The figure confirms our earlier observations.
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Table 2. Mean Style and Quantile Style for Fidelity Magellan Fund. Estimation
Std. Err.
ˆ LG ()
ˆ LV ()
ˆ SG ()
ˆ SV ()
LS-SE1 LS-SE2
0.40 0.07 0.07
0.30 0.08 0.08
0.18 0.06 0.06
0.21 0.08 0.07
0.45 0.11 0.11
Q-SE1 Q-SE2 Q-SE3
0.49 0.11 0.15 0.15
0.21 0.13 0.19 0.21
0.01 0.09 0.10 0.10
0.36 0.12 0.16 0.16
−1.43 0.17 0.19 0.19
Q-SE1 Q-SE2 Q-SE3
0.45 0.08 0.10 0.10
0.30 0.09 0.12 0.12
0.11 0.07 0.08 0.09
0.19 0.09 0.08 0.08
−0.25 0.13 0.16 0.16
Q-SE1 Q-SE2 Q-SE3
0.40 0.08 0.07 0.07
0.38 0.09 0.09 0.09
0.18 0.06 0.08 0.08
0.12 0.09 0.09 0.09
0.44 0.12 0.13 0.13
Q-SE1 Q-SE2 Q-SE3
0.35 0.08 0.06 0.06
0.39 0.10 0.08 0.08
0.24 0.07 0.06 0.06
0.10 0.09 0.07 0.07
1.07 0.13 0.14 0.13
Q-SE1 Q-SE2 Q-SE3
0.27 0.12 0.15 0.13
0.20 0.14 0.20 0.20
0.32 0.10 0.09 0.08
0.26 0.13 0.12 0.11
2.49 0.19 0.25 0.25
Least squares
Quantile = 0.1
Quantile = 0.3
Quantile = 0.5
Quantile = 0.7
Quantile = 0.9
␣() ˆ
Note: LS-SE1 = Conventional standard errors for the least squares estimates. LS-SE2 = White’s heteroskedasticity-consistent standard errors for the least squares estimates.
The pattern of the quantile style (as a function of ) we have found is qualitatively similar to the findings in Bassett and Chen (2001) except for certain large values of , but we can now provide confidence intervals around the quantile style weights that are robust to the potential misspecification of the conditional quantile function. In order to examine the potential for misspecification, we apply our quantile specification test using the selection function h(X t ) = vec h(JX t X t J ) with J = [04×1 I 4×4 ]. The test statistics for the selected values for and for the three alternatives to compute the bandwidth are given in Table 3. The results are fairly robust to the choice of the bandwidth. The overall conclusion is that for most quantiles we do not have evidence strong enough to reject at the 5% level the null that the linear quantile model in (7.1) is correctly specified. The last row in the
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Fig. 2. Quantile Style for Fidelity Magellan Fund.
table, however, indicates that when = 0.9, the linear specification in (7.1) may be misspecified. It is worth noting again that the standard errors in Table 2 and the confidence intervals in Fig. 2 are still valid under this potentially misspecified circumstance.2
Table 3. Information Matrix Specification Test Statistics. Quantile ()
0.1 0.3 0.5 0.7 0.9
Bandwidth Choice: 1.06 T −1/5
Bandwidth Choice: 0.79R T −1/5
12.49 14.52 15.62 16.33 22.10
15.79 14.42 15.42 16.58 22.13
Bandwidth Choice: Least-Squares Cross-Validation 12.40 14.48 15.42 16.58 22.13
Note: The critical value at the 5%-level for the 2 distribution with 10 degrees of freedom is 18.31.
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8. CONCLUSION We have obtained the asymptotic normality of the quantile estimator for a possibly misspecified model and provided a consistent estimator of the asymptotic covariance matrix. This covariance estimator is misspecification-consistent, that is, it is still valid under misspecification. If researchers confine themselves to a parametric world, then our results are useful when there is uncertainty about correct specification or when one wishes to maintain a model that does not pass a specification test. Although we have restricted our discussion to the linear conditional quantile model for iid data, our methods extend straightforwardly to non-linear conditional quantile models with dependent and possibly heterogeneous data. Investigation of these cases is a promising direction for future research. White (1994, esp. pp. 74–75) provides some consistency results for such cases. See Komunjer (2001) for some asymptotic distribution results in this direction.
NOTES 1. An excellent survey on recent developments in quantile regression, particularly with regard to empirical applications of quantile regression, is provided in Buchinsky (1998). For readers who would like to approach quantile regression from a historical perspective, Koenker (2000) is worth reading. An introductory exposition of quantile regression can be also found in books such as Peracchi (2001), Birkes and Dodge (1993). 2. All MATLAB programs and the data set used in the paper can be obtained at the following website: http://www.nottingham.ac.uk/∼leztk/publications.html
ACKNOWLEDGMENTS The authors would like to thank Douglas Stone for providing the data used in the paper and Clive Granger, Patrick Fitzsimmons, Alex Kane, Paul Newbold, Christophe Muller, Christian Gourieroux, Adrian Pagan, and Jin Seo Cho for their helpful comments. White’s research was supported by NSF grants SBR-9811562 and SES-0111238 and Kim’s research was supported by a grant from the University of Nottingham.
REFERENCES Andrews, D. W. K. (1989). Asymptotics for semiparametric econometric models: II. Stochastic equicontinuity and nonparametric kernel estimation. Cowles Foundation discussion paper No. 909R.
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Amemiya, T. (1982). Two stage least absolute deviation estimators. Econometrica, 50, 689–711. Bai, J. (1995). Least absolute deviation estimation of a shift. Econometric Theory, 11, 403–436. Bassett, G., & Chen, H. L. (2001). Portfolio style: Return-based attribution using regression quantiles. Empirical Economics, 26, 293–305. Bassett, G., & Koenker, R. (1978). Asymptotic theory of least absolute error regression. Journal of the American Statistical Association, 73, 618–622. Bhattacharya, P. K., & Gangopadhyay, A. K. (1990). Kernel and nearest-neighbor estimation of a conditional quantile. The Annals of Statistics, 18, 1400–1415. Billingsley, P. (1986). Probability and measure. New York: Wiley. Birkes, D., & Dodge, Y. (1993). Alternative methods of regression. New York: Wiley. Bloomfield, P., & Steiger, W. L. (1983). Least absolute deviations: Theory, applications, and algorithms. Boston: Birkhauser. Buchinsky, M. (1995). Estimating the asymptotic covariance matrix for quantile regression models: A Monte Carlo study. Journal of Econometrics, 68, 303–338. Buchinsky, M. (1998). Recent advances in quantile regression models: A practical guideline for empirical research. The Journal of Human Resources, 33, 88–126. Buchinsky, M., & Hahn, J. (1998). An alternative estimator for the censored quantile regression model. Econometrica, 66, 653–671. Doob, J. L. (1953). Stochastic processes. New York: Wiley. Hahn, J. (1995). Bootstrapping quantile regression estimators. Econometric Theory, 11, 105–121. Herce, M. A. (1996). Asymptotic theory of LAD estimation in a unit root process with finite variance errors. Econometric Theory, 12, 129–153. Horowitz, J. L. (1998). Bootstrap methods for median regression models. Econometrica, 66, 1327–1351. Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, 221–233. Koenker, R. (2000). Galton, Edgeworth, Frisch, and prospects for quantile regression in econometrics. Journal of Econometrics, 95, 347–374. Koenker, R., & Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33–50. Koenker, R., & Zhao, Q. (1996). Conditional quantile estimation and inference for ARCH models. Econometric Theory, 12, 793–813. Komunjer, I. (2001). The ‘␣-quantile’ distribution function and its applications to financial modelling. Working Paper, Department of Finance, HEC School of Management. MacKinnon, J. G., & White, H. (1985). Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties. Journal of Econometrics, 29, 305–325. Peracchi, F. (2001). Econometrics. New York: Wiley. Phillips, P. C. B. (1991). A shortcut to LAD estimator asymptotics. Econometric Theory, 7, 450–464. Powell, J. (1983). The asymptotic normality of two-stage least absolute deviations estimators. Econometrica, 51, 1569–1575. Powell, J. (1984). Least absolute deviations estimation for the censored regression model. Journal of Econometrics, 25, 303–325. Powell, J. (1986). Censored regression quantiles. Journal of Econometrics, 32, 143–155. Pollard, D. (1985). New ways to prove central limit theorems. Econometric Theory, 1, 295–314. Pollard, D. (1991). Asymptotics for least absolute deviation regression estimators. Econometric Theory, 7, 186–199. Ruppert, D., & Carroll, R. J. (1980). Trimmed least squares estimation in the linear model. Journal of the American Statistical Association, 75, 828–838.
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Sharpe, W. F. (1988). Determining a fund’s effective asset mix. Investment Management Review (December), 59–69. Sharpe, W. F. (1992). Asset allocation: Management style and performance measurement. Journal of Portfolio Management, 18, 7–19. Sheather, S. J., & Marron, J. S. (1990). Kernel quantile estimators. Journal of the American Statistical Association, 85, 410–416. Silverman, B. W. (1986). Density estimation for statistics and data analysis. London: Chapman & Hall. Sakata, S., & White, H. (1995). An alternative definition of finite sample breakdown point with applications to regression model estimators. Journal of the American Statistical Association, 90, 1099–1106. Sakata, S., & White, H. (1998). High breakdown point conditional dispersion estimation with application to S & P 500 daily returns volatility. Econometrica, 66, 529–567. Weiss, A. A. (1990). Least absolute error estimation in the presence of serial correlation. Journal of Econometrics, 44, 127–158. White, H. (1980a). Nonlinear regression on cross-section data. Econometrica, 48, 721–746. White, H. (1980b). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48, 817–838. White, H. (1992). Nonparametric estimation of conditional quantiles using neural networks. In: H. White (Ed.), Artificial Neural Networks: Approximation and Learning Theory (pp. 191–205). Oxford: Blackwell. White, H. (1994). Estimation, inference and specification analysis. New York: Cambridge University Press. Zheng, J. X. (1998). A consistent nonparametric test of parametric regression models under conditional quantile restrictions. Econometric Theory, 14, 123–238.
MATHEMATICAL APPENDIX ¯ Proof of Lemma 1: We define Q T () ≡ S T () − S T (∗ ) and Q() ≡ E(Q T ()). ¯ Note that Q() does not depend on T as a result of the iid assumption. Since S T (∗ ) does not depend on , ˆ T ∈ arg min∈B Q T () = T −1 Tt=1 q(Z t , ) where q(Z t , ) ≡ (Y t − X t ) − (Y t − X t ∗ ) and Z t ≡ (Y t , X t ). In order to apply Lemma 2.2 and Lemma 2.3 in White (1980a), we need to establish the following: (1) q(·, ) is measurable for each  ∈ B. (2) q(Zt , ·) is continuous on B almost surely. (3) There exists a measurable function m : R k+1 → R such that (i) |q(z, )| ≤ m(z) for all  ∈ B. (ii) There exists ␦ > 0 such that E|m(Z t )|1+␦ ≤ M < ∞. ¯ (4) Q() has a unique minimum at ∗ . Conditions (1) and (2) are trivially satisfied by inspection of the functional ¯ + ||∗ ||)( + 1) where ¯ is a solution to form of q. Let m(Z t ) ≡ ||X t ||(||||
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max∈B ||||. The existence of such a solution is guaranteed by Assumption 5. It is obvious that m is measurable. It is easily seen that |q(z, )| ≤ m(z) for all  ∈ B. Condition (3)(ii) is satisfied by Assumption 3 . The last step is to verify ¯ ∗ ) = 0 by Assumption 4 and Q() ¯ = condition (4). Let ␦ ≡ −∗ . Since Q( E(q(Z t , )) by Assumption 1, it is sufficient to show that E(q(Z t , )) > 0 for any ␦ = 0. Note that Assumptions 2 and 6 imply that there exists a positive number f0 such that || < f 0 ⇒ f |X (|x) > f 0 for all x. Using this property, one can show that Xt ␦
E(q(Z t , )) = E 0
Xt ␦ 0 ( − X t ␦)1[−f 0 0 and d 0 > 0 such that ||()|| ≥ a|| − ∗ || for || − ∗ || ≤ d 0 . There exist b > 0 and d 0 > 0 such that E[u(Z t , , d)] ≤ bd for || − ∗ || + d ≤ d0. There exist c > 0 and d 0 > 0 such that E[u(Z t , , d)2 ] ≤ cd for || − ∗ || + d ≤ d 0 . E||X t (t )||2 < ∞.
First, note that condition (1) is just Assumption 8, and conditions (2) and (3) have been proved in Lemma 1 and Lemma 2 using Assumptions 1, 2, 3 , 4–6. Condition (4) is easily checked using the equivalent definition of separability in Billingsley (1986). This condition ensures that the function u(·, , d) is measurable. Condition (5) is satisfied given Assumption 4 by letting 0 = ∗ . We now verify condition; (6). Let a be the smallest eigenvalue of E[f |X (∗t |X t )X t X t ]. We can make E[f |X (∗t |X t )X t X t ] sufficiently close to E[f |X (0|X t )X t X t ] which is positive-definite by Assumption 9 by choosing a sufficiently small d 0 > 0. We choose such a d 0 > 0 so that a is positive. We have by (3.2) ||()|| = ||Q ∗ ( − ∗ )|| ≥ a|| − ∗ ||
for || − ∗ || ≤ d 0 ,
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which verifies condition (6). Some simple algebra gives the inequality u(Z t , , d) ≤ ||X t ||1[|t −X t (−∗ )|≤||X t ||d] , which implies by the law of iterated expectations and Assumption 10 that E[u(Z t , , d)] ≤ 2f 1 E||X t ||2 ]d. Let b ≡ 2f 1 E[||X t ||2 ]. Then b is positive and finite by Assumption 3 , which verifies condition (7). Condition (8) can be verified in a similar fashion by letting c ≡ 2f 1 E[||X t ||3 ] > 0, which is finite by Assumption 3 . Note that E||X t (t )||2 = E[ (t )2 X t X t ] ≤ E[X t X t ] < ∞ by Assumption 3 . All the conditions in Theorem 3 in Huber (1967) are thus satisfied. We therefore have the desired result: T −1/2
T
X t (t ) + T 1/2 (ˆ T ) = o p (1).
t=1
Proof of Theorem 1: Using the Lindeberg-Levy CLT, Assumptions 1, 3, 4 and 11 imply that T −1/2
T
d
X t (t )→N(0, V).
(A.1)
t=1
Lemma 3, Lemma 4 and (A.1) together imply that d
T 1/2 (ˆ T − ∗ )→N(0, C) −1 where C ≡ Q −1 0 VQ 0 .
Proof of Lemma 5: Consider Q T ≡ (2c T T)−1 Tt=1 1[−c T ≤t ≤c T ] X t X t . Using the mean value theorem, one can show that E(Q T ) = E[T −1 Tt=1 f |X (T |X t )X t X t ] where −c T ≤ T ≤ c T . Hence, T = o(1) by Assumption 12(ii). Let Q 0 ≡ E[T −1 Tt=1 f |X (0|X t )X t X t ]. Then it is easily checked that |E(Q T ) − Q 0 | → 0 by the triangle inequality and Assumptions 3 and 7. Using a LLN for double arp rays (e.g. Theorem 2 in Andrews, 1989), we have that Q T → E(Q T ). Therefore, p we have that Q T → Q 0 . The rest of the proof is carried out in two steps: p ˜ 0T ≡ (2c T T)−1 T 1[−ˆcT ≤ˆt ≤ˆcT ] X t X t ; and ˜ 0T − Q T | → 0 where Q (1) |Q t=1 p ˜ 0T | → ˆ 0T − Q 0. (2) |Q
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˜ 0T − Q T | is given by The (i, j)th element of |Q T −1 1[−ˆcT ≤ˆt ≤ˆcT ] − 1[−c T ≤t ≤c T ] )X ti X tj (2c T T) t=1
≤ (2c T T)−1
T
1[|t +c T |≤d T ] + 1[|t −c T |≤d T ] )|X ti ||X tj | ≡ U 1T + U 2T
t=1
where d T ≡ ||X t ||ˆ T − ∗ || + |ˆcT − c T | (by the fact that |1[x≤0] − 1[y≤0] | ≤ 1[|x|≤|x−y|] |), U 1T ≡ (2c T T)−1 Tt=1 1[|t −c T |≤d T ] |X ti ||X tj |, and U 2T ≡ (2c T T)−1 Tt=1 1[|t −c T |≤d T ] |X ti ||X tj |. p
First, we show that U 1T → 0. For this, let > 0 and consider
T −1 1[|t −c T |≤d T ] |X ti ||X tj | > ≡ P(A), P(U 1T > ) = P (2c T T) t=1
where A ≡ (2c T T)
−1
T
1[|t −c T |≤d T ] |X ti ||X tj | >
and
t=1
P(A) ≤ P(A ∩ B ∩ C) + P(C c ) + P(B c )
for any events B and C.
−1 ∗ ˆ Now let B ≡ {c −1 cT − T ||T −  || ≤ z} for a constant z > 0 and C ≡ {c T |ˆ c c T | ≤ z}. Note that as T → ∞, (1) P(B ) → 0 by Assumption 12(iii) and T 1/2 ||ˆ T − ∗ || = O p (1); and (2) P(C c ) → 0 by Assumption 5(i). Now
P(A ∩ B ∩ C) ≤ (2cT T )−1 ≤ (2cT
T )−1
T
T
(by the Markov inequality)
t=1 E t=1 E
(||Xt ||+1)zcT +cT −(||Xt ||+1)zcT +cT f |X (|X t )d|X ti ||X tj | (||Xt ||+1)zcT +cT −(||Xt ||+1)zcT +cT f 1 d|X ti ||X tj |
(by Assumptions 10) = zf1 −1 E[(||Xt || + 1)Xti ||Xtj |] < ∞ (by Assumptions 1 and 3 ). We can choose z arbitrarily small, so P(U 1T > ) → 0 which implies that p p U 1T → 0 because U 1T ≥ 0. It can be shown in the same fashion that U 2T → 0, ˜ 0T = ˆ 0T − Q which completes the first step. To show the second step, consider Q ˆ 0T . Note that Q ˜ 0T = O p (1) and (c T /ˆcT − 1) = o (1) by Assump(c T /ˆcT − 1)Q tion 5(i). Hence, the second step follows, which delivers the desired result: p ˆ 0T → Q Q0.
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Proof of Lemma 6: Since the proof is quite similar to the proof of Lemma 5, we do not provide the details. Let V T ≡ T −1 Tt=1 (t )2 X t X t . By the law of large p
numbers for iid random variables, we have V T → V by Assumption 1 and 3 . p Next we show that Vˆ T − V T → 0. Consider the (i, j)th element of |Vˆ T − V T |: T T −1 2 2 2 −1 T ≤ ( + 1) [ (ˆ ) − ( ) ]X X T 1[|t |≤d T ] |X ti ||X tj |, ti tj t t t=1
t=1
(where d T ≡ ||X t ||||ˆ T − ∗ ||) by the triangle inequality and the Cauchy-Schwarz inequality. Let U T ≡ T −1 Tt=1 1[|t |≤d T ] |X ti ||X tj |. By the same argument as in the proof p
of Lemma 5, one can show that U T → 0 using Assumptions 1, 3 , 10, and T 1/2 ||ˆ T − ∗ || = O p (1).
Proof of Lemma 7: First, note that 2u ≡ (1 − ). Hence, E[(u 2t − 2u )X t X t ] = E[(u 2t −(1 − ))X t X t ] = E[cE(u t |X t )X t X t ] where c ≡ 2 − 1, which is zero because of the correct specification assumption, E(u t |X t ) = 0. Proof of Theorem 3: Since T 1/2 (ˆ T − ∗ ) = O p (1), Lemma 8 implies that T 1/2 m(ˆ T ) = T 1/2 m(∗ ) + T −1/2
T
h(X t )[F t (ˆ T ) − F t (∗ )] + o p (1). (A.2)
t=1
It is straightforward to show that T
1/2
∗
m( ) = T
−1/2
T
h(X t ) (t )
t=1
T −1/2
T t=1
h(X t )[F t (ˆ T ) − F t (∗ )] = T −1
T
f |X (0|X t )h(X t )X t T 1/2 (ˆ T − ∗ )
t=1
+ o p (1). −1/2 = A 0 Q −1 0 T
T t=1
X t (t ) + o p (1).
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Plugging these expressions into (A.2) and collecting terms gives T 1/2 m(ˆ T ) = T −1/2
T
a t (t ) + o p (1),
(A.3)
t=1
where a t ≡ A 0 Q −1 0 X t − h(X t ). Under the assumption that the conditional quantile model is correctly specified, the Lindeberg-Levy CLT delivers that T −1/2
T
d
a t (t )→N(0, 20 ),
t=1 −1 (1 − )(A 0 Q −1 0 QQ 0 A 0
−1 − A 0 Q −1 where 0 ≡ 0 A − A Q 0 A 0 + D), which completes the proof of (i). Furthermore, if there is no conditional heteroaltitudinality in f, then it can be shown that A 0 = f(0)A as well as that Q 0 = f(0)Q. Then the asymptotic variance 0 in (i) simplifies to
≡ (1 − )(D − A QA), which completes the proof of (ii).
QUASI–MAXIMUM LIKELIHOOD ESTIMATION WITH BOUNDED SYMMETRIC ERRORS Douglas Miller, James Eales and Paul Preckel ABSTRACT We propose a quasi–maximum likelihood estimator for the location parameters of a linear regression model with bounded and symmetrically distributed errors. The error outcomes are restated as the convex combination of the bounds, and we use the method of maximum entropy to derive the quasi–log likelihood function. Under the stated model assumptions, we show that the proposed estimator is unbiased, consistent, and asymptotically normal. We then conduct a series of Monte Carlo exercises designed to illustrate the sampling properties of the quasi–maximum likelihood estimator relative to the least squares estimator. Although the least squares estimator has smaller quadratic risk under normal and skewed error processes, the proposed QML estimator dominates least squares for the bounded and symmetric error distribution considered in this paper.
1. INTRODUCTION We consider the problem of estimating the location parameters for the linear regression model Y = x + ⑀ Maximum Likelihood Estimation of Misspecified Models: Twenty Years Later Advances in Econometrics, Volume 17, 133–148 Copyright © 2003 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(03)17006-5
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For present purposes, we assume the explanatory variables stated in the (n × k) matrix x are linearly independent and fixed in repeated samples. The n uncorrelated elements of the error process are assumed to be symmetrically distributed on bounded support [−, ] such that Pr[−v < e i < v] = 1 for some v such that v ≥ > 0. The bounding and symmetry properties imply that the error process has mean zero and finite variance. Otherwise, we do not specify a particular probability model for the error process so that the linear regression model specification is semiparametric. We believe there are some important reasons to consider alternative estimators to the least squares (LS) estimator ˆ = (x x)−1 x Y for this case. First, many observed economic variables (e.g. prices, quantities, asset values, expenditure shares) have some finite upper and lower bounds that reflect the range of economic and institutional conditions for sample periods in the recent past and the relevant future. Consequently, Gaussian (normal) or other error distributions with unbounded support are not fully representative of the unexplained components in the regression model. Second, our use of a semiparametric regression model implies that a symmetric distribution on the bounded error support is appropriate given that we have limited information about the distributional character of the unexplained components. If we only know that the error term is bounded and has mean zero, then we are unlikely to have information that would allow us to assign different probabilities to complementary events in the error space. In particular, the principle of insufficient reason would imply that we should assign symmetric weight to complementary events as Pr[−v < e i < −¯v] = Pr[¯v < e i < v]
(2)
for any v¯ such that 0 < v¯ < v. Although we recognize that the bounded and symmetric error specification is not commonly adopted in applied econometric research, this specification has been used in economics and a wide range of other fields. For example, Gstach (1998) presents an extension of Data Envelopment Analysis (DEA) known as DEA+ that explicitly relies on the bounded noise assumption for consistent estimation of technical efficiency measures in a stochastic frontier model of economic production. The bounded noise assumption also appears in many engineering and physical science applications, and a recent survey is presented by Bai, Nagpal and Tempo (1996). Finally, statisticians have adopted the bounded noise specification to derive a variety of useful results. For example, Chan and Tong (1994) discuss the ergodicity properties of nonlinear time series processes with bounded error support. Under the standard semiparametric specification of the linear regression model, the LS estimator is the best linear unbiased estimator (BLUE) by the
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Gauss–Markov theorem. We can also show that the LS estimator is consistent and asymptotically normal, and the LS estimator is widely used in practice for these reasons. However, we also have the additional information regarding the symmetry and boundedness of the error process. The purpose of this paper is to examine two questions: (1) can we derive an approximation to the error distribution that may be used for likelihood–based estimation, and (2) what are the relative sampling properties of the resulting estimator? To address the first question, we use Jaynes’ method of maximum entropy (Jaynes, 1957a, b) to form an approximate or quasi–log likelihood function that may be used to derive a quasi–maximum likelihood (QML) estimator of the model parameters. Then, we address the second question by using known results to establish the sampling properties of the QML estimator. We also use Monte Carlo simulation exercises to compare the relative sampling performance of the QML and LS estimators under three distinct error processes.
2. A MAXIMUM ENTROPY QUASI–LOG LIKELIHOOD FUNCTION Our objective in this section is to derive an approximate probability model for the bounded error process conditional on the model parameters . Given this model, the QML estimator of  is the parameter vector that maximizes the quasi–log likelihood function. Following Golan, Judge and Miller (1996), Boschmann, Preckel and Eales (2001) represent outcomes of the error process (e i ) by assigning positive discrete weights wi1 > 0 and wi2 = 1 − wi1 > 0 (i.e. wi1 + wi2 = 1) to the interval endpoints (−v and v) for each i = 1, . . . , n. Given the stated properties of i , we know there exist wi1 and wi2 such that each e i may be stated as the convex combination e i = −wi1 v + wi2 v
(3)
We know E[i ] = 0 for each i given that the error distribution is symmetric on a finite interval. In the absence of additional ex ante information, we would assign uniform weights, wi1 = wi2 = 0.5 to the error bounds to represent the mean–zero property of the errors. We derive our approximate probability model for the error process by choosing wi1 and wi2 that are as close to uniform as possible while also satisfying the regression equations, i () = −wi1 v + wi2 v where i ≡ y i − x i  (for some ). To recover discrete weights on the error bounds that exhibit these properties, we use Jaynes’ method of maximum entropy. The objective of the maximum entropy
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problem is the Shannon entropy functional H(w) = −
n 2
wij ln(wij )
(4)
i=1 j=1
which is a pseudo–distance function that measures the uniformity of distribution wi . The maximal value of Shannon’s entropy functional is achieved under uniform discrete distributions (wi1 = wi2 = 0.5 for each i), and H(w) = 0 if each distribution is degenerate on one of the boundary points. The maximum entropy problem is formally solved by maximizing (4) by choice of w subject to the regression constraints, i () = −wi1 v + wi2 v (for some ), and the additivity constraints, wi1 + wi2 = 1, for each i. The Lagrange equation is L(w, , ) = −
2 n
wij ln(wij ) +
i=1 j=1
n
i [i () + wi1 v − wi2 v]
i=1
+ i [1 − wi1 − wi2 ]
(5)
and the necessary conditions for the discrete weights wij are ∂L ˆ i1 ) + vˆ i − ˆ i = 0 = −1 − ln(w ∂wi1
(6)
∂L ˆ i2 ) − vˆ i − ˆ i = 0 = −1 − ln(w ∂wi2
(7)
and
where ˆ i is the Lagrange multiplier for the ith regression constraint and ˆ i is the Lagrange multiplier for the additivity constraint. After solving the necessary conditions for the maximum entropy weights ˆ i1 = exp(−1 + vˆ i − ˆ i ) w
(8)
ˆ i2 = exp(−1 − vˆ i − ˆ i ) w
(9)
and ˆ i1 + w ˆ i2 = 1 to determine we can use the additivity constraint w exp(−1 − ˆ i ) = [exp(vˆ i ) + exp(−vˆ i )]−1
(10)
By substitution, ˆ i1 = w
exp(vˆ i ) exp(vˆ i ) + exp(−vˆ i )
(11)
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and ˆ i2 = w
exp(−vˆ i ) exp(vˆ i ) + exp(−vˆ i )
(12)
and the maximum entropy weights are now functions of the Lagrange multipliers ˆ i , which are implicit functions of . The solutions to most maximum entropy problems must be stated in this intermediate form because there is no explicit solution for the Lagrange multipliers. However, due to the special structure of this problem, we can solve for each ˆ i as an explicit function of  from the necessary conditions for the Lagrange multipliers ∂L ˆ i1 v + w ˆ i2 v = −v tanh(vˆ i ) = i () = −w ∂i
(13)
The function tanh(z) = (exp(z) − exp(−z))/(exp(z) + exp(−z)) is the hyperbolic tangent, which is invertible as tanh−1 (z) = ln((1 + z)/(1 − z))/2. We can use the inverse function to solve for the Lagrange multiplier from Eq. (13) as 1 v + i () (14) ˆ i = − ln 2v v − i () which is now an explicit function of . The computational burden of the maximum entropy problem may be further reduced by concentrating the objective function. First, we multiply the numerators and denominators of (11) and (12) by exp(ˆ i i ()) to derive equivalent maximum entropy weights ˆ i1 = w
exp(ˆ i (v + i ())) exp(ˆ i (v + i ())) = ∗i (i ()) exp(ˆ i (v + i ())) + exp(−ˆ i (v − i ()))
(15)
and ˆ i2 = w
exp(−ˆ i (v − i ())) exp(−ˆ i (v − i ())) (16) = ∗i (i ()) exp(ˆ i (v + i ())) + exp(−ˆ i (v − i ()))
where ∗i (i ()) ≡ exp(ˆ i (v + i ())) + exp(−ˆ i (v − i ())) is the partition function (normalizing constant) for the error weights. The maximum entropy solution (given ) is a saddle point defined by the minima of the Lagrange equation (5) with respect to and the maxima of (5) with respect to w. If we substitute (15) and (16) into the Lagrange expression (5), we derive a concentrated objective function n m ∗ () = ln[ ∗i (i ())] (17) i=1
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DOUGLAS MILLER, JAMES EALES AND PAUL PRECKEL
which is only a function of ˆ and . By substitution of ˆ from Eq. (14) into ∗i , the concentrated objective function reduces to m() =
n
ln[ i (i ())]
(18)
i=1
where
v + i () v + i () i (i ()) ≡ exp − ln 2v v − i () v − i () v + i () + exp ln 2v v − i ()
(19)
The maximum entropy estimator of  is ˜ = argmax m(). For estimation purposes, we view the concentrated objective function m() as a quasi–log likelihood function based on an approximate density function for the error process g i (i ) ∝ i (i )
Fig. 1. Quasi–Density Function, g() for v = 5.
(20)
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The only piece of the actual quasi–log likelihood function missing from m() is the natural log of the normalizing constant for this kernel function v i () d (21) κi (v) = −v
which does not depend on  due to the symmetry of i . The quasi–density function is plotted in Fig. 1 for v = 5, and g i (i ) is symmetric about zero on the support [−v, v] with mean zero and finite variance. These properties are compatible with our assumptions for the underlying error process. The concentrated entropy objective function (18) is the quasi–log likelihood function based on this approximate density function after excluding the normalizing terms that do not involve . Thus, the maximum entropy estimator ˜ is also the QML estimator of  under this approximate density function for the error process. We derive the sampling properties of the QML estimator in the next section.
3. SAMPLING PROPERTIES OF THE QML ESTIMATOR Based on the model assumptions stated in Section 1, we can show that the QML estimator of  is unbiased, consistent, and asymptotically normal. To demonstrate these claims, we discuss the key steps involved in the proof of each proposition. Proposition 1. ˜ is an unbiased estimator of the true parameter vector 0 . Proof: The property follows directly from the distributional symmetry result presented in Theorem 1 by Andrews (1986). The theorem is based on the following condition: Assumption A1: m() is an even function of (). i We confirm that Andrews’ Assumption A1 holds for m() by referring to the partition function (19), which is an even function such that i (i ()) = i (−i ()). The concentrated objective function is the sum of natural logarithms of the quasi–density components, so m() is an even function of i (). Thus, Andrews’ Theorem 1 implies that the finite sample ˜ = 0 . distribution of ˜ is symmetric about 0 and further that E[] Proposition 2. The QML estimator is
√ p n–consistent such that ˜ → 0 .
Proof: The consistency proof for the QML estimator follows Theorem 3.5 in White (1994). White’s theorem is based on several regularity conditions:
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Assumption 2.3: The kernel of the quasi–density function g ( ) in Eq. (20) is i i measurable and continuous in . Assumption 3.1: The expected value of the quasi–log likelihood function (18) exists, and n −1 m() is continuous in  and converges under a strong or weak uniform law of large numbers (ULLN). Assumption 3.2: The QML estimation problem has a unique interior solution. White’s Assumption 2.3 is met by Eq. (20), which is continuous by construction and measurable under the definition provided by Royden (1988, pp. 66–67). Also, we can show that Assumption 3.1 is satisfied for (18) under the stated assumptions for the error process (symmetry and boundedness). Finally, Assumption 3.2 holds as a consequence of the saddle–point property of the maximum entropy problem and the strict concavity of (18) with respect to . Proposition 3. The QML estimator is asymptotically normal such that √ d An n(˜ − 0 )→N(0, I k ) B −1/2 n where An = n
−1
E
∂2 m() ∂∂ 0
and
Bn = n
−1
E
∂m() ∂m() ∂ 0 ∂ 0
Proof: The proof of the asymptotic normality property for the QML estimator follows Theorem 6.4 in White (1994). White’s theorem is based on the assumptions required for Proposition 2 plus some additional regularity conditions: Assumption 3.6: The quasi–density functions g (i ) in Eq. (20) are i twice–continuously differentiable in . Assumption 3.7(a): The gradient of m() has finite expectation. Assumptions 3.8 and 3.9: The Hessian matrix of m() has finite expectation, is uniformly continuous in , and converges under a strong or weak ULLN. Assumption 6.1: The gradient of m() is asymptotically normal as d −1/2 −1/2 ∂m() Bn n →N(0, I k ) ∂ 0 As in the proof outline for Proposition 2, we can use the properties of functions (18) and (20) to show that the regularity conditions are met. In particular, we note that the gradient function of m() has null expectation (i.e. White’s Assumption 3.7(a)) when evaluated at 0 due to the symmetry of the error process. The stated form of White’s Assumption 6.1 also reflects the mean–zero property of the gradient function.
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4. EXAMPLES In this section, we compare the sampling properties of the QML estimator ˆ and the least squares (LS) estimator, ˆ = (x x)−1 x Y , under three distinct error processes with unitary variance. First, we generate centered and scaled U(0, 8) outcomes √ 3(U(0, 8) − 4) i ∼ 4 which have a symmetric and bounded distribution. Second, we consider a symmetric but unbounded case, the standard normal error process with outcomes i ∼ N(0, 1). Finally, we consider the centered and scaled Chi-square(3) error process i ∼
(23 − 3) √ 6
which is unbounded and skewed. The LS estimator is unbiased, consistent, and asymptotically normal under each error process. Further, ˜ is the maximum likelihood (ML) estimator in the N(0, 1) case and is the minimum variance unbiased (MVU) or efficient estimator. The QML estimator exhibits the unbiasedness, consistency, and asymptotic normality properties in the first two cases but is biased under the skewed Chi-square distribution. To compare the performance of ˆ and ˜ under these scenarios, we conduct a set of Monte Carlo experiments designed to simulate the bias Bias(˜ k ) = E[˜ k ] − k and variance var(˜ k ) = E[(˜ k − E[˜ k ])2 ] associated with the QML and LS estimators of each of the k = 1, . . . , 4 elements of . We also simulate the expected squared error loss or quadratic risk ˜ = E[(˜ − ) (˜ − )] = QR()
4 k=1
var(˜ k ) +
4
(Bias(˜ k ))2
k=1
for the LS and QML estimators in each case. To reduce the summary statistics to manageable form, we report the relative simulated variance of the QML ˆ = (x x)−1 under each estimators for each k for k = 1, . . . , 4 (note that cov() of the error processes). We also report the relative simulated quadratic risk for the QML estimator under the separate experimental designs, and the benchmark is the variance and risk of the LS estimator.
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The sample size is set at n = 30 for each simulation experiment, and the sampling process is replicated for 2,000 Monte Carlo trials. The true parameter vector is  = (1 2 3 4) , and the elements of x are generated as pseudo–random N(0, 1) outcomes and held fixed across the replicated samples. The error bounds for the centered √ and scaled U(0, 8) error process are v = 3, 5, 10, 15, and 20 such that v > = 3 in each case. The error bounds are v = 5, 10, 15, and 20 for the N(0, 1) error process, which is supported on the real line. Finally, the error bounds are v = 10, 15, and 20 for the centered √ and scaled Chi-square(3) error process, which is supported on the interval (− 3/2, ∞). As such, there is a very small probability that e i ∈ / (−v, v) in the unbounded cases. The Monte Carlo results for the centered U(0, 8) error process and five levels of v are stated at the top of Table 1. The simulated variance of the QML estimator is uniformly smaller than the LS variance. Further, the simulated relative quadratic risk for the QML estimator is smaller for each v, and the risk reduction provided by the QML estimator is about 14% in the case v = 3. We do not report the simulated bias results because the QML and LS estimates are of comparable magnitude and are all very small (on the order of 10−3 ). As v increases, the bias estimates remain comparable but the differences between the variance and quadratic risk estimates for the two estimators diminish. For v = 20, the risk reduction provided by the QML estimator declines to less than 1%, but it appears that the QML Table 1. Simulated Relative Variance and Risk of QML versus LS. QML Bound v
Relative Variance 1
Relative Quadratic Risk
2
3
4
Centered and scaled U(0, 8) errors 3 0.8473 5 0.9495 10 0.9883 15 0.9945 20 0.9971
0.8603 0.9640 0.9896 0.9954 0.9975
0.8704 0.9581 0.9892 0.9954 0.9977
0.8732 0.9527 0.9890 0.9954 0.9972
0.8641 0.9563 0.9891 0.9952 0.9974
Standard normal errors 5 1.0050 10 1.0014 15 1.0000 20 1.0001
1.0041 0.9994 0.9998 0.9999
1.0047 1.0010 1.0003 1.0000
1.0044 1.0016 0.9998 0.9999
1.0044 1.0009 1.0000 1.0000
Centered and scaled chi-square(3) errors 10 1.0617 1.0324 15 1.0173 1.0144 20 1.0123 1.0118
1.0378 1.0150 1.0101
1.0499 1.0153 1.0098
1.0453 1.0154 1.0110
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estimator uniformly dominates LS under squared error loss based on this evidence. In general, the QML estimator behaves more like the LS estimator as v increases due to the locally quadratic character of the quasi–log likelihood function (18). Preckel (2001) discusses the relationship between the entropy and least squares objective functions as the bound magnitude increases in a related context. The Monte Carlo simulation results for the N(0, 1) error process are stated in the center of Table 1. In this case, the simulated relative variances of the QML estimator are now larger than one because the LS estimator is the MVUE for this error process. Accordingly, the relative risk for the QML estimator is greater than or equal to one for each v (i.e. the LS estimator dominates QML under squared error loss). However, the difference declines from about 0.4% in the v = 5 case to zero as v increases. The convergence of the relative variance and risk estimates also implicitly illustrates the unbiasedness of the QML estimator. Thus, the LS estimator has only a slight advantage over the QML estimator when ˆ is the MVUE. The simulated outcomes from the centered and scaled Chi-square(3) distribution are presented at the bottom of Table 1. Under this scenario, the QML estimator is biased but the LS estimator is unbiased. Accordingly, the unreported bias estimates for the QML estimator are roughly two to four times larger (in absolute magnitude) than the LS bias estimates, but the magnitudes are still quite small. The simulated variance and quadratic risk of the QML estimator are about 3–6% larger than the LS estimates in the case v = 10. As expected, the LS estimator risk dominates the biased QML estimator, but the relative differences in sampling performance decline as v increases. As before, the QML and LS estimators become more similar as v increases, and the relative risk advantage of LS is only about 1% for the case v = 20. The finite sample properties of ˆ are widely known (the unbiasedness and covariance properties are stated above). Under the error processes with unitary variance, the asymptotic distribution of the LS estimator is a ˆ ∼ N(, (x x)−1 )
(22)
which is an exact distribution under the standard normal error process. To evaluate ˜ we conduct Monte Carlo experiments based on the finite sample properties of , the QML estimator with smallest bounds for each error process (i.e. v = 3 for the U(0, 8) case, v = 5 for the N(0, 1) case, and v = 10 for the Chi-square case). Based on the findings stated in Table 1, the QML estimators with small v behave least like the LS estimator and provide good cases for contrast. We also limit the sample sizes to n = 10 and n = 50 to further restrict our attention to the finite sample ˜ properties of .
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Table 2. Simulated Mean and Variance of the QML Estimator. Sample Moment
Location Parameters 1 = 1
2 = 2
Quadratic Risk
3 = 3
4 = 4
Centered and scaled U(0, 8) errors, v = 3, n = 10 Mean 0.9998 1.9995 Variance 0.1328 0.4677
2.9996 0.0728
3.9865 0.2306
0.9041
Centered and scaled U(0, 8) errors, v = 3, n = 50 Mean 0.9978 1.9943 Variance 0.0168 0.0149
3.0042 0.0119
3.9988 0.0176
0.0613
Standard normal errors, v = 5, n = 10 Mean 0.9894 1.9990 Variance 0.1119 0.0776
2.9967 0.0647
3.9887 0.0820
0.3365
Standard normal errors, v = 5, n = 50 Mean 1.0029 2.0072 Variance 0.0223 0.0209
3.0008 0.0252
4.0059 0.0319
0.1004
Centered and scaled chi-square(3) errors, v = 10, n = 10 Mean 1.0054 2.0130 3.0022 Variance 0.1173 0.3102 0.1386
4.0131 0.1805
0.7470
Centered and scaled chi-square(3) errors, v = 10, n = 50 Mean 1.0143 1.9996 2.9979 Variance 0.0235 0.0187 0.0252
3.9995 0.0192
0.0868
The estimated mean and variance of each ˜ k from the second set of Monte Carlo exercises are reported in Table 2. The simulation results do not provide evidence of any systematic or large bias, and the variance estimates decline with n (as expected). Thus, the stated results demonstrate (but do not prove) the convergence in probability property of the QML estimator. We also report the simulated quadratic risk of ˜ for each case. Although the relatively large simulated quadratic risk for the U(0, 8) model with n = 10 is unexpected, we note that the QML estimator has smaller risk under the U(0, 8) case (i.e. symmetric and bounded) when n = 50. The other error processes represent departures from the model assumptions, and we expect ˜ to exhibit relatively larger quadratic risk in these cases (as observed). In Fig. 2, we plot the kernel density estimates of the sampling distributions of ˜ 2 based on the Monte Carlo replicates. The solid line is the estimated density function, the dashed line is the normal asymptotic distribution, and the true value of the parameter (2 = 2) is indicated with the vertical line. The kernel density estimates (based on the Epanechnikov kernel function with optimal bandwidth)
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Fig. 2. Kernel Density Estimates of the ˜ 2 Sampling Distribution.
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exhibit roughly Gaussian and unbiased character, even in the small sample cases (n = 10). Further, the slight differences between the kernel density estimates and the asymptotic distributions diminish as the sample size increases to n = 50. Thus, the sampling performance of the QML estimator appears to be well approximated by the large sample results stated in the preceding section, even if v or n are small.
5. EXTENSIONS The proposed estimator was originally developed by Boschmann, Preckel and Eales (2001) for an AIDS model of an aggregate demand system, and the QML estimator is used as an alternative to the seemingly unrelated regression (SUR) estimator. They refer to this estimation procedure as the maximum entropy SUR (MESUR) estimator. The sampling results presented in Section 3 may be directly extended to the MESUR case if we view the regression model (1) as the feasible GLS transformation of the stacked SUR model equations ˜ −1/2 Y = ˜ −1/2 x + ˜ −1/2 ⑀ ⇒ Y ∗ = x ∗  + ⑀∗ ,
(23)
where ≡ ( ⊗ I n ) and is the contemporaneous covariance matrix of ⑀. Under additional regularity conditions, we can also show that the MESUR estimator proposed by Boschmann, Preckel and Eales (2001) is unbiased, consistent, ˜ and asymptotically normal. In particular, the unbiasedness property holds if is an even function of i (), which is true for the standard estimators of the contemporaneous covariance matrix. For more details, refer to Assumption A3 associated with Andrews’ Theorem 1 (Andrews, 1986). Boschmann, Preckel and Eales (2001) also extend the QML estimation procedure to include bounds on the location parameters, and the bounds are selected as in the generalized maximum entropy (GME) procedure described by Golan, Judge and Miller (1996). Accordingly, Boschmann, Preckel and Eales (2001) refer to their GME–based estimator of the SUR model as GMESUR. In general, QML estimators based on GME–type parameter bounds are biased because Assumption A1 fails unless the parameter bounds are centered on the true parameter vector. As noted by Andrews (1986, p. 691), the bias property is also shared by most Bayesian point estimators (i.e. those based on proper prior distributions) and other shrinkage estimators. We must also assume that the true location vector 0 is interior to the parameter bounds in order to establish consistency and asymptotic normality of GME–type estimators. However, Bayes, GME, and other shrinkage estimators may exhibit smaller variance and risk than the unbiased alternatives if the parameter bounds or other prior information specifications are appropriately selected.
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6. CONCLUSION The QML estimator proposed in this paper is based on a reparameterized version of the linear regression model. The error outcomes are stated as convex combinations of the error bounds, and we use the method of maximum entropy to derive the most uniform set of weights on these discrete points given the observed data and the location parameters (). We interpret the concentrated objective function for the maximum entropy problem as a quasi–log likelihood function, and the QML estimator is the vector ˜ that maximizes m(). The class of QML estimators for linear regression models with bounded and symmetric error distributions includes the MESUR estimator introduced by Boschmann, Preckel and Eales (2001). Under the stated model assumptions, we show that the QML estimator is unbiased, consistent, and asymptotically normal. We use a set of Monte Carlo exercises to demonstrate the comparative sampling properties of the QML and LS estimators. The LS estimator has smaller risk than QML under the normal error process (as expected) and the skewed and unbounded Chi-square error process, which do not satsify the stated model assumptions. The QML estimator dominates LS under the centered uniform error process, which is bounded and symmetric. The simulation results point to an interesting empirical issue regarding the optimal value of v. If the underlying error process is bounded and symmetric, the results indicate that v should be relatively close to (but not smaller than) the actual error bound. Otherwise, the QML estimator behaves more like the LS estimator, and the relative risk gains diminish. If the actual error process is unbounded or skewed, the LS estimator is clearly preferred to QML, but the relative advantage is slight as v increases. Thus, the QML estimator appears to be relatively efficient for error processes satisfying the stated regularity conditions and may be relatively robust if the error distribution deviates from the assumptions.
REFERENCES Andrews, D. (1986). A note on the unbiasedness of feasible gls, quasi-maximum likelihood, robust, adaptive, and spectral estimators of the linear model. Econometrica, 54, 687–698. Bai, E.-W., Nagpal, K., & Tempo, R. (1996). Bounded error parameter estimation: Noise models and recursive estimation. Automatica, 32, 985–999. Boschmann, A., Preckel, P., & Eales, J. (2001). Entropy-based seemingly unrelated regressions. Working Paper, Purdue University. Chan, K., & Tong, H. (1994). A note on noisy chaos. Journal of the Royal Statistical Society, Series B, 56, 301–311. Golan, A., Judge, G., & Miller, D. (1996). Maximum entropy econometrics: Robust estimation with limited information. Chichester, UK: Wiley.
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Gstach, D. (1998). Another approach to data envelopment analysis in noisy environments: DEA+. Journal of Productivity Analysis, 9, 161–176. Jaynes, E. (1957a). Information theory and statistical mechanics I. Physical Review, 106, 620–630. Jaynes, E. (1957b). Information theory and statistical mechanics II. Physical Review, 108, 171–190. Preckel, P. (2001). Least squares and entropy: A penalty function perspective. American Journal of Agricultural Economics, 83, 366–377. Royden, H. (1988). Real analysis. Englewood Cliffs, NJ: Prentice-Hall. White, H. (1994). Estimation, inference, and specification analysis. New York, NY: Cambridge University Press.
CONSISTENT QUASI-MAXIMUM LIKELIHOOD ESTIMATION WITH LIMITED INFORMATION Douglas Miller and Sang-Hak Lee ABSTRACT In this chapter, we use the minimum cross-entropy method to derive an approximate joint probability model for a multivariate economic process based on limited information about the marginal quasi-density functions and the joint moment conditions. The modeling approach is related to joint probability models derived from copula functions, but we note that the entropy approach has some practical advantages over copula-based models. Under suitable regularity conditions, the quasi-maximum likelihood estimator (QMLE) of the model parameters is consistent and asymptotically normal. We demonstrate the procedure with an application to the joint probability model of trading volume and price variability for the Chicago Board of Trade soybean futures contract.
1. INTRODUCTION The solutions to many economic problems require information about the joint stochastic character of observable economic outcomes. Although a wide range of parametric joint probability models are available for empirical economic research, researchers may encounter problems for which the available distributions do not Maximum Likelihood Estimation of Misspecified Models: Twenty Years Later Advances in Econometrics, Volume 17, 149–164 Copyright © 2003 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(03)17007-7
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adequately represent the assumed or observed characteristics of the economic outcomes. In such cases, one of the key tasks researchers face is to specify an approximate joint probability model based on their partial or limited information. The purpose of this chapter is to describe a method for deriving joint probability models based on limited available information about the marginal distributions and the marginal and joint moments of the random variables. Due to our lack of complete knowledge, the model may be misspecified, and we view the resulting probability models as quasi-density approximations to the true but unknown probability model. Accordingly, we view the parameter estimators for these approximate models as quasi-maximum likelihood (QML) estimators. For present purposes, we consider a multivariate economic process that generates outcomes for m random variables, X 1 , . . . , X m . The problem is to derive the joint probability density function f (x 1 , . . . , x m ) from limited information about the random variables. The first source of information is the set of marginal distributions, G j (x j , ␣j ), where ␣j is a vector of unknown model parameters. We view the associated marginal density functions g j as potentially misspecified, but we use the approximations as reference densities for the problem of recovering f. If the random variables are mutually independent, an approximation to the joint density function may be formed as the product of the marginal reference density functions g(x 1 , . . . , x m ; ␣1 , . . . , ␣m ) =
m
g j (x j , ␣j )
(1)
j=1
To simplify notation, we suppress the parameters of the marginal distributions unless required for our discussion. The second source of information is a set of moment conditions that represent the joint stochastic character of the multivariate process. The vector of moment conditions is stated as = ␥(x 1 , . . . , x m ) f (x 1 , . . . , x m ) dx (2) and may include marginal moments (e.g. mean or variance) and joint moments (e.g. correlation or covariance). In the following section, we use information theoretic methods based on Jaynes’ maximum entropy principle (Jaynes, 1957a, b) to derive a quasi-density function f from the reference marginals (1) and the moment conditions (2). We then discuss the asymptotic properties of the associated QML estimators of the model parameters and present an application of the entropy-based procedure to daily price variability and trading volume for a soybean futures contract.
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2. MINIMUM CROSS-ENTROPY JOINT DISTRIBUTIONS The entropy-based approach to univariate density recovery problems has been examined in the econometrics literature by Zellner and Highfield (1988), Ryu (1993), and Ormoneit and White (1999). In this paper, we extend the multivariate density recovery procedure outlined by Miller and Liu (2002, Section 4.1) in the context of QML estimation. The information theoretic criterion used to choose the joint density function f is the mutual information functional f (x 1 , . . . , x m ) I( f : g) = f (x 1 , . . . , x m ) ln dx (3) g 1 (x 1 )g 2 (x 2 ) . . . g m (x m ) If the marginal distributions G j are correctly specified and the random variables are mutually independent, then f and g contain the same information and I( f : g) = 0. Otherwise, f contains information distinct from g, and the two density functions diverge such that I( f : g) > 0. It is widely known that I( f : g) is not a true distance function or measure because I( f : g) = I(g : f ). Consequently, the mutual information functional is known as a directed divergence or pseudo-distance measure of the information contained in f relative to g. However, the mutual information functional is a useful modeling criterion despite the asymmetric character of I( f : g) because g is a fixed reference distribution for the problem of recovering f. The minimum cross-entropy approach to solving the density recovery problem from the available information is to choose f to minimize I( f : g) subject to the moment conditions (2) and the required additivity condition 1 = f (x 1 , . . . , x m ) dx (4) Following Eqs (16) and (17) presented by Miller and Liu, the calculus of variations solution to the minimum cross-entropy problem is ˆ fˆ (x 1 , . . . , x m ) = g 1 (x 1 ) . . . g m (x m ) exp (ˆ ␥(x 1 , . . . , x m ))−1 ()
where the normalizing constant or partition function is ˆ ≡ g 1 (x 1 ) . . . g m (x m ) exp (ˆ ␥(x 1 , . . . , x m )) dx ()
(5)
(6)
The vector ˆ is the set of optimal Lagrange multipliers for the moment constraints (2). If the reference distribution (1) satisfies the moment conditions (2), the constraints are non-binding such that ˆ = 0 and fˆ = g. Otherwise, the exponential term in Eq. (5) acts as a density weighting function that exponentially tilts the reference distribution to reflect the moment constraints.
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The computational burden of the minimum cross-entropy density recovery problem may be reduced by concentrating the problem. By substitution of the minimum cross-entropy solution (5) back into the Lagrange expression, we derive the concentrated objective function M() = − ln[()]
(7)
which is strictly concave in due to the saddle-point property of the constrained estimation problem. Thus, we can compute the optimal Lagrange multipliers ˆ by unconstrained maximization of M() with respect to . The gradient vector of M is (8) ∇λ M() = − ␥(x 1 , . . . , x m ) f (x 1 , . . . , x m ) dx and the Hessian matrix is ∇λλ M() = − ␥(x 1 , . . . , x m )␥(x 1 , . . . , x m ) f (x 1 , . . . , x m ) dx
(9)
The partition function, gradient vector, and Hessian matrix may be evaluated with numerical integration software, and we may compute ˆ with familiar iterative optimization algorithms (e.g. Newton-Raphson, quasi-Newton, or steepest descent).
2.1. Asymptotic Properties In practice, the parameter vectors ␣j in Eq. (1) and the moment vector in Eq. (2) will not be known and must be estimated from the available sample information. Although fˆ may be misspecified relative to the true probability model f, we can derive the asymptotic sampling properties of ˆ from conditions on the sampling ˆ j and . ˆ The key assumption in this derivation properties of the estimators ␣ is the existence of true parameter vectors ␣j0 for the marginal quasi-density approximations (i.e. reference densities) and a vector of true moments 0 for (2). Then, there exists a vector of Lagrange multipliers 0 for the minimum cross-entropy distribution based on the reference distributions G j (x j , ␣j0 ) and moments 0 . As a result, we may combine the arguments presented by Miller and Liu with the QMLE and 2SQMLE theorems in White (1994) to derive conditions under which ˆ is a consistent and asymptotically normal estimator. First, we consider the case in which the reference densities are not functions of estimated parameters (e.g. uniform reference densities). Miller and Liu (Section 4.2) sketch the arguments required to prove consistency and asymptotic ˆ In particular, normality of ˆ given about the sample moments, . √ assumptions ˆ p 0 and n( ˆ − 0 )→d N(0, ) (see Assumptions A1–A3 stated by if →
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√ Miller & Liu), we can use White’s Theorem 3.5 to prove n-consistency such p ˆ ˆ 0 . Under the conditions of White’s Theorem 6.10, ˆ is asymptotically that → normal as √ d n(ˆ − 0 )→N(0, −1 −1 ) where ≡ ∇λλ M()|λ=λ0 (10) In this case, the limiting covariance matrix represents the variability in ˆ as it is ˆ mapped directly from . Next, we consider the case in which the true parameters of the marginal reference densities ␣0 = (␣10 , . . . , ␣m0 ) are estimated in a first-stage QML estimation problem. In this sense, we view the optimal Lagrange multipliers ˆ as the second-stage component of a two-stage quasi-maximum likelihood estimator ˆ m ) is consistent such that ␣→ ˆ p ␣0 and ˆ = (␣ ˆ 1 , . . . , ␣ (2SQMLE). Suppose ␣ asymptotically normal as √ d ˆ − ␣0 )→N(0, −1 −1 ) n(␣ (11) √ Under the conditions of White’s Theorem 3.10, ˆ is n-consistent such that ˆ p 0 . White’s Theorem 6.11 provides conditions for the asymptotic normality → ˆ and the limiting covariance matrix is of , −1 −1 − −1 A−1 B 1 −1 − −1 B 2 −1 A −1 + −1 A−1 −1 A −1
(12)
where A ≡ ∇αλ M()|α=α0 ,λ=λ0 and B 1 and B 2 are defined in the theorem. In ˆ in this case, the sampling variability in the data influences ˆ directly through ˆ in the reference densities. the moment constraints and indirectly through ␣ The efficiency of ˆ and other QML estimators is largely dependent on the accuracy of the approximate model specification. For example, Engle and Gonzalez-Rivera (1991) note that QML estimators based on the multivariate normal quasi-likelihood function may be relatively inefficient if the true probability model is distinctly non-Gaussian. Accordingly, we expect that the relative asymptotic efficiency of the entropy-based QML estimators should improve as the approximations provided by the marginal quasi-density or reference density functions approach the true marginal distributions. However, we leave further consideration of the asymptotic efficiency issue to later research.
2.2. Copula Probability Models An alternative to the entropy-based quasi-density models is the class of copulabased joint probability models. In words, a copula is a function that may be used
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to form a joint probability model from the marginal distributions of a group of dependent random variables. The key property of a copula-based joint probability model is that the marginal distributions are fixed – the degree of dependence imposed on the component random variables may change, but the marginal distributions associated with the joint probability model remain the same. Thus, if researchers can specify good approximations to the marginal distributions but do not know the joint distribution, they can use a copula to build a joint probability model that has these exact marginal distributions and imposes dependence among the component random variables. Formally, a copula is a function C(u 1 , . . . , u m ; ) : [0, 1]m → [0, 1]
(13)
that maps the uniform marginal distribution functions for each u j ∼ U(0, 1) where j = 1, . . . , m to an m-variate joint cumulative distribution function (CDF). The parameters control the degree of dependence among the m variables in the multivariate process, and different subclasses of copula models are based on alternative measures of covariation (e.g. the correlation coefficient) or concordance (e.g. Spearman’s rho and Kendall’s tau). The copula model may be used to construct joint distributions with non-uniform marginals, G 1 (x 1 ), . . . , G m (x m ), under the converse to the probability integral transformation theorem. The CDF transformation of the marginal random variables satisfies G j (X j ) ∼ U(0, 1), and a joint distribution with these marginals is formed by substitution F (x 1 , . . . , x m ) = C(G 1 (x 1 ), . . . , G m (x m ); )
(14)
For example, one of the most widely used copula models is the multivariate normal copula F (x 1 , . . . , x m ) = m ( −1 (G 1 (x 1 )), . . . , −1 (G m (x 1 )); )
(15)
where m is the multivariate normal CDF with null mean and covariance matrix and −1 is the inverse standard normal CDF (i.e. the N(0, 1) quantile function). Lee (1983) presented one of the earliest applications of copula models in econometric research, and several recent articles and books summarize the important copula classes and their properties. Two prominent and comprehensive references include the books by Nelsen (1999) and by Joe (1997), and Miller and Liu examine the links between the entropy literature and the copula literature. If the marginal distributions G j (x j , ␣j ) are functions of unknown parameters ␣j , we can maximize the quasi-log likelihood function to estimate each ␣j plus the dependence parameters . As noted by Joe (Section 10.1), the QML estimation procedure may be based on simultaneous choice of (␣1 , . . . , ␣m ) and or a two-stage estimation procedure (i.e. estimate given the first-stage QML
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ˆ 1, . . . , ␣ ˆ m ) and ˆ estimates of each ␣j ). In either case, the QML estimators (␣ are consistent and asymptotically normal under the conditions provided by White (1994). As noted in the discussion of the minimum cross-entropy models, the asymptotic efficiency of the entropy-based approach depends on the accuracy of the quasi-density approximations. Given that the copula models have fixed marginal distributions, the copula-based QML estimators should be relatively efficient if the marginal distributions G j (x j , ␣j ) are correctly specified. Although the minimum cross-entropy probability model presented in the preceding section will not have fixed marginals, the entropy-based approach has some practical advantages relative to the copula-based models. First, the range of dependence that may be imposed on the component random variables under a copula model may be limited, and the upper and lower limits are provided by the Fr´echet bounds (see Theorem 3.1 in Joe (1997) or Section 2.2 in Nelsen, 1999 for details). Miller and Liu show that the entropy-based joint probability models are less restricted by the Fr´echet bounds on joint dependence because the marginals of (5) are not fixed. Second, the entropy-based approach may be easier to implement because we only require the marginal density functions for estimation purposes. Many copula-based joint density functions are explicit functions of the cumulative distribution functions, which may not exist in closed-form and may be difficult to use in computation. For example, some bivariate distribution functions in the Archimedean copula class may be stated as C(u 1 , u 2 ) = −1 ((u 1 ) + (u 2 ))
(16)
where is known as an additive generating function. For all Archimedean class members with twice continuously differentiable additive generating functions, Nelsen (p. 103) shows that the joint PDF is f (u 1 , u 2 ) = −
(C(u 1 , u 2 )) (u 1 ) (u 2 ) [ (C(u 1 , u 2 ))]3
(17)
where and are the first and second derivatives of . Note that the joint PDF requires the marginal CDF’s as arguments to this density function. Third, the entropy-based joint probability model is readily suited for numerical simulation of integral functions by importance sampling from the reference marginal distributions (see the discussion surrounding Eq. (23) in Miller & Liu and the comments in the following section). Finally, although the marginal distributions of the entropy-based joint density functions are not fixed, Miller and Liu show that the minimum cross-entropy models may have marginal distributions that are very close to the reference marginals. In such cases, the efficiency of the QML estimators for the copula-based and entropy-based models should be similar if the marginal distributions are correctly specified.
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3. SOYBEAN FUTURES PRICE VARIABILITY-VOLUME To demonstrate the entropy-based method for deriving joint quasi-density functions, we consider the problem of modeling futures price variability and trading volume. Tauchen and Pitts (1983) derive the joint probability distribution of daily price changes P t = P t − P t−1 and daily trading volume V t conditional on the number of traders I t . The observable variables are assumed to be distributed as P t ∼ N(0, 21 I t ) and V t ∼ N(2 I t , 22 I t ) and are conditionally independent. The correlation between futures price variability and volume is driven by the number of traders. Given that I t is unobservable, Tauchen and Pitts represent the number of traders with a lognormal probability model and integrate with respect to this variable to derive the joint distribution, f ( P t , V t ). The purpose of the Tauchen and Pitts study is to explain a potential anomally in the observed behavior of the U.S. Treasury bill futures market. After the futures contract was introduced on January 6, 1976, trading volume expanded but price variability (as measured with the sample variance of P t over subsamples) declined, which contradicts the established theory of variability-volume behavior on speculative markets. Tauchen and Pitts use the estimated joint distribution to derive the conditional variance of P t given V t 2 x f (x, V t ) dx var[ P t |V t ] = (18) f (x, V t ) dx where E[ P t ] = 0 is imposed on their model. The conditional variance function is increasing and convex, and the observed price and volume relationship may be explained by considering the impact of increasing numbers of traders. For our purposes, we consider the recent variability-volume behavior of the nearby soybean futures contract traded on the Chicago Board of Trade (CBOT). The sample period is August 1, 1995 to August 20, 1999, and the sample consists of 988 daily observations on the closing price P t and the daily trading volume V t . We also divide the sample into two regimes – the first ends on Friday, September 19, 1997, and the second regime begins on Monday, September 22, 1997. From August 1995 to September 1997, soybean trading volume and price variability were relatively large due to recent declines in soybean acreage, poor crop yields, historically low grain stocks, and favorable economic growth in importing nations. During the fall harvest period in 1997, the pattern reversed and trading volume and price volatility declined due to increased soybean acreage resulting from changes in the federal farm programs, improved yields and stock levels, and declining demand from Asian soybean customers following the East Asian financial crisis. In particular, the observed average trading volume declined by over 50%, and the sample
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variance of daily price changes declined by nearly one-third. As implied by the work of Tauchen and Pitts, we should not use the observed changes in volume and price variability across major market regimes to represent the volume-variability relationship within a given regime. Our objective is to compute separate estimates of the conditional variance function (18) for the 1995–1997 and 1997–1999 subsample periods. One practical difficulty associated with making direct use of the Tauchen and Pitts model f ( P t , V t ) is that the observed behavior of daily trading volume in the CBOT soybean contract is distinctly non-normal. To examine the distributional characteristics of daily trading volume, we estimated ARCH models of V t and conducted normality tests on the standardized model residuals. We find that the tests for excess skewness and kurtosis strongly reject the Gaussian null hypothesis. Further, the evidence from additional diagnostic tests suggests that a lognormal model is an appropriate approximation for the distribution of daily trading volume on the CBOT soybean futures contract.
3.1. Model Estimation Based on the prelimary estimation results, our minimum cross-entropy joint probability model is based on a lognormal reference density for V t and a normal reference density for P t . The moderate degree of correlation between the two random variables is imposed with the covariance constraint 212 = (x 1 − 1 )(x 2 − 2 ) f (x 1 , x 2 ) dx 1 dx 2 (19) As noted by Zellner and Highfield and by Ormoneit and White, computational algorithms used to derive minimum cross-entropy estimates of continuous density functions may be hampered by numerical problems such as exponential overflow. One effective way to avoid computational difficulties is to rescale the variables before solving the density estimation problem. In our application, we rescale the price change data as p t = (P t − P t−1 )/10 to represent ten cent price increments. We also rescale the volume data to represent 100,000 contract units (vt = V t /100,000). After rescaling the daily price change and volume data, we did not experience any computing problems associated with exponential overflow. The observed marginal sample mean and variance for p t and vt plus the sample covariance statistics for the two sample regimes are reported in Table 1. Although the sample means for p t are not significantly different from zero, we do not impose the E[ p t ] = 0 restriction on the probability model in order to demonstrate the impact this has on the associated Lagrange multiplier (Tauchen and Pitts impose
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Table 1. Sample Moments and Estimated Lagrange Multipliers. Population Moment
August 1, 1995 to September 19, 1997 Sample Moment
E[ p t ] E[vt ] var[ p t ] var[vt ] cov[ p t , vt ]
0.0287536 (0.04973) 0.8087219 (0.03141) 1.3355817 (0.14677) 0.5326572 (0.07720) −0.0755015 (0.01405)
September 22, 1997 to August 20, 1999
Lagrange Multiplier −7.8384 ×10−5 (0.02635) −1.4246 ×10−5 (0.03612) −0.003019 (0.02619) −0.007570 (0.01053) −0.106902 (0.01642) n = 540
Sample Moment −0.06328125 (0.04623) 0.3718599 (0.02670) 0.9573735 (0.33389) 0.3194901 (0.06615) −0.00176684 (0.04587)
Lagrange Multiplier −8.4387 ×10−6 (0.03187) −1.1009 ×10−7 (0.03571) −5.3481 ×10−6 (0.01249) −1.6223 ×10−5 (0.00297) −0.005823 (0.14791) n = 448
Note: Approximate standard errors appear in parentheses.
this restriction on their model). Also, the sample covariance for the 1997–1999 subsample is not significantly different from zero, but we retain this constraint in the minimum cross-entropy problem. We use the GAUSS programming environment to solve the minimum cross-entropy problems, and the integral expressions are evaluated with the Gaussian-Legendre quadrature procedure for double integrals (intquad2). We provide an abridged version of the GAUSS code at the end of the chapter to illustrate the basic programming steps (secondary operations such as minor rescaling to account for small inaccuracies in the integral approximations, required inputs for the quadrature procedure, and iteration counters are omitted). Given starting values 0 = 0, we use a steepest descent algorithm to iteratively compute the minimum cross-entropy estimates ˆ based on the marginal sample mean and variance and the sample covariance statistic. The convergence criterion is the squared norm of the gradient ⑀, and the stopping rule is ⑀ < 1e − 12. The steepest descent algorithm converged to the minimum cross-entropy solutions for the 1995–1997 and 1997–1999 cases in less than 10 seconds on a 1.6 GHz PC.
3.2. Estimation Results The estimated Lagrange multipliers ˆ for the 1995–1997 and 1997–1999 regimes are stated in Table 1 with the estimated asymptotic standard errors. In both cases, the Z-test statistics for H 0 : j = 0 for j = 1, . . . , 4 (i.e. the marginal mean and
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variance constraints) are small and do not give reason to reject any of the null hypotheses. The Lagrange multiplier for the covariance constraint ˆ 5 in the first subsample period is large relative to its standard error (Z = −6.51), and we reject H 0 for this moment restriction. In contrast, the value of ˆ 5 for the second subsample regime is not significantly different from zero, which is logically consistent with our earlier finding that the sample covariance statistic is not significantly different from zero. To graphically illustrate the estimated joint probability model, we present the isodensity contour plots of the joint reference density function (dashed lines) and the estimated minimum cross-entropy density function (solid lines) for the 1995–1997 case in Fig. 1. The reference density is the product of the marginal densities for p t and vt , and the isodensity contours are symmetric about the
p t = 0 line. As discussed earlier in this chapter, the minimum cross-entropy procedure exponentially tilts the reference density to satisfy the moment constraints. In this case, p t and vt exhibit moderate negative correlation, and the mass under the reference density is shifted counter-clockwise to achieve a minimum cross-entropy density function with this property. Given the fitted joint probability models for p t and vt , we use importance sampling to estimate the conditional variance function for the daily price change (18). Following Miller and Liu, the importance sampling algorithm is initiated by drawing N pseudo-random outcomes of p t from the N(0, σˆ 12 ) reference marginal distribution. For a given value of vt = v, the estimated value of (18) is v ar[ p t |vt ] =
N −1
N
2 ˆ i=1 ( p i − E[ p t ]) exp ( ␥( p i , v)) ˆ N −1 N i=1 exp ( ␥( p i , v))
(20)
For present purposes, we use N = 5,000 importance samples and trading volume levels v ∈ {0.05, 0.10, . . . , 4}, which coincides with the observed range of daily volume. The estimated conditional variance functions v ar[ p t |vt ] for 1995–1997 (solid line) and 1997–1999 (dashed line) are presented in Fig. 2. First, we note that the conditional variance functions are roughly centered on the sample variance of daily price changes (ˆ 21 = 1.336 for 1995–1997 and ˆ 21 = 0.957 for 1997–1999 from Table 1). Second, the conditional variance function for the earlier subsample regime is increasing and convex and has similar appearance to the functions for the T-bill futures contract presented by Tauchen and Pitts. As expected, an increase in trading volume increases the expected variability of soybean futures prices. Finally, the estimated conditional variance function for the second subsample regime lies below the 1995–1997 function and has a very slight positive slope (perhaps not visually evident in the figure). Recall that the sample covariance
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Fig. 1. Isodensity Plots for the 1995–1997 Soybean Data.
between p t and vt was not significantly different from zero for this sample period. Accordingly, the Lagrange multiplier for the covariance constraint ˆ 5 is not significantly different from zero in the second regime. Thus, the estimated joint probability model exhibits limited joint character, and the results imply that daily
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Fig. 2. Conditional Variability in CBOT Soybean Futures Prices.
trading volume does not have a significant impact on daily soybean futures price variability from 1997 to 1999. The result is logically consistent with the Tauchen and Pitts model in which p t and vt are unconditionally independent (i.e. the number of active traders has little or no impact on the volume-variability relationship).
4. SUMMARY In this chapter, we present a minimum cross-entropy method for deriving quasidensity approximations for joint probability models. Given information about the marginal distributions and the joint moments of the multivariate process, we select the joint density function that minimizes the mutual information functional subject to moment and additivity constraints. We compare the advantages and disadvantages of the entropy-based procedure to copula-based joint probability models. Under suitable regularity conditions, the QML estimator of the model parameters is consistent and asymptotically normal. We apply the procedure to the problem of estimating the conditional variance of daily soybean futures prices given the level of daily trading volume. The minimum cross-entropy approach may be extended in numerous ways, and there are several unresolved research
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questions. For example, we note that one of the key topics for further research is the relative efficiency of the entropy-based QML estimators.
5. GAUSS PROGRAM /* Enter moment constraints */ mu = {0.0287536, 0.8087219, 1.3355817, 0.5326572, -0.0755015}; /* Set integration bounds and starting values */ bdsdp = {10, -10}; bdsv = {30, 0}; lhat = zeros(5,1); muhat = zeros(5,1); eps = 1; step = 10; /* Steepest descent algorithm */ do while eps > 1e-12; temp = intquad2(&denom, bdsx, bdsy); muhat[1] = intquad2(&expdp, bdsdp, bdsv)/temp; muhat[2] = intquad2(&expv, bdsdp, bdsv)/temp; muhat[3] = intquad2(&vardp, bdsdp, bdsv)/temp; muhat[4] = intquad2(&varv, bdsdp, bdsv)/temp; muhat[5] = intquad2(&covar, bdsdp, bdsv)/temp; dm = mu - muhat; lhat = lhat + dm/step; eps = dm’dm; endo; /* Normal and Lognormal PDF’s */ proc npdf(x); retp(exp(-(x-mu[1]).∗(x-mu[1])/ (2∗mu[3]))/ sqrt(2∗pi∗mu[3])); endp; proc lnpdf(y); local a, b; b = ln(mu[2]∗mu[2] + mu[4]) - 2∗ln(mu[2]); a = ln(mu[2]) - b/2;
Consistent Quasi-Maximum Likelihood Estimation with Limited Information
retp(exp(-(ln(y)-a).∗(ln(y)-a)/(2∗b))./ (y∗sqrt(2∗pi∗b))); endp; /* Kernel function */ proc denom(x, y); local arg; arg = x∗lhat[1]+y∗lhat[2]+(x-mu[1]) .∗(x-mu[1])∗lhat[3]; arg = arg + (y-mu[2]).∗(y-mu[2])∗lhat[4]; arg = arg + (x-mu[1]).∗(y-mu[2])∗lhat[5]; retp(npdf(x).∗lnpdf(y).∗exp(arg)); endp; /* Expected value of dP */ proc expdp(x, y); retp(x.∗denom(x, y)); endp; /* Expected value of V */ proc expv(x, y); retp(y.∗denom(x, y)); endp; /* Variance of dP */ proc vardp(x, y); retp((x-mu[1]).∗(x-mu[1]).∗denom(x, y)); endp; /* Variance of V */ proc varv(x, y); retp((y-mu[2]).∗(y-mu[2]).∗denom(x, y)); endp; /* Covariance of dP and V */ proc covar(x, y); retp((x-mu[1]).∗(y-mu[2]).∗denom(x, y)); endp;
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REFERENCES Engle, R., & Gonzalez-Rivera, G. (1991). Semiparametric arch models. Journal of Business and Economic Statistics, 9, 345–359. Jaynes, E. (1957a). Information theory and statistical mechanics I. Physical Review, 106, 620–630. Jaynes, E. (1957b). Information theory and statistical mechanics II. Physical Review, 108, 171–190. Joe, H. (1997). Multivariate models and dependence concepts. New York, NY: Chapman & Hall. Lee, L.-F. (1983). Generalized econometric models with selectivity. Econometrica, 51, 507–512. Miller, D., & Liu, W.-H. (2002). On the recovery of joint distributions from limited information. Journal of Econometrics, 107, 259–274. Nelsen, R. (1999). An introduction to Copulas. New York, NY: Springer-Verlag. Ormoneit, D., & White, H. (1999). An efficient algorithm to compute maximum entropy densities. Econometric Reviews, 18, 127–140. Ryu, H. (1993). Maximum entropy estimation of density and regression functions. Journal of Econometrics, 56, 397–440. Tauchen, G., & Pitts, M. (1983). The price variability-volume relationship on speculative markets. Econometrica, 51, 485–506. White, H. (1994). Estimation, inference, and specification analysis. New York, NY: Cambridge University Press. Zellner, A., & Highfield, R. (1988). Calculation of maximum entropy distributions and approximation of marginal posterior distributions. Journal of Econometrics, 37, 195–209.
AN EXAMINATION OF THE SIGN AND VOLATILITY SWITCHING ARCH MODELS UNDER ALTERNATIVE DISTRIBUTIONAL ASSUMPTIONS Mohamed F. Omran and Florin Avram ABSTRACT This paper relaxes the assumption of conditional normal innovations used by Fornari and Mele (1997) in modelling the asymmetric reaction of the conditional volatility to the arrival of news. We compare the performance of the Sign and Volatility Switching ARCH model of Fornari and Mele (1997) and the GJR model of Glosten et al. (1993) under the assumption that the innovations follow the Generalized Student’s t distribution. Moreover, we hedge against the possibility of misspecification by basing the inferences on the robust variance-covariance matrix suggested by White (1982). The results suggest that using more flexible distributional assumptions on the financial data can have a significant impact on the inferences drawn.
1. INTRODUCTION There is growing evidence that the response of current volatility to past shocks is asymmetric with negative shocks having more impact on current volatility than Maximum Likelihood Estimation of Misspecified Models: Twenty Years Later Advances in Econometrics, Volume 17, 165–176 Copyright © 2003 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(03)17008-9
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positive shocks (see Engle & Ng, 1993; Fornari & Mele, 1997). One explanation for this asymmetry is the leverage effect discussed in Black (1976) and Christie (1982). The leverage effect implies that a reduction in the stock price will lead to an increase in the debt to equity ratio measured in terms of market values which might cause an increase in the riskiness of the firm’s stocks, and subsequently higher returns’ volatility. In order to model this asymmetry in the response of the conditional volatility to past positive and negative returns shocks, Glosten et al. (1993) proposed a model (GJR hereafter) essentially equivalent to the following: r t = 0 + 1 r t−1 + u t 2t = + ␣u 2t−1 + 2t−1 + ␦0 u 2t−1 s t−1 where rt denotes the logarithmic returns, the innovations ut follow a distribution with mean 0 and variance 2t , and s t = 1 if u t > 0 and s t = −1 if u t < 0. Engle and Ng (1993) provided evidence that the GJR model was the best parametric model available at the time to model the asymmetry in the volatility of stock returns. Fornari and Mele (1997) introduced a more general Sign and Volatility-Switching ARCH model (VS hereafter): 2t = + ␣u 2t−1 + 2t−1 + (␦0 u 2t−1 − ␦1 2t−1 − ␦2 )s t−1 which reduces to the GJR model when ␦1 = ␦2 = 0. The VS model allows volatility to be influenced not only by the sign of the previous shock, as in GJR model, but also by the size of it. The size of the shock is defined as the unexpected volatility at t − 1, given the set of information available at time t − 2 (u 2t−1 − 2t−1 ). A small negative shock at time t − 1 which generates lower level of volatility than expected at time t − 2 would not lead to a higher volatility at time t, at the same time, a positive shock which produces lower than expected volatility would not lead to a higher volatility at time t. Using the fact that the GJR model is nested into the VS model, Fornari and Mele (1997) tested whether the MLE estimates of the new coefficients ␦1 , ␦2 were signifficantly different from 0 and found that these coefficients were signifficantly different from 0 in five out of six data sets (which consisted of daily returns in six international markets: U.S., Germany, Japan, U.K., France, and Italy; the exception was Germany). The tests were performed however under the assumption of normal innovations and without taking into account the possibility of misspecification of the model. Since normality is not supported by the data, Fornari and Mele (1997) suggested a re-examination of the data, but under more general distributional assumptions. Note that the distributional modelling of financial assets has important
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consequences for the pricing and hedging options, as recently demonstrated in Pinn (1999). We undertook this study, generalizing in two directions: We used the assumption (suggested by Bollerslev et al., 1994) of a Generalized Student’s t distribution for the innovations. (This includes the Generalized Error and the Student’s t distributions as particular cases.) Furthermore, we hedge against the possibility of model misspecification by using the robust covariance estimator suggested by White (1982).1 Our results indicate that both the distributional assumption and the use of White’s robust covariance matrix estimator affect significantly the inference. As expected, the estimate of the leverage coefficient ␦0 suggested by Glosten et al. (1993) has a negative sign and is always signifcant. However, in contrast with the findings of Fornari and Mele (1997), the estimates of ␦1 and ␦2 are insignificant in the case of U.S., and marginally significant in the case of Germany. With regard to the other four countries of Japan, U.K., France, and Italy, our results are in line with the finding of Fornari and Mele (1997) that the VS model outperforms the GJR model. Another difference is that the estimates of ␦1 are always positive and the estimates of ␦2 are always negative. (Fornari & Mele, 1997 also found that ␦1 and ␦2 take opposite signs from each other for each country, but without any pattern emerging across countries.) Our findings are in line with those of other researchers like Bera and Jarque (1982), Baillie and DeGennaro (1990) and Duan (1997) that failure to take into account the distributional properties of stock returns and misspecification may lead to the possibility of wrong inferences being drawn. Finally, we observe from the continuous limit of the VS model (derived under the more general distributional assumptions, and including a correction to the original derivation), that it may lead to negative volatilities for nonzero values of the parameter ␦2 ; and, as such, the model needs to be modified somehow. The paper is organized as follows. The next section describes the data and methodology. The third section presents the empirical results. Conclusions are given in the fourth section and the continuous limit is derived in the Appendix.
2. DATA AND METHODOLOGY The data set is the same as that of Fornari and Mele (1997), and was obtained from the Journal of Applied Econometrics Data Archive. The sample has 1494 daily logarithmic returns during the period from 1/1/1990 to 16/10/1995. The
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series are the S&P 500 (U.S.), Topix (Japan), CAC 40 (France), FTSE 100 (U.K.), FAZ (Germany) and MIB (Italy). We use as a probability density function for the innovations ut the Generalized Student’s t distribution, with density: −1 ( + −1 ) −1 −1 |u| −(+ ) f(u) = 1 + k t 2 ()(−1 ) t k where is the gamma function and k = (()(−1 )/(3−1 )(−2−1 ))1/2 Note that the constant k equals b(1/) in the notation of Bollerslev et al. (1994), formula (9.6); however, the formula for b there contains a typo. This class includes the Student’s t and Generalized Error densities as special cases, obtained by putting = 2, = d/2 (d being the degrees of freedom), and = ∞ respectively. √ The Student’s t distribution for example is obtained by noting that k reduces to d − 2 in the case = 2, = d/2, yielding: −((d+1)/2) |u|2 (d + 1)/2 −1/2 −1 1+ 2 t f(u) = √ (d − 2) (d/2) t (d − 2)
3. EMPIRICAL RESULTS The BHHH maximisation routine of Berndt et al. (1974) was used to obtain the estimates of the parameters of the VS model using the Generalized Student’s t. The SIMPLEX algorithm (see Press et al., 1988) was used to refine the initial values of the parameters finally used as input for the BHHH routine. The parameter estimates reported in Fornari and Mele (1997) were used as starting values for the SIMPLEX algorithm. Different starting values were also tried to ensure that the algorithm reached a global maximum. All of our inferences (including the Wald test) are based on the robust covariance estimator proposed by White (1982). For comparison, we estimated the VS and GJR models. The results are in Tables 1 and 2. The most important findings are: (1) Comparison with the results of Fornari The results show that the estimates of ␦0 are significant at the 5% level for all six countries under the VS and GJR models. This indicates the existence of significant leverage effects in these markets. An interesting feature of the results is that ␦1 has a positive sign, and ␦2 has a negative sign for all countries. This result is not observed in the Fornari and Mele (1997) paper. The estimates of ␦1 are significant for all countries except for the U.S. This is in contrast
Parameter 0 1 ␣  ␦0 ␦1 ␦2 Log like Wald test H0 : ␦ 1 = ␦2 = 0 SBT NSBT PSBT
U.S. 3.3E-4 (1.99) −0.006 (−0.23) 7.4E-7 (3.47) 0.048 (7.45) 0.942 (275.8) −0.027 (−3.34) 0.049 (1.21) −1.6E-6 (−1.23) 1.47 s.e. = 0.07 6.14 s.e. = 0.67 5196.41 1.55 1.56 0.11 1.09
Germany 1.3E-4 (0.56) 0.045 (1.50) 1.2E-6 (2.35) 0.057 (2.71) 0.933 (50.12) −0.045 (−2.36) 0.043 (2.40) −3.3E-6 (−2.81) 2.45 s.e. = 0.05 2.23 s.e. = 0.45 4785.22 17.42 0.62 1.26 0.29
Japan −7.3E-4 (−2.83) 0.134 (4.78) 5.0E-6 (5.18) 0.15 (15.79) 0.859 (55.80) −0.097 (−6.86) 0.194 (10.24) −1.2E-5 (−5.64) 2.50 s.e. = 0.38 1.57 s.e. = 0.76 4371.14 126.6 0.093 0.027 0.95
U.K. 1.08E-4 (30.3) 0.073 (2.39) 1.6E-6 (2.32) 0.057 (8.22) 0.922 (51.31) −0.033 (−2.68) 0.092 (6.37) −2.8E-6 (−2.65) 3.20 s.e. = 0.31 1.39 s.e. =0.35 5139.39 40.71 1.43 3.75 1.06
France −1.9E-4 (−0.94) 0.03 (1.11) 5.9E-6 (15.47) 0.044 (19.79) 0.919 (660.8) −0.044 (−4.92) 0.269 (7.41) −2.5E-5 (−5.98) 2.58 s.e. = 0.16 2.49 s.e. = 0.59 4459.04 57.77 −0.77 1.09 0.11
Italy −2.1E-4 (−0.79) 0.279 (10.47) 2.2E-6 (5.66) 0.073 (19.48) 0.922 (296.9) −0.026 (−2.38) 0.15 (8.25) −1.3E-5 (−6.57) 3.07 s.e. = 0.14 1.71 s.e. =0.15 4372.1 111.17 −1.42 0.62 −1.19
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Table 1. Parameters of the Volatility Switching Model with the Generalized Student’s Distribution (t-Ratios in Parentheses).
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Table 2. Parameters of the GJR Model with the Generalized Student’s Distribution (t-Ratios in Parentheses). Parameter
2.9E-4 (1.99) −7.2E-2 (−0.29) 0.13E-6 (5.81) 0.073 (14.5) 0.909 (214.36) −0.04 (−3.79) 1.41 s.e. = 0.06 6.89 s.e. = 1.38 5195.08 1.79 0.63 1.10
Germany 1.4E-4 (0.79) 0.044 (1.58) 1.1E-6 (32.27) 0.057 (37.3) 0.933 (657.02) −0.043 (−6.86) 2.32 s.e. = 0.07 2.55 s.e. = 0.16 4784.87 0.25 1.10 −0.05
Japan −7.0E-4 (−2.85) 0.131 (4.71) 6.5E-6 (10.79) 0.17 (18.5) 0.824 (125.4) −0.12 (−10.2) 2.5 s.e. = 0.35 1.47 s.e. = 0.50 4364.67 0.98 0.56 0.92
U.K. 1.4E-4 (0.86) 0.072 (2.29) 2.2E-6 (3.60) 0.064 (9.82) 0.899 (75.18) −0.039 (−2.94) 3.10 s.e. = 0.33 1.46 s.e. = 0.41 5136.3 1.84 3.96 0.97
France −1.7E-4 (−0.76) 0.027 (1.10) 1.2E-5 (5.20) 0.069 (5.57) 0.841 (41.4) −0.062 (−5.47) −2.55 s.e. = 0.10 2.47 s.e. = 1.5 4445.8 −0.19 1.86 0.16
Italy −1.9E-4 (−0.76) 0.244 (9.16) 1.7E-6 (5.27) 0.058 (19.58) 0.933 (358.9) −0.027 (−3.07) 2.79 s.e. = 0.13 2.05 s.e. = 0.20 4376.39 −1.15 0.21 −1.65
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0 1 ␣  ␦0 Log like SBT NSBT PSBT
U.S.
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to Fornari and Mele (1997) where the estimates of ␦1 are significant for all countries except Germany. The estimates of ␦2 are significant for all countries except U.S., while they were significant in Fornari and Mele (1997) for U.S. and Japan only. Using the 5% level of significance, and based on the individual t-statistics and the Wald test for the joint hypothesis that ␦1 = ␦2 = 0, we reject the proposition that the VS model outperforms the GJR model for the U.S., and we do not reject the same proposition for Germany. The two last conclusions regarding U.S. and Germany are in contrast to the findings of Fornari and Mele (1997). These contradictions between our findings and those of Fornari and Mele (1997) clearly show that using more exible distributional assumptions can have a significant impact on the inference.2 (2) Interesting country specific variation In the U.S. case there is no evidence of outperformance for the VS model using tests based either on t-statistics of individual parameters, or on Wald and Likelihood Ratio (LR) tests.3 In the case of Germany, the Wald and LR tests give conflicting results; also, the t-tests on the individual parameters are significant at the 5% level but not the 1% level. The asymmetric behaviour is apparently marginal in the case of Germany. For the remaining countries (excluding U.S. and Germany) our findings support those of Fornari and Mele that the VS model outperforms the GJR model. (3) The rejection of more specific models It is well known that normality is always rejected by the data, which led authors to propose various more general distributions. For example, the Student’s t distribution has been suggested as a candidate for modelling stock returns (see Blattberg & Gonedes, 1974). If we were to apply a t-test for the hypothesis that = 2 (the value taken by the Student’s t distribution), we would reject this hypothesis for five countries (the exception being Japan) under both the VS and the GJR models. Finally, the tables report the t-statistics for the Sign Bias Test (SBT), Negative Sign Bias Test (NSBT), and the Positive Sign Bias Test (PSBT), for the residuals from both models, the VS and GJR. These tests were suggested by Engle and Ng (1993) to test for the presence of asymmetric behaviour of volatility. SBT, NSBT, and PSBT are the t-statistics for the coefficients of a linear regression of − + − the squared residuals on S − t−1 , S t−1 u t−1 , S t−1 u t−1 , respectively, with S t being a dummy variable which equals plus one if sign (u t ) = −1 and 0 otherwise, and − S+ t = 1 − S t . Fornari and Mele (1997) applied these tests to the original data before estimating their VS model, and found significant asymmetric effects in all series. The same tests were applied to the residuals from the VS and GJR models to check whether there are any asymmetries left. All the tests are not significant at the
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5% level of significance, except for the NSBT test for the U.K. under the VS and GJR models.
4. CONCLUSIONS The paper extended the VS model of Fornari and Mele (1997) by using a more flexible distribution, the Generalized Student’s t, and by employing White’s robust covariance estimator to draw inferences. Our results support in general the conclusion of Fornari and Mele (1997) that the size of the shock is important in all countries examined except for the U.S. However, country specific contradictions between our results and theirs as well as the discovered patterns in the signs of ␦1 and ␦2 illustrate the importance of the more general distributional assumptions. With regard to the distribution of the innovations of stock returns, the hypothesis that stock returns can be described by the Student’s t distribution was rejected for all countries except Japan. It has always been difficult to approximate the distribution of stock returns. Therefore, it is recommended that inferences should be based on the robust covariance matrix estimator as suggested by White (1982).
NOTES 1. Under misspecification, the MLE, now called “quasi” maximum likelihood estimator (QMLE) is to be interpreted as an estimator of the “closest” model from the data in the adopted parametric family (with respect to the Kullback-Leibler distance). However, under misspecification the covariance matrix of the QMLE may no longer be estimated 2 ˆ −1 by the inverse of Fisher’s information matrix −A n (where A = (∂ /∂i ∂j )log f(X, ) ˆ and An denotes averaging with respect to the empirical measure). Instead, as shown by ˆ −1 ˆ ˆ −1 White (1982), it should be estimated by A n Bn An where B is the outer square of the gradient B = ((∂/∂i )log f(X, ) · (∂/∂j )log f(X, )) (the classical asymptotic equivalence between B and −A does not hold under misspecification). Also, under misspecification the likelihood ratio test is no longer asymptotically 2 distributed and so should preferably be replaced by a Lagrange multiplier or Wald test (see White, 1982). 2. The values of the likelihood for our models are lower than those of Fornari and Mele (1997), even though they should be higher since we are employing a more flexible distribution. Based on attempts to replicate their results assuming the normal distribution, we conjecture that the difference is due to their omitting the constant −0.5 log(2) in the log-likelihood function. 3. The LR test is twice the difference in the likelihood values of the VS and GJR models. If the model is not misspecified, the Wald and LR tests are equivalent, and each
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of them is asymptotically distributed as 22 . However, the Wald test is preferred to the LR test if misspecification is present.
ACKNOWLEDGMENTS We would like to thank Professors Halbert White, Thomas Fomby, Carter Hill, and participants at the Advances in Econometrics Conference, Louisiana State University, November 1–3, 2002, for useful comments and suggestions. Earlier versions of this paper were presented at the 2000 Mid West Finance association annual meeting in Chicago, and the 1998 International Symposium on Forecasting in Edinburgh.
REFERENCES Baillie, R. T., & DeGennaro, R. P. (1990). Stock returns and volatility. Journal of Financial and Quantitative Analysis, 25, 203–214. Bera, A. K., & Jarque, C. M. (1982). Model specification tests. Journal of Econometrics, 20, 59–82. Berndt, E. K., Hall, B. H., Hall, R. E., & Hausman, J. A. (1974). Estimation and inference in nonlinear structural models. Annals of Economic and Social Measurement, 4, 653–665. Black, F. (1976). Studies of stock market volatility changes. Proceedings of the American Statistical Association, Business and Economic Statistics Section, 177–181. Blattberg, R. C., & Gonedes, N. J. (1974). A comparison of the stable and student distributions as statistical models for stock prices. Journal of Business, 47, 244–280. Bollerslev, T., Engle, R., & Nelson, D. (1994). ARCH models. In: R. Engle & D. McFadden (Eds), Handbook of Econometrics (Vol. 4, pp. 2959–3038). Christie, A. A. (1982). The stochastic behavior of common stock variances: Value, leverage and interest rate effects. Journal of Financial Economics, 10, 407–432. Duan, J. C. (1997). Augmented GARCH (p, q) process and its diffusion limit. Journal of Econometrics, 79, 97–127. Engle, R. F., & Ng, V. K. (1993). Measuring and testing the impact of news on volatility. Journal of Finance, 43, 1749–1778. Ethier, S. N., & Kurtz, T. G. (1986). Markov processes. Characterization and convergence. New York: Wiley. Fornari, F., & Mele, A. (1997). Sign- and volatility-switching ARCH models: Theory and applications to international stock markets. Journal of Applied Econometrics, 12, 49–65. Glosten, L. R., Jagannathan, R., & Runkle, D. (1993). On the relationship between expected value and the volatility of nominal excess return on stocks. Journal of Finance, 43, 1779–1801. Nelson, D. (1990). ARCH models as diffusion approximations. Journal of Econometrics, 45, 7–38. Pinn, K. (1999). Minimal variance hedging of options with student-t underlying. Preprint, http://xxx.lanl.gov/abs/cond-mat/9907421 Press, W., Teukolsky, S., Vettering, W., & Flannery, B. (1988). Numerical recipes in C. New York: Cambridge University Press. White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50, 1–25.
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APPENDIX: THE CONTINUOUS LIMIT OF THE VS MODEL We sketch here the derivation of the stochastic differential equation(SDE) satisfied by the variance (or volatility) t = 2t obtained by taking the continuous limit of the VS model, under the assumption of general innovations with a symmetric distribution and at least four moments. The qualitative structure of the limit will turn out to be the same as that of the VS model, the only change being the appearance of a parameter c 2 = E(z 2t − 1)2 in the limit (which is two in the normal case). Let dt = (t ) dt + (t ) dW t
(1)
denote the limiting SDE describing the evolution of the volatility. Under appropriate assumptions on and the volatility process will have a stationary density p(), given by the well known formula p() = k(es() /2 ()) where s() = 2 (/2 ) d and k is the proportionality constant. It turns out however (after correcting an error in the Fornari and Mele (1997) paper) that when ␦2 = 0 the coefficient of the Brownian perturbation () in (1) may be strictly positive when = 0; as a consequence of that, the state space of t and its stationary distribution are not anymore supported on the positive axis. Thus, t is not a proper volatility process and it should be modified somehow (say by replacing it with its absolute value t ). Let now h denote the length of the time interval between two observations and let zt denote the standardised innovations u t /t . The VS recursion for the volatility is given by: t+h − t = + ( − 1)t + ␣t z 2t + k t s t where k t = ␦0 t z 2t − ␦1 t −␦2 and st is the sign of zt . Under appropriate scaling conditions (see the diffusion scaling assumptions below), this difference equation will give rise in the continuous limit h → 0 to a stochastic differential equation (diffusion). The standard method to establish convergence to a diffusion is to show convergence of the first two conditional moments of the increments to the drift and volatility of the diffusion and to show convergence of the conditional higher order cumulants to 0 (for more details see for example Duan, 1997; Nelson, 1990; or Fornari & Mele, 1997). Below we only sketch the derivation of the drift and variance parameters. It is convenient to rewrite the difference equation as a sum of terms entirely known at time t − h, which will yield the drift and of zero mean “innovations” (terms with conditional expectation 0), each of which will yield in the limit Wiener
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processes. t+h − t = + t (+␣ − 1) + ␣t (z 2t − 1) + ␦0 t s t (z 2t − 1) + s t (t (␦0 − ␦1 ) − ␦2 )
(2)
− 1) and st above are zero mean innovations. The condiThe terms − 1, tional mean of the increments is thus: t+h − t = + t ( + ␣ − 1) E F t−h z 2t
s t (z 2t
The conditional variance of the increments is just the sum of the variances, since the symmetry of the Generalized Student’s t distribution implies that the innovations z 2t − 1, s t (z 2t − 1), st are uncorrelated. Thus, the variance is given by t+h − t = 2t c 2 (␣2 + ␦20 ) + (t (␦0 − ␦1 ) − ␦2 )2 (3) Var F t−h where c 2 = E(z 2t − 1)2 (under the Generalized Student’s t distribution). To obtain a diffusion in the limit we need to assume that the order of magnitude of the terms which produce the drift √ is h and that the order of magnitude of the terms which produce the√diffusion is √ h: This requires diffusion scaling assumptions: √ the following √ ␣ = ␣ h, ␦0 = ␦ 0 h, ␦1 = ␦ 1 h, ␦2 = ␦ 2 h, = h, ␣ +  − 1 = −h, where the primed quantities are asymptotically constant as h → 0 (thus,√the original quantities are assumed to be of the order of magnitude O(h) or O( h) respectively). Under these assumptions, the limiting conditional mean and variance of the increments t+h − t are asymptotically of the form t h and 2t h where t = − t 2t = 2t k 2 (␣ )2 + 2t c 2 (␦ 0 )2 + (t (␦ 0 − ␦ 1 ) − ␦ 2 )2 Standard diffusion convergence results (see for example Ethier & Kurtz, 1986, Theorem 7.4.1) imply that for small h the discrete GARCH volatility will be well approximated by the solution of the SDE: dt = (t ) dt + (t ) dW t
(4)
where () = − () = 2 k2 (␣ )2 + 2 c2 (␦ 0 )2 + ((␦ 0 − ␦ 1 ) − ␦ 2 )2
(5) (6)
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Note: When ␦ = 0 we find by plugging in (6) that the SDE (4) reduces to one with 2 linear coefficients dt = ( − t ) dt + t dW t
(7)
where we put 2 = k 2 (␣ )2 + c 2 (␦ 0 )2 + (␦ 0 − ␦ 1 )2 . In this well known limiting form of the GJR (and NGARCH) model, the diffusion coefficient t becomes 0 when t = 0: As a consequence, the state space of vt may be taken to be the positive numbers. When ␦ = 0 the conditional variance is 2 = 2 2 + ␦2 − 2d (␦ − ␦ ). t t 2 2 1 2 t 0 This corrects the formulas (27)–(28) in Fornari and Mele, who omit the cross term −2d 2 t (␦ 0 − ␦ 1 ) (which results when expanding the square in the last term under the square rootin (4)). More significantly, in this case the conditional variance 2t may be strictly positive when t = 0. As a consequence of this, when ␦ 2 = 0 the Fornari model loses the desirable feature of a positively supported stationary distribution for the volatility of the continuous limit, which is exhibited by the standard GARCH models like GJR, NGARCH and EGARCH and needs to be further modified.
ESTIMATING A LINEAR EXPONENTIAL DENSITY WHEN THE WEIGHTING MATRIX AND MEAN PARAMETER VECTOR ARE FUNCTIONALLY RELATED Chor-yiu Sin ABSTRACT Most economic models in essence specify the mean of some explained variables, conditional on a number of explanatory variables. Since the publication of White’s (1982) Econometrica paper, a vast literature has been devoted to the quasi- or pseudo-maximum likelihood estimator (QMLE or PMLE). Among others, it was shown that QMLE of a density from the linear exponential family (LEF) provides a consistent estimate of the true parameters of the conditional mean, despite misspecification of other aspects of the conditional distribution. In this paper, we first show that it is not the case when the weighting matrix of the density and the mean parameter vector are functionally related. A prominent example is an autoregressive moving-average (ARMA) model with generalized autoregressive conditional heteroscedasticity (GARCH) error. As a result, the mean specification test is not readily modified as heteroscedasticity insensitive. However, correct specification of the conditional variance adds conditional moment conditions for estimating the parameters in conditional mean. Based on the recent
Maximum Likelihood Estimation of Misspecified Models: Twenty Years Later Advances in Econometrics, Volume 17, 177–197 © 2003 Published by Elsevier Ltd. ISSN: 0731-9053/doi:10.1016/S0731-9053(03)17009-0
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literature of efficient instrumental variables estimator (IVE) or generalized method of moments (GMM), we propose an estimator which is modified upon the QMLE of a density from the quadratic exponential family (QEF). Moreover, GARCH-M is also allowed. We thus document a detailed comparison between the quadratic exponential QMLE with IVE. The asymptotic variance of this modified QMLE attains the lower bound for minimax risk.
1. INTRODUCTION Since the publication of White (1982), a vast literature in econometrics and statistics has been devoted to the development of misspecification analysis on quasi- or pseudo-maximum likelihood estimator (QMLE or PMLE). As far as parametric modelling is concerned, researchers become more and more aware of the possible misspecification in one aspect or the other. This can partially be reflected by the frequent reporting of the so-called White standard errors, first introduced to the field of economics in a preceding paper White (1980). White (1982) and many other related papers amount to a new way of thoughts on the one hand, and imply a big impact on applied work on the other hand. In sum, it is fair to say that White (1982) changes some of our interpretations of empirical results. Most economic models in essence specify the mean of some explained variables, conditional on a number of explanatory variables. In another seminal paper Gourieroux, Monfort and Trognon (1984), they investigate the consistency of the parameters in the conditional mean, ignoring or misspecifying other features of the true conditional density. They show that it suffices to have a QMLE of a density from the linear exponential family (LEF). Conversely, a necessary condition for a QMLE being consistent (for the parameters in the conditional mean) is that the likelihood function belongs to the LEF. As a natural extension, Chapter 5 of White (1994) show in detail Gourieroux et al.’s (1984) results carry over to dynamic models with possibly serially correlated and/or heteroscedastic errors. In this paper, we first show that it is not the case when the weighting matrix of the density and the mean parameter vector are functionally related. Denote the mean parameter vector by  and the weighting matrix by t (), where = [ , ␣ ] . We say that they are functionally related if for some , P{∇ t () = 0} < 1.
(1.1)
A prominent example is an autoregressive moving-average (ARMA) model with generalized autoregressive conditional heteroscedasticity (GARCH) error. A
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detailed discussion on this model will be found in Lemma 3.1. Should it be the case, the mean specification test is not readily modified as heteroscedasticity insensitive. See, for instance, Wooldridge (1990) in which he considers a weighted least squares (WLS) rather than QMLE. Next we show that, if the conditional variance is also correctly specified, the QMLE of a density from a quadratic exponential family (QEF), which is a straightforward extension of LEF, may in fact be more efficient than WLS. Intuitively, as the weighting matrix and the mean parameter vector are functionally related, there is one more conditional moment and in virtue of the recent literature of instrumental variable estimation (IVE) or generalized method of moments (GMM), we come up with a more efficient estimator. Further, based on the literature of efficient IVE developed in Bates and White (1988, 1993), we propose a modified QMLE (MQMLE) which is more efficient than QMLE. Lastly, we argue that the MQMLE attains a lower bound for minimax risk, using the results in Chamberlain (1987). It is worth mentioning that our efficient MQMLE is different substantively with those suggested in Cragg (1983) and MaCurdy (1999). In these two papers, the conditional variance is unspecified and an “instrument” is used to construct an estimator, which is more efficient than the ordinary least squares (OLS). See also Gourieroux, Monfort and Renault (1996) who exploit the zero odd-order conditional moments. First proposed and discussed in Gourieroux et al. (1984), QEF has been found to be a natural family of density that models means, variances as well as covariances. Recent examples include Zhao and Prentice (1991), McCullagh (1994) and Pommeret (2002). Both LEF and QEF contain normal density as a leading example. Following the lines in White (1982, 1994), throughout this paper, we consider a class of stochastic parametric specification of Y t ∈ Rl , conditional on an increasing -algebra Ft−1 : S = { f t () = exp ((W t , ), (W t , ))},
(1.2)
where lies in a compact subset of Rp , and W t is measurable-Ft−1 . Moreover, following the literature on LEF and QEF, (W t , ) is often interpreted as the locational or mean specification, while (W t , ) is often interpreted as the weighting or variance specification. Detailed regularity conditions can be found in Appendix A. For sake of notational simplicity, apart from the time-invariant functions (·, ·), (·) and (·), in Appendix A, we assume that the observables {X t ∈ R } are stationary. Throughout, →L denotes convergence in distribution, while →P denotes convergence in probability.
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The paper proceeds as follows. Section 2 argues that the QMLE of a LEF density can be inconsistent, when the weighting matrix is functionally related to the mean parameter vector, and the weighting matrix does not correctly specify the variance. Section 3 first discusses the consistency and efficiency of the QMLE of a QEF density; followed by proposing a MQMLE. The algorithm of this MQMLE is also provided. This MQMLE is shown to attain the lower bound for minimax risk. Conclusions can be found in Section 4. Appendix A contains the other essential regularity conditions not stated in the main text. All technical proofs are relegated to Appendix B.
2. INCONSISTENT LINEAR EXPONENTIAL QMLE WHEN CONDITIONAL VARIANCE IS MISSPECIFIED In this section, we consider a conditional density from the linear exponential family (LEF). In particular, we refer to the stochastic specification S = { f t (·) = exp (·, ·)} in (1.2), where (t (), t ()) ≡ t () = a(t (), t ()) + b( y t , t ()) + c(t (), t ()) y t ,
(2.1)
where we suppress the argument W t , as no ambiguity arises. The following is similar to Theorem 5.4, p. 67 in White (1994). Theorem 2.1. Suppose Assumption A.1 applies to S specified in Eq. (2.1). Given some legitimate initial values of observables and possibly latent variables, let ˆ be the QMLE generated by S. Then ˆ →P ∗ , where E[∇ t (∗ )] = 0 and ∇ t () = ∇ t ()∇ c(t (), t ())[y t − t ()] + ∇ t ()[∇ a(t (), t ()) + ∇ b( y t , t ()) + ∇ c(t (), t ())y t ].
(2.2)
In our context, we partition = [ , ␣ ] , where  lies in a compact subset B of Rp 1 and ␣ lies in a compact subset A of Rp 2 , p 1 + p 2 = p. Suppose the function of expectation t () = t (), so that such models as GARCH-M are ruled out. As in White (1994), we interpret t () as a nuisance parameter function. Suppose there exists 0 ∈ B such that the conditional expectation of y t is t (0 ). As one can see from the second term in Eq. (2.2), possibly ∇ t () = 0 and thus the weighting matrix is functionally related to the mean parameter vector. See Jobson and Fuller (1980) and Carroll and Ruppert (1982) who consider a normal density. As a result, E[∇ t ([0 , ␣∗ ] )] = 0 in general. If that is the case, ∗ = 0 .
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It should be noted that in Gourieroux et al. (1984) and Chapter 5 of White (1994), the linear exponential density does not include a weighting matrix t (). On the other hand, in Theorem 6.13 of White (1994), although there is a weighting matrix, the parameter involved (denoted as ␣) and the mean parameter (denoted as ) are estimated separately and in the second stage of estimating , the estimator for ␣ is not altered. The next lemma is about an example of multivariate GARCH, in which if the conditional variance is misspecified, possibly E[∇ t ([0 , ␣∗ ] )] = 0, even though the conditional mean is correctly specified. In particular, we assume that: t () = diag1/2 (h t ())diag1/2 (h t ()), h t () = 0 +
r
i ε˜ t−i () +
i=1
s
r+i h t−i (),
(2.3)
i=1
where is an l × l time-invariant positive-definite matrix of correlations, and h t () is an l × 1 time-varying vector of variances, and ε˜ t () ≡ (ε21t (), . . . , ε2lt ()) , εt () ≡ (ε1t (), . . . , εlt ()) ≡ y t − t (). Model (2.3) is an extension of the commonly used time-invariant-correlation multivariate GARCH model (Bollerslev, 1990), first proposed by Jeantheau (1998). See Ling and McAleer (2003) for the asymptotic properties of estimating the correctly specified Model (2.3). See also Sin and Ling (2003) for a more detailed discussion on a simplified version of it. Lemma 2.1. Suppose the assumptions in Theorem 2.1 hold with 1 1 1 t () = − ln(2) − εt ()−1 t ()εt () − ln |t ()|, 2 2 2
(2.4)
where t () is as defined in Eq. (2.3). Then ˆ →P ∗ = [∗ , ␣∗ ] , ∗ ∗ ∗ ∗ E[∇ t (∗ )−1 t ( )εt ( )] = −E[∇ t ( )∇ t ( )],
(2.5)
where ∇ t (∗ ) = ∇ h t (∗ )l [diag1/2 (h t (∗ )) ⊗ I l + I l ⊗ diag1/2 (h t (∗ ))], (2.6) −1 r s Il − ∗r+i L i ∗i L i [diag(εt (∗ ))∇ t (∗ )] , ∇ h t (∗ ) = 2 i=1
i=1
(2.7) 1 ∗ ∗ ∗ −1 ∗ ∇ t (∗ ) = vec{−1 t ( )[εt ( )εt ( )t ( ) − I l ]}, 2
(2.8)
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where for every diagonal l × l matrix , l is an l × l 2 matrix with the property that l () = vec(). (·) is an l × 1 vector containing the diagonal elements of an l × l matrix. (See Definition 7.1, p. 109 of Magnus (1988).) L is a backward-shift operator such that for any variable U t , L i U t = U t−i . Note in Lemma 2.1, typically, ∇ t () is measurable-Ft−1 . Thus, so is ∇ t (). If, in addition to correct specification in conditional mean, the conditional variance is correctly specified in the sense that ∃␣0 ∈ A such that for 0 = [0 , ␣0 ] , E[εt (0 )εt (0 )|Ft−1 ] = t (0 ). Then by (2.8), E[∇ t (0 )|Ft−1 ] = 0 ⇒
0 0 0 0 E[∇ t (0 )−1 t ( )εt ( )] = −E[∇ t ( )∇ t ( )] = 0.
In other words, in view of (2.5), ˆ is consistent for the “true” parameter 0 . However, if the conditional variance is misspecified, and ∀␣ ∈ A, P{E[εt (0 )εt (0 )|Ft−1 ] = t [0 , ␣ ] } < 1, then there is no guarantee that ∇ t ([0 , ␣ ] ) and ∇ t ([0 , ␣ ] ) are uncorrelated. As a result, ˆ may not be consistent for 0 .
3. CONSISTENT QUADRATIC EXPONENTIAL QMLE WHEN CONDITIONAL VARIANCE IS CORRECTLY SPECIFIED In this section, we consider a conditional density from the quadratic exponential family (QEF). In particular, we refer to the stochastic specification S = { f t (·) = exp (·, ·)} in (1.2), with (t (), t ()) ≡ t () = A(t , t ) + B( y t ) + C(t , t ) y t + y t D(t , t )y t , (3.1) where A(·, ·) and B(·, ·) are scalar functions, C(·, ·) is an l × 1-vector function and D(·, ·) is an l × l-matrix function. As in the previous section, t = t () and t = t (). Unlike Lemma 2.1, here we do not restrict that t () only depends on . As a result, models such as GARCH-M may be allowed. The next theorem shows consistency as well as asymptotic normality of the quadratic exponential QMLE. This theorem is essentially a complete version of Theorem 6 of Gourieroux et al. (1984), which proof only considers l = 1. At the same time, along the lines in White (1994), we consider dynamic models. See
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also Theorem 2.1 of Bollerslev and Wooldridge (1992), who consider a normal density, a particular case in the QEF. For ease of exposition, we define the folt () lowing variables: t () ≡ , M t () ≡ (I l ⊗ t ()) + (t () ⊗ I l ). When t () no ambiguity arises, we write = t () and M = M t (). Theorem 3.1. Suppose Assumption A.2 applies to S specified in Eq. (3.1). In addition, ∃0 ∈ int() such that E[y t |Ft−1 ] = t (0 ) and E[( y t − t (0 ))( y t − t (0 )) |Ft−1 ] = t (0 ). Given some legitimate initial values of observables and possibly latent variables, let ˆ be the QMLE generated by S. Then √ n(ˆ − 0 ) →L N(0, J 0−1 I 0 J 0−1 ), (3.2)
∇ C + (∇ D)M ∇ D J 0 = E 0 ∇ λ ∇ λ , (3.3) ∇ D ∇ D I 0 = E 0 [∇ ∇ ], ∇ = ∇ [∇ C( y − ) + ∇ Dvec(yy − − )].
(3.4)
As usual, we can estimate J 0 and I 0 by their sample analogues, defined as:
n ˆ t + (∇ D ˆ t )M ˆt ˆ t ∇ D ∇ C ∇ ˆ t ∇ ˆ t , Jˆ 0 = n −1 ˆt ˆt ∇ D ∇ D t=1
Iˆ 0 = n −1
n
ˆ t ∇ ˆ [∇ t ],
t=1
ˆ t = ∇ ˆ t [∇ C ˆ t ( yt − ˆ t vec( y t y t − ˆ t ) + ∇ D ˆ t ˆ t − ˆ t )], ∇ ˆ ∇ C ˆ t = ∇ C t ( ˆ t = ∇ D t ( ˆt = ˆ t , ˆ t ), ∇ D ˆ t , ˆ t ), ∇ D with ∇ ˆ t = ∇ t (), ˆ ˆ ∇ D t ( ˆ t , ˆ t ), ˆ t = t () and ˆ t = t (). √ (0) Given some n-consistent initial estimator ˆ , the quadratic exponential QMLE ˆ can be obtained by the following iterative Newton-Raphson algorithm:
−1 n ∇ Ct + (∇ Dt )Mt ∇ Dt (k+1) (k) ˆ ∇ t = ˆ − ∇ t | ˆ (k) ∇ Dt ∇ Dt t=1 n × ∇ t (∇ C t ( y t − t ) + ∇ D t vec( y t y t − t t − t )) | ˆ (k) . t=1
(3.5)
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In the next lemma, we consider the QMLE of a normal density, which belongs to the QEF. Based on this lemma, we will argue that the QMLE of the mean parameter can be more efficient than the corresponding WLS. Lemma 3.1. Suppose the assumptions in Theorem 3.1 hold with 1 1 1 t () = − ln(2) − ( y t − t ()) −1 t ()( y t − t ()) − ln |t ()|. (3.6) 2 2 2 Consider ˆ = (ˆ , ␣ˆ ) , the QMLE generated by S given some legitimate initial values of observables and possibly latent variables. Suppose in addition, (i) the information matrix equality holds, in the sense that I 0 = J 0 ; (ii) E 0 [∇ (−1 )∇␣ ] + (1/2)E 0 [∇ (−1 ⊗ −1 )∇␣ ] = 0p 1 xp 2 . Then √ ˆ 0  n( −  ) →L N(0, I˜ 0−1 ), where
1 I˜ 0 = E 0 [∇ (−1 )∇ ] + E 0 [∇ (−1 ⊗ −1 )∇ ]. 2
(3.7)
Moreover, {E 0 [∇ (−1 )∇ ]}−1 − I˜ 0−1 is positive semi-definite (p.s.d.), where the first term is the asymptotic variance of the WLS. It should be noted that in Lemma 3.1, the equality of information matrix may not imply that the data generating process (DGP) is normal. It suffices to have correctly specified third and fourth conditional moments. For condition (ii), it suffices that ∇␣ t (0 ) = 0; and t (0 ) is a symmetric function of the deflated error −1/2 s (0 )( y s − s (0 )), s = . . . , t − 2, t − 1, which in turn is symmetrically distributed. See Engle (1982) for the argument applied to a univariate system, and Sin and Ling (2003) for a multivariate example. Refer to the general case in Theorem 3.1. There are two conditional moment restrictions. More precisely, ∃0 ∈ int() such that: E[y t − t (0 )|Ft−1 ] = 0,
(3.8) 0
E[( y t − t ( ))( y t − t ( )) − t ( )|Ft−1 ] = 0. 0
0
(3.9)
Moreover, it is shown in the proof of Theorem 3.1 that the information matrix I 0 can be written as: I 0 = E 0 [∇ (∇ C∇ C + ∇ C3 ∇ D + ∇ D3 ∇ C + ∇ D4 ∇ D)∇ ], 3 = 3t () ≡ E[y t ⊗ y t y t |Ft−1 ] − (vec[t ()t () + t ()])t (),
(3.10)
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4 = 4t () ≡ E[y t y t ⊗ y t y t |Ft−1 ] − (vec[t ()t () + t ()]) × (vec[t ()t () + t ()]) . Based on the two conditional moment restrictions (3.8) and (3.9), if we are able to exploit the information about the higher-order conditional moments E[y t ⊗ y t y t |Ft−1 ] and E[y t y t ⊗ y t y t |Ft−1 ], the following estimator is constructed: ˜ = argmin∈ s n ()V −1 n s n (),
(3.11)
where s n () = n −1
u t () =
n
∇ t ()u t (),
t=1
εt () , vec[εt ()εt () − t ()]
εt () = y t − t (), V n →P V 0 = E 0 ,0 [∇ t ()−1 t (, )∇ t ()],
t (0 , 0 ) = E[u t (0 )u t (0 )|Ft−1 ]. Note in the above formulation, presumably we can correctly specify the higherorder conditional moments E[y t ⊗ y t y t |Ft−1 ] and E[y t y t ⊗ y t y t |Ft−1 ] such that ∃0 , t (0 , 0 ) = E[u t (0 )u t (0 )|Ft−1 ]. Here for concreteness, we consider a parametric model for the higher-order moments. In certain cases, we may consider a non-parametric model, following the lines in Newey (1993) and Ai and Chen (2000) who√focus on the estimation of mean parameter vector. Given a n-consistent estimator for the nuisance parameters , denoted as ˜ , as √ (0) well as a n-consistent initial estimator ˜ , the MQMLE ˜ can be obtained by the following iterative Newton-Raphson algorithm: −1 n n (k+1) (k) −1 −1 = ˜ + ∇ t t (·, ˜ )∇ t | (k) ∇ t t (·, ˜ )u t | (k) . ˜
t=1
˜
t=1
˜
(3.12)
The estimator in Eq. (3.11) is essentially a particular case of the efficient instrumental variables estimator suggested in Theorem 2.8 of Bates and White (1988). Here, for simplicity sake, we assume away the possible autocorrelation and confine our attention to the first two conditional moments. In turn, this estimator extends one of the examples in Bates and White (1988) (see pp. 13–15) to a multivariate system with possibly non-zero third-order conditional moment.
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The next theorem summarizes some of the properties of this estimator. Theorem 3.2. Suppose the assumptions in Theorem 3.1 hold. In addition, Assumption A.3 holds. Consider the modified QMLE (MQMLE) ˜ in (3.11). Then √ (i) n(˜ − 0 ) →L N(0, V 0−1 ); (ii) V 0−1 attains the minimax bound, in the sense that for any loss function l ∈ L, √ liminfn→∞ sup(F,)∈ E F {l[ n(T j − j )]} ∞ −1/2 ≥ (2) l(0j v)exp(−(1/2)v2 ) dv; −∞
J 0−1 I 0 J 0−1
− V 0−1
(iii) is p.s.d.; (iv) Suppose, in addition, (3.1) correctly specifies 3t (0 ) and 4t (0 ) defined around Eq. (3.10), I 0−1 = J 0−1 = V 0−1 ; where j is the j-th element of , T j is any arbitrary estimator for j , 02 j is the ( j, j)-th element of V 0−1 . F is a well-defined distribution. and the set of loss functions L are discussed and defined in Assumption A.3(b) and Theorem 3 of Chamberlain (1987). Parts (i) and (ii) in the above theorem are standard. They directly come from Bates and White (1988) and Chamberlain (1987), respectively. (iii) shows that, if one correctly specify the third and fourth conditional moments, the MQMLE cannot be worse than the QMLE of any QEF density, at least asymptotically. (iv) is essentially the converse of (iii). It states that if the QEF density correctly specify the third and fourth moments (apart from the first two), the corresponding QMLE is as efficient as the MQMLE.
4. CONCLUSIONS Most economic models in essence specify the mean of some explained variables, conditional on a number of explanatory variables. Since the publication of White’s (1982) Econometrica paper, a vast literature has been devoted to the quasi- or pseudo- maximum likelihood estimator (QMLE or PMLE). Among others, it was shown that QMLE of a density from the linear exponential family (LEF) provides a consistent estimate of the true parameters of the conditional mean, despite misspecification of other aspects of the conditional distribution. In this paper, we first show that it is not the case when the weighting matrix of the density and
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the mean parameter vector are functionally related. A prominent example is an autoregressive moving-average (ARMA) model with generalized autoregressive conditional heteroscedasticity (GARCH) error. As a result, the mean specification test is not readily modified as heteroscedasticity insensitive. However, correct specification of the conditional variance adds conditional moment conditions for estimating the parameters in conditional mean. Based on the recent literature of efficient instrumental variables estimator (IVE), we propose an estimator which is modified upon the QMLE of a density from the quadratic exponential family (QEF). We thus document a detailed comparison between the quadratic exponential QMLE with IVE. The asymptotic variance of this modified QMLE attains the lower bound for minimax risk. Throughout, we assume the absence of dynamic misspecification. In a time-series parametric context, Sin and White (1996) may be used to determine the number of lags such that there will be no dynamic misspecification, at least asymptotically. A rigorous treatment of the weak or strong consistency of picking the correct number of lags parsimoniously is needed. In this paper, we focus on the first two moments, conditional on a σ-algebra Ft−1 . Needless to say, there may be further efficiency gain if we include more conditioning variables, in the spirit of seemingly unrelated regression (SUR) first proposed in Zellner (1962). On the other hand, given Ft−1 , as far as efficiency gain is concerned, we may exploit the fact that, for any natural number q, q
q
E 0 [εt () − t ()|Ft−1 ] = 0. In particular, Gourieroux et al. (1996) assume that some of the odd-order conditional moments are zero. While we are skeptical of this assumption, this paper shows that a linear exponential or quadratic exponential QMLE for the mean parameter may be inconsistent, if the conditional variance is misspecified. Moreover, the efficiency of our modified QMLE (of a QEF density) hinges on the correct specification of higher-order moments. All in all, the tradeoff between efficiency and misspecification-insensitivity is always an interesting issue, especially when the sample size is fairly small. Some preliminary analysis is undertaken by the author, along the lines in Vuong (1986) and Monfort (1996).
ACKNOWLEDGMENTS I thank helpful comments from Wai-yan Cheng, Thomas Fomby, Carter Hill, Halbert White, Hector O. Zapata and other participants at the 2002 Advances in Econometrics Conference. The hospitality of the two editors and the local organizers at Louisiana State University is much appreciated.
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REFERENCES Ai, C., & Chen, X. (2000). Efficient estimation of models with conditional restrictions containing unknown functions. In: C. Ai & X. Chen (Eds), Econometrics: Semiparametric Econometric Methods (pp. 125–163). China: Peking University Press. Bates, C. E., & White, H. (1988). Efficient instrumental variables estimation of systems of implicit heterogeneous nonlinear dynamic equations with nonspherical errors. In: W. A. Barnett, E. R. Berndt & H. White (Eds), Proceedings of the Third International Symposium in Economic Theory and Econometrics (pp. 3–26). New York: Cambridge University Press. Bates, C. E., & White, H. (1993). Determination of estimators with minimum asymptotic covariance matrices. Econometric Theory, 9, 633–648. Bollerslev, T. (1990). Modelling the coherence in the short-run nominal exchange rates: A multivariate generalized ARCH approach. Review of Economics and Statistics, 72, 498–505. Bollerslev, T., & Wooldridge, J. M. (1992). Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econometric Reviews, 11, 143–172. Carroll, R. J., & Ruppert, D. (1982). A comparison between maximum likelihood and generalized least squares in a heteroscedastic linear model. Journal of the American Statistical Association, 77, 878–882. Chamberlain, G. (1987). Asymptotic efficiency in estimation with conditional moment restrictions. Journal of Econometrics, 34, 305–334. Cragg, J. G. (1983). More efficient estimation in the presence of heteroscedasticity of unknown form. Econometrica, 51, 751–763. Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50, 987–1008. Gourieroux, C., Monfort, A., & Renault, E. (1996). Two-stage generalized moment method with applications to regressions with heteroscedasticity of unknown form. Journal of Statistical Planning and Inference, 50, 37–63. Gourieroux, C., Monfort, A., & Trognon, A. (1984). Pseudo-maximum likelihood methods: Theory. Econometrica, 52, 681–700. Jeantheau, T. (1998). Strong consistency of estimators for multivariate ARCH models. Econometric Theory, 14, 29–52. Jobson, J. D., & Fuller, W. A. (1980). Least squares estimation when the covariance matrix and parameter vector are functionally related. Journal of the American Statistical Association, 75, 176–181. Ling, S., & McAleer, M. (2003). Asymptotic theory for a new vector ARMA-GARCH model. Econometric Theory, 19, 280–310. MaCurdy, T. E. (1999). Using information on the moments of disturbances to increase the efficiency of estimation. In: C. Hsiao, K. Morimune & J. L. Powell (Eds), Nonlinear Statistical Modeling – Proceedings of the Thirteenth International Symposium in Economic Theory and Econometrics: Essays in Honor of Takeshi Amemiya (pp. 281–319). New York: Cambridge University Press. Magnus, J. R. (1988). Linear structures. New York: Oxford University Press. McCullagh, P. (1994). Exponential mixtures and quadratic exponential families. Biometrika, 81, 721–729. Amendments and corrections (1996). Biometrika, 83, 248. Monfort, A. (1996). A reappraisal of misspecified econometric models. Econometric Theory, 12, 597–619. Newey, W. K. (1993). Efficient estimation of models with conditional moment restrictions. In: G. S. Maddala, C. R. Rao & H. D. Vinod (Eds), Handbook of Statistics: Econometrics (Vol. 11, pp. 419–457). Amsterdam: North-Holland.
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Pommeret, D. (2002). Information inequalities for the risks in simple quadratic exponential families. Statistics and Decisions, 20, 81–93. Sin, C.-y., & Ling, S. (2003). Estimation and testing for partially nonstationary vector autoregressive models with GARCH. Hong Kong Baptist University Discussion Paper, Hong Kong. Sin, C.-y., & White, H. (1996). Information criteria for selecting possibly misspecified parametric models. Journal of Econometrics, 71, 207–225. Vuong, Q. H. (1986). Cramer-Rao bounds for misspecified models. California Institute of Technology Social Science Working Paper 652, California. White, H. (1980). A heteroskedasticity consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48, 817–838. White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50, 1–25. Corrigendum (1983). Econometrica, 51, 513. White, H. (1994). Estimation, inference and specification analysis. New York: Cambridge University Press. Wooldridge, J. M. (1990). A unified approach to robust, regression-based specification tests. Econometric Theory, 6, 17–43. Zellner, A. (1962). An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. Journal of the American Statistical Association, 57, 348–368. Zhao, L. P., & Prentice, R. L. (1991). Use of a quadratic exponential model to generate estimating equations for means, variances, and covariances. In: V. P. Godambe (Ed.), Estimating Functions (pp. 103–117). Oxford: Clarendon Press.
APPENDIX A: REGULARITY ASSUMPTIONS The following assumptions are essentially modified upon those in White (1994). Assumptions such as those about uniform weak law of large numbers (UWLLN) and central limit theorem (CLT) are far from primitive. For primitive assumptions of a special case of an ARMA-GARCH model, we refer the readers to Ling and McAleer (2003). In that model, there are latent variables such as εt () and h t () in our main text. In Ling and McAleer (2003), the UWLLN and CLT of the essential variables (such as the derivatives of the likelihood function) are proved with primitive assumptions on the infinite past of the observables and the latent variables, which in turn approximate those essential variables and the legitimate initial values. Assumption A.1. When no ambiguity arises, we write t () = ( y t , t , t ), where t = (W t , ) and t = (W t , ). (a) The observed data are a realization of a stationary stochastic process X ≡ {X t : → Rν , t = 1, 2, . . .} on a complete probability space (, F, P), where ν ≥ l and X t = (Y t , Z t ) . Further, W t ∈ Rk t is measurable-Ft−1 , ˜ t−1 ≡ (X , . . . , X , Z t ) . where W t is a sub-vector of X 1 t−1
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(b) For t = 1, 2, . . ., and for all W t ∈ Rk t , the functions (W t , ·) and (W t , ·) are continuously differentiable of order 1 on . (c) ( y, , ) = a(, ) + b( y, ) + c(, ) y, where a(·, ·) and c(·, ·) are continuously differentiable of order 1 on int(M × K); for each y in M, b( y, ·) is continuously differentiable of order 1 on int(K), and for each in int(K), b(·, ) is a measurable function. ∇ c(, ) is non-singular for each (, ) in M × K, and for each ∈ int(M), ∇ c(, ·) is one-to-one. Further, for a given -finite measure , the functions a, b and c satisfy exp ( y, , ) d( y) = 1, and y exp ( y, , ) d( y) = , (A.1) for all (, ) in M × K. (d) (i) For each ∈ , E[1 ()] exists and is finite; (ii) E[1 (·)] is continuous on ; (iii) {t ()} satisfies a UWLLN. (e) E[1 ()] has an identifiably unique maximizer ∗ ∈ int(). Assumption A.2. As in Assumption A.1, when no ambiguity arises, we write t () = ( y t , t , t ), where t = (W t , ) and t = (W t , ). (a) Assumption A.1(a). (b) For t = 1, 2, . . ., and for all W t ∈ Rk t , the functions (W t , ·) and (W t , ·) are continuously differentiable of order 2 on . (c) ( y, , ) = A(, ) + B( y) + C(, ) y + yD(, ) y, where A(·, ·) is continuously differentiable of order 1 on int(M × K), C(·, ·) and D(·, ·) are continuously differentiable of order 2 on int(M × K); and B(·) is a measurable function. Write = ( , ) . For a given -finite measure , the functions A, B, C and D satisfy exp[A() + B( y) + C() y + y D()y] d( y) = 1, (A.2)
y exp[A() + B( y) + C() y + y D()y] d( y) = ,
(A.3)
and vec(yy )exp[A() + B( y) + C() y + y D()y] d( y) = vec( + ), (A.4) for all in M × K. (d) Assumption A.1(d). (e) Assumption A.1(e).
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2 ()] exists and is finite; (ii) E[∇ 2 (·)] is (f) (i) For each ∈ , E[∇ 1 1 2 ()} satisfies a UWLLN; (iv) E[∇ 2 (∗ )] continuous on ; (iii) {∇ t 1 is negative definite.
(g) {n −1/2 nt=1 ∇ t (∗ )} satisfies a CLT with covariance matrix J ∗ ≡ E[∇ 1 (∗ )∇ 1 (∗ )].
Assumption A.3. 0 0 0 0 (a) Define Z t = −1 t ( , )∇ t ( ), t () = u t (), and e t = u t ( ). Recall that t () is measurable-Ft−1 , there is no dynamic misspecification, and the first two conditional moments are correctly specified. It is legitimate to assume conditions (i)–(iv) in Definition 2.6, pp. 9–10 in Bates and White ˜ ˜ )], n () ≡ [t ()], and e n ≡ [e t ]. ˆ n ≡ [Z t (, (1988), with Z n ≡ [Z t ], Z (b) Refer to Condition (C 2 ) and the set of loss function L defined respectively in p. 320 and p. 314 in Chamberlain (1987). Following the lines in Lemma 4.1 of Vuong (1986), given 0 , define a parametric distribution function F 0 , assume that (F 0 , 0 ) satisfies Condition (C 2 ). is defined as the subset of the neighbourhood of (F 0 , 0 ) which elements satisfy Condition (C 2 ).
APPENDIX B: TECHNICAL PROOFS Proof of Theorem 2.1: Assumption A.1 is analogous to those for Theorem 3.5 in White (1994). As a result, ˆ → ∗ . Given the smoothness assumption in Assumption A.1(c), in view of the proof of Lemma 6.7 in pp. 118–120 in White (1994), ∇ E[t ()] = E[∇ t ()]. Thus, the first order condition of maximization yields E[∇ t (∗ )] = 0. Straightforward differentiation yields the expression for ∇ t (∗ ) as in Eq. (2.2). The second term in Eq. (2.2) is due to the dependence of a(·, ·), b(·, ·) and c(·, ·) on . Proof of Lemma 2.1: Consider ∇ t in Eq. (2.2), Theorem 2.1. Given the conditional log-likelihood function in Eq. (2.4), it follows that (for details, see Ling & McAleer, 2003, or Sin & Ling, 2003) the first p 1 element of ∇ t () is: ∇ t ()−1 t ()εt () + ∇ t ()∇ t ().
(B.1)
ˆ into (B.1). In view of Plug in ∗ = [∗ , ␣∗ ] , the probability limit of , Theorem 2.1, it follows that: ∗ ∗ ∗ ∗ E[∇ t (∗ )−1 t ( )εt ( )] + E[∇ t ( )∇ t ( )] = 0.
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Thus (2.4) is proved. The expressions for ∇ t (∗ ) and ∇ t (∗ ) can be derived, following the lines in Ling and McAleer (2003) or Sin and Ling (2003). Proof of Theorem 3.1: Given Assumption A.2, ∗ exists. ∗ = 0 follows by Property 6, p. 691 in Gourieroux et al. (1984). The rest of this proof extends that in Appendix 5 of Gourieroux et al. (1984) to a multivariate system. To economize the notation, we write = t (). In view of (3.1), ∇ = ∇ [∇ A + ∇ Cy + ∇ Dvec(yy )].
(B.2)
However, given (A.2) in Appendix A, similar to (A5.1) in Gourieroux et al. (1984), we have: ∇ A = −∇ C − ∇ Dvec( + ).
(B.3)
Putting (B.3) into (B.2), ∇ = ∇ [∇ C( y − ) + ∇ Dvec(yy − − )].
(B.4)
Thus the expression for I 0 in Eq. (3.4) is shown. On the other hand, J = E 0 (∇ ) = E 0 ∇ 0
= E 0 ∇ = E 0 ∇
Il M ∇ C + ∇ D ∇ 0l2 xl Il 2
∇ C + M(∇ D) ∇ C + M(∇ D)
∇ D
∇ D
∇ C + (∇ D)M ∇ D
∇ D
∇ D
∇
∇
,
(B.5)
in which the last equality is due to the symmetry of J 0 . Thus (3.3) is also shown. Lastly, we show (3.10). For brevity of exposition, for any variable U t which is measurable-Ft , denote E −1 [U] = E 0 [U t |Ft−1 ]. From (3.4), I 0 = E 0 {∇ [∇ C( y − )( y − ) ∇ C + ∇ C( y − ) × [vec(yy − − )] ∇ D + ∇ Dvec(yy − − ) × ( y − ) ∇ C + ∇ Dvec(yy − − ) × [vec(yy − − )] ∇ D]∇ }.
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However, E −1 [( y − )( y − ) ] = , E −1 [vec(yy − − )( y − ) ] = E −1 [vec(yy )y − vec(yy ) ] = 3 , E −1 [vec(yy − − )[vec(yy − − )] ] = E −1 [vec(yy )[vec(yy )] − vec( + )[vec( + )] ] = 4 .
Thus (3.10) is also proved.
Proof of Lemma 3.1: Consider the conditional log-likelihood function in Eq. (3.6). Refer to the general QEF density in Eq. (3.1), 1 A(, ) = − ( + ln||), 2 1 B( y) = − ln(2), 2 C(, ) = −1 , 1 D(, ) = − −1 . 2 Consider the terms in the expression for J 0 , (3.3). ∇ C = −1 ,
∇ = (∇ , ∇ ),
∇ D = 0lxl 2 ,
∇ D =
1 −1 ( ⊗ −1 ). 2 (B.6)
Putting (B.6) into (3.3),
J = E 0 (∇ , ∇ ) 0
−1 0l2 xl
= E 0 [∇ (−1 )∇ +
0lxl2 ∇ 1 −1 ∇ ( ⊗ −1 ) 2
1 ∇ (−1 ⊗ −1 )∇ ] 2
1 ∇ (−1 ⊗ −1 )∇ 2 = E 0 1 ∇␣ (−1 )∇ + ∇␣ (−1 ⊗ −1 )∇ 2 1 ∇ (−1 )∇ + ∇ (−1 ⊗ −1 )∇  2 = E 0 0p2 xp1 ∇ (−1 )∇ +
1 ∇ (−1 ⊗ −1 )∇␣ 2 1 ∇␣ (−1 )∇␣ + ∇␣ (−1 ⊗ −1 )∇␣ 2 0p1 xp2 , 1 −1 −1 −1 ∇␣ ( )∇␣ + ∇␣ ( ⊗ )∇␣ 2 ∇ (−1 )∇␣ +
as E 0 [∇ (−1 )∇␣ ] + (1/2)E 0 [∇ (−1 ⊗ −1 )∇␣ ] = 0p 1 xp 2 .
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Given the information matrix equality, the asymptotic variance of ˆ is = J 0−1 , which equals:
I 0−1
−1 1 −1 −1 −1 0p1 xp2 E 0 [∇ ∇ + 2 ∇ ⊗ ∇ ] . 1 −1 −1 −1 −1 0p2 xp1 (E0 [∇␣ ∇␣ + ∇␣ ⊗ ∇␣ ]) 2
In particular, the asymptotic variance of ˆ is 0−1 , where 1 0 −1 −1 −1 = E 0 ∇ ( )∇ + ∇ ( ⊗ )∇ . 2 It is clear that (E 0 [∇ (−1 )∇ ])−1 − 0−1 is p.s.d., since 0 − E 0 [∇ (−1 )∇ ] is p.s.d. Lemma B.1. Suppose the assumptions in Theorem 3.1 hold.
Il ˜3 = (∇ C) + (∇ D) . 0l2 xl
M ˜ 3 + (∇ D) ˜4 = (∇ C) , Il 2
(B.7)
(B.8)
˜ 3 and ˜ 4 are the conditional moments similar to 3 and 4 for the where conditional density specified in Eq. (3.1). Proof of Lemma B.1: Differentiate (A.3) w.r.t. ,
˜ −1 ( y ⊗ yy ) = ∇ A + ∇ C( + ) + ∇ D E
Il 0l2 xl
,
(B.9)
˜ −1 (·) denotes the expectation for the conditional density specified in where E Eq. (3.1). Differentiate (A.4) w.r.t. ,
M ˜ −1 ( y ⊗ yy ) + ∇ D E ˜ −1 ( yy ⊗ yy ) = ∇ A[vec( + )] + ∇ C E . Il2 (B.10) ˜ 3, ˜ 4 and M. The Put (B.3) into (B.9) and (B.10). Recall the definitions of lemma is proved.
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Proof of Theorem 3.2: Given Assumption A.3(a), Theorem 3.2(i) follows by Theorem 2.8 in Bates and White (1988). Here we assume away the possible autocorrelation and under the assumptions in Theorem 3.1, E 0 [∇ u t ()|Ft−1 ] = −∇ t (0 ). Given Assumption A.3(b), Theorem 3.2(ii) comes directly from Theorem 3, pp. 322–323 in Chamberlain (1987). Next we prove Theorem 3.2(iii). Recall the Hessian for the QMLE (denoted as J 0 ), and the “generalized” scores for the QMLE and for the MQMLE as expressed in Eqs (3.3), (3.4) and (3.12) respectively. In view of the proof of Theorem 2.6 in Bates and White (1993), it suffices to show that: 0 E 0 ,0 [∇ t ()−1 t (, )u t ()∇ t ] = J .
As no ambiguity arises, we write Eq. (3.4),
∇
=
∇ t .
(B.11)
Also write ε = y − . From
∇ = (ε ∇ C + [vec(εε + ε + ε − )] ∇ D)∇ . Write = t (, ). Therefore, (B.11) can be expressed as: E 0 ,0 [∇ −1 u(ε ∇ C + [vec(εε + ε + ε − )] ∇ D)∇ ] = J 0 . (B.12)
12 Partition such that ≡ . Refer to (B.12). Consider 12 22 E −1 {u(ε , [vec(εε + ε + ε − )] )} first. The upper-left and the lower-left sub-matrices are E −1 [εε ] = ,
E −1 [vec(εε − )ε ] = 12 .
Next consider the upper-right sub-matrix. E −1 {ε[vec(εε − )] + ε[vec(ε )] + ε[vec(ε )] } = 12 + M, where the derivation of the second term involves much tedious algebra which is not reported here. Then consider the lower-right sub-matrix. E −1 {vec(εε − )[vec(εε − )] + vec(εε − )ε[vec(ε )] + vec(εε − )ε[vec(ε )] } = 22 + 12 M, where, again, the derivation of the second term involves much tedious algebra which is not reported here.
11 12 . Partition −1 such that −1 ≡ 12 22
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Thus −1 E −1 {u(ε , [vec(εε + ε + ε − )] )} can be expressed as:
0lxl (11 + 12 12 )M 0lxl M I l+l 2 + , = I l+l 2 + 0l2 xl 0l2 xl2 0l2 xl (12 + 22 12 )M where the last equality is due to the properties of the inverse of a partitioned matrix. Therefore, the LHS of (B.12) can be expressed as:
M ∇ C Il E 0 ∇ ∇ , 0l2 xl Il2 ∇ D which equals J 0 , as one can see from Line 2, (B.5) in the proof of Theorem 3.1. Therefore (B.12) holds and Theorem 3.2(iii) is proved. Lastly, we prove Theorem 3.2(iv). As (3.1) correctly specifies 3t (0 ) and 4t (0 ), the information matrix equality holds and J 0−1 I 0 J 0−1 = J 0−1 . Next recall the definitions of V 0 and J 0 . Around (3.11), we define V 0 as E 0 ,0 [∇ −1 ∇ ]. From Line 3, (B.5) in the proof of Theorem 3.1, due to the symmetry of J 0 ,
∇ D C + (∇ D)M ∇ J 0 = E 0 ∇ ∇ . ∇ C + (∇ D)M ∇ D Therefore, to prove J 0 = V 0 , it suffices to show that:
∇ C + (∇ D)M ∇ D = I l+l 2 . ∇ C + (∇ D)M ∇ D Next consider the partition of and the definition of ε above. 12 = E −1 [vec(εε − )ε ] = E −1 [vec(εε )ε ] − vec()E −1 [ε ] = E −1 [ε ⊗ εε ] = E −1 [( y − ) ⊗ ( y − )( y − ) ] = 3 − M , where the last equality involves much tedious algebra which is not reported here. On the other hand, E −1 {vec(εε − )[vec(εε − )] } = E −1 {vec(εε )[vec(εε )] − vec(εε )[vec()] − vec()[vec(εε )] + vec()[vec()] } = E −1 {vec(εε )[vec(εε )] } − vec()[vec()] = E −1 {vec(( y − )( y − ) )[vec(( y − )( y − ) )] } − vec()[vec()]
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= E −1 {( y − )( y − ) ⊗ ( y − )( y − ) } − vec()[vec()] = 4 − 3 M − M 3 + M M, where the last equality involves much tedious algebra which is not reported here. Therefore, it remains to show that:
∇ C + (∇ D)M ∇ D ∇ C + (∇ D)M ∇ D
3 − M × = I l+l 2 . 3 − M 4 − 3 M − M 3 + M M First consider the upper-left sub-matrix: (∇ C) + (∇ D)M + (∇ D)3 − (∇ D)M = (∇ C) + (∇ D)3 = I l , where the last equality follows by (B.7). Next consider the lower-left sub-matrix: (∇ C) + (∇ D)M + (∇ D)3 − (∇ D)M = (∇ C) + (∇ D)3 = 0l 2 xl , where, once again, the last equality follows by (B.7). Finally consider the lowerright sub-matrix. By (B.8) and (B.7), (∇ C)3 − (∇ C)M + (∇ D)M 3 − (∇ D)M M + (∇ D)4 −(∇ D)3 M − (∇ D)M 3 + (∇ D)M M = (∇ C)3 + (∇ D)4 −[(∇ C) + (∇ D)3 ]M = I l 2 − 0l 2 xl M = I l 2 . Thus Theorem 3.2(iv) is also proved.
TESTING IN GMM MODELS WITHOUT TRUNCATION Timothy J. Vogelsang ABSTRACT This paper proposes a new approach to testing in the generalized method of moments (GMM) framework. The new tests are constructed using heteroskedasticity autocorrelation (HAC) robust standard errors computed using nonparametric spectral density estimators without truncation. While such standard errors are not consistent, a new asymptotic theory shows that they lead to valid tests nonetheless. In an over-identified linear instrumental variables model, simulations suggest that the new tests and the associated limiting distribution theory provide a more accurate first order asymptotic null approximation than both standard nonparametric HAC robust tests and VAR based parametric HAC robust tests. Finite sample power of the new tests is shown to be comparable to standard tests.
1. INTRODUCTION The generalized method of moments (GMM) estimation method has now become one of the standard methodologies in econometrics since it was first introduced to the econometrics literature by the influential paper Hansen (1982). GMM is widely used in empirical macroeconomics and finance. GMM is appealing because it can deliver consistent estimates of parameters in models where likelihood functions are either hard or impossible to write down. The class of GMM models Maximum Likelihood Estimation of Misspecified Models: Twenty Years Later Advances in Econometrics, Volume 17, 199–233 Copyright © 2003 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(03)17010-7
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is large and includes such special cases as linear regression, nonlinear regression, instrumental variables (IV), and maximum likelihood. In macroeconomic and finance applications, heteroskedasticity and/or autocorrelation of unknown form is usually an important specification issue. In many cases, GMM estimators are consistent in spite of heteroskedasticity and/or autocorrelation and can have certain optimality properties. Therefore, heteroskedasticity and/or serial correlation is not a problem, per se, for estimation, but it does affect inference in that standard errors robust to heteroskedasticity and/or serial correlation are required. Such standard errors are often called HAC robust standard errors because they are computed using heteroskedasticity autocorrelation consistent (HAC) covariance matrix estimators. A key component in constructing HAC robust standard errors is the estimation of the spectral density at frequency zero of the random process that underlies the moment conditions that define a GMM estimator. The HAC robust standard error literature in econometrics has grown from and extended the spectral density estimation literature in time series statistics. Usually, nonparametric estimators have been proposed although parametric estimators have been receiving some attention recently. Some key contributions to the nonparametric approach include White (1984), Newey and West (1987), Gallant (1987), Gallant and White (1988), Andrews (1991), Andrews and Monahan (1992), Hansen (1992) and Robinson (1998). Recent papers by den Hann and Levin (1997, 1998) have argued that parametric estimators based on vector autoregression (VAR) approximations can have certain advantages over the nonparametric approach. A practical problem with both the nonparametric and parametric approach to constructing HAC robust standard errors is the need to choose “tuning parameters” such as a truncation lag or bandwidth in the nonparametric approach or the lag length in the parametric approach. In both cases, asymptotic theory designed to determine conditions under which HAC robust standard errors will be consistent only provides conditions on the rates at which truncation lags or VAR lags must grow as the sample size increases. However, an empirical researcher faced with a finite sample could choose any truncation lag or VAR lag and justify that choice using clever asymptotic arguments. Thus, no standard has emerged for the computation of HAC robust standard errors in practice. This situation contrasts models with only heteroskedasticity where the approach proposed by White (1980) has become an empirical standard of practice. The tuning parameter problem has not gone unnoticed in the HAC literature and many data dependent methods have been proposed for choosing the tuning parameters. The practical drawback to the data dependent approach is that the choice of truncation lag or VAR lag length is replaced with choices such as approximating model of serial correlation (nonparametric approach) or maximal
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VAR lag length (parametric approach). Data dependent methods, while important improvements over the basic case, ultimately require practical choices and do not establish a standard of practice for the computation of HAC robust standard errors. The goal of this paper is to propose a different approach to the computation of HAC robust standard errors in GMM models that could potentially lead to a standard of practice in empirical applications. The results in this paper build upon and extend to the GMM framework the approach proposed by Kiefer and Vogelsang (2002a). The results in this paper also generalize to the GMM framework the approach of Kiefer, Vogelsang and Bunzel (2000) which is a special case of Kiefer and Vogelsang (2002a) as shown by Kiefer and Vogelsang (2002b). The basic idea is to consider an alternative view to the notion that valid standard errors can only be obtained by searching for consistent covariance matrix estimators. Consistency is a sufficient, but not necessary condition, for obtaining valid standard errors. The class of standard errors considered here are those constructed using nonparametric spectral density estimators but without truncation; the truncation lag is chosen to be the sample size. This new approach requires a new asymptotic theory for HAC robust tests. Kiefer and Vogelsang (2002a) developed the required distribution theory for linear regression models. It was not obvious that the results obtained by Kiefer and Vogelsang (2002a) extend to over-identified GMM models. However, the new asymptotic theory does go through smoothly for over-identified GMM models. This result is the main theoretical contribution of the paper. An additional advantage of the new approach is a more accurate first order asymptotic approximation of the finite sample null distributions of t and F tests. It has been well documented that tests based on traditional HAC robust standard errors can perform badly in finite samples and are subject to substantial size distortions. See, for example, Andrews (1991), Andrews and Monahan (1992), den Haan and Levin (1997), and, with a focus on GMM, a special issue of the Journal of Business and Economic Statistics (Vol. 14, July 1996). Size distortions associated with the new tests (presented here) are smaller and yet power remains comparable to standard tests. The remainder of the paper is organized as follows. The next section describes the model and gives assumptions sufficient for the main theoretical results. Section 3 reviews the standard approach to testing in the GMM framework. The new approach is given in Section 4 along with the theoretical results of the paper. Section 5 illustrates the new tests in the special case of linear IV estimation. A finite sample simulation study compares and contrasts the new tests with standard tests. Section 6 concludes. Mathematical proofs and kernel formulas are given in two appendices.
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The following notation is used throughout the paper. The symbol ⇒ denotes weak convergence, B j (r) denotes a j-vector of standard Brownian motions ˜ j (r) = B j (r) − rB j (1) denotes a (Wiener processes) defined on r ∈ [0, 1], B j-vector of standard Brownian bridges, and [rT] denotes the integer part of rT for r ∈ [0, 1].
2. THE MODEL AND ASSUMPTIONS Suppose we are interested in estimating the p × 1 vector of parameters, , where 0 denotes the true value of . Let vt denote a vector of observed data for t = 1, 2, . . . , T. Assume that q moment conditions hold for the data and that these moment conditions can be written as E[f(vt , 0 )] = 0,
(1)
where f(·) is a q × 1 vector of functions with q ≥ p. The moment conditions given by (1) are often derived from economic models. The basic idea of GMM estimation is to find a value for that most closely satisfies an empirical analog of themoment conditions (1). Define g t () = T −1 tj=1 f(vj , ) where g T () = T −1 Tt=1 f(vt , ) can be viewed as the sample analog to (1). Then, the GMM estimator of based on a sample of T observations is defined as ˆ T = arg min g T () W T g T () ∈
where W T is a q × q positive definite weighting matrix. ˆ T can also be defined in terms of the solution to the first order conditions (FOC) of the minimization problem G T (ˆ T ) W T g T (ˆ T ) = 0,
(2)
T −1 tj=1 ∂f(vj , )/∂ .
When the model is exactly identified and where G t () = q = p, the weighting matrix becomes irrelevant and ˆ T is defined by the equation g T (ˆ T ) = 0. A function related to G t () will be important for technical developments. Application of the mean value theorem for each t = 1, 2, . . . , T implies that g t (ˆ T ) = g t (0 ) + G t (¯ t,T )(ˆ T − 0 ),
t = 1, 2, . . . , T
(3)
(i) (i) (i) (i) where for i = 1, . . . , q, ¯ t,T = t,T 0 + (1 − t,T )ˆ T for some 0 ≤ t,T ≤ 1 and (i)
t,T is the q × 1 vector with ith element t,T . Because the focus of this paper is on hypothesis testing, it is taken as given that ˆ T is consistent and asymptotically normally distributed. So, rather than
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focus on well known regularity conditions under which ˆ T is consistent and asymptotically normally distributed (consult Hansen, 1982; Newey & McFadden, 1994), attention is given to high level assumptions that are sufficient for obtaining the main results of the paper. To that end, the following assumptions are used: Assumption 1. p lim ˆ T = 0 . 1/2 g Assumption 2. T −1/2 [rT] q (r) where is [rT] (0 ) ⇒ B t=1 f(vt , 0 ) = T the matrix square root of , i.e. = , where = ∞ j=−∞ j , j = E[f(vt , 0 ) f(vt−j , 0 ) ]. Assumption 3. p lim G [rT] (ˆ T ) = rG 0 and p lim G [rT] (¯ [rT],T ) = rG 0 uniformly in r ∈ [0, 1] where G 0 = E[∂f(vt , )/∂ ] and G 0 has full row rank. Assumption 4. W T is positive semi-definite and p lim W T = W ∞ where W ∞ is a matrix of constants. These assumptions are fairly standard with the exception of Assumption 2. Assumption 2 requires that a functional central limit theorem apply to T 1/2 g t (0 ). Asymptotic normality of ˆ T requires the less stringent assumption that a central limit theorem apply to T 1/2 g T (0 ). In the standard approach, however, because must be consistently estimated in order to construct asymptotically valid tests regarding 0 , regularity conditions needed to obtain a consistent estimate of are more than sufficient for Assumption 2 to hold. For example, a typical regularity condition for consistent estimation of using spectral kernel methods is that f(vt , 0 ) be a mean zero fourth order stationary process that is ␣-mixing (see, e.g. Andrews, 1991). Assumption 2 holds under the milder assumption that f(vt , 0 ) is a mean zero 2 + ␦ order stationary process (for some ␦ > 0) that is ␣-mixing (see, e.g. Phillips & Durlauf, 1986). Assumption 3 does not hold for moment conditions that exhibit deterministic trending behavior. When there are deterministic trends in the model, a modified version of Assumption 3 could be made and the distribution theory would be different.
3. ASYMPTOTIC NORMALITY AND COVARIANCE MATRIX ESTIMATION The typical starting point for inference regarding the parameter vector 0 is an asymptotic normality result for ˆ T . The following lemma provides the foundation upon which the test statistics proposed in this paper are built. A proof is given in the appendix.
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Lemma 1. Under Assumptions 1–4, as T → ∞, T 1/2 (ˆ T − 0 ) ⇒ −(G 0 W ∞ G 0 )−1 G 0 W ∞ B q (1) ≡ −(G 0 W ∞ G 0 )−1 ∗ B p (1), where ∗ ∗ = G 0 W ∞ W ∞ G 0 . Given that B p (1) is a p-vector of standard normal random variables, it immediately follows from the lemma that d T 1/2 (ˆ T − 0 ) → N(0, V),
V = (G 0 W ∞ G 0 )−1 ∗ ∗ (G 0 W ∞ G 0 )−1 .
Exploiting the asymptotic normality of ˆ T , asymptotically valid test statistics regarding 0 can be constructing in the usual way (i.e. t and Wald tests) provided a consistent estimator of the asymptotic covariance matrix, V, is available. The (G 0 W ∞ G 0 )−1 term of V can be consistently estimated using [G T (ˆ T ) W T G T (ˆ T )]−1 . The middle term of V is more difficult to estimate. If we write, ∗ ∗ = G 0 W ∞ W ∞ G 0 = G 0 W ∞ W ∞ G 0 , ˆ T G T (ˆ T ) then the middle term can be consistently estimated using G T (ˆ T ) W T W ˆ where is a consistent estimator of . Therefore, V can be consistently estimated using ˆ T G T (ˆ T )[G T (ˆ T ) W T G T (ˆ T )]−1 . Vˆ = [G T (ˆ T ) W T G T (ˆ T )]−1 G T (ˆ T ) W T W Recall from Assumption 2 that is the infinite sum of the autocovariances of f(vt , 0 ). It is well known that is equal to 2 times the spectral density matrix of f(vt , 0 ) evaluated at frequency zero. Therefore, it has become standard in the GMM framework to use spectral density estimators from the time series literature as a way of estimating . One of the most popular classes of spectral density estimators, the nonparametric class, take the form T−1 j ˆ ˆ = k (4)
j M j=−(T−1)
with
ˆ j = T −1 Tt=j+1 f (vt , ˆ T )f (vt−j , ˆ T )
ˆ j = T −1 Tt=−j+1 f (vt+j , ˆ T )f (vt , ˆ T )
for j ≥ 0, for j < 0,
where k(x) is a kernel function ∞ satisfying k(x) = k(−x), k(0) = 1, |k(x)| ≤ 1, k(x) continuous at x = 0 and −∞ k 2 (x) dx < ∞. Often k(x) = 0 for |x| > 1 so M ˆ to be “trims” the sample autocovariances and acts as a truncation lag. For
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consistent, M → ∞ and M/T → 0 as T → ∞. The technical requirement that M grows at a suitable rate has long been recognized as an important practical limitation to nonparametric spectral density estimators. The problem in practice is that for any finite sample of size T, any choice of M ≤ T can be justified as consistent with any rate of increase in M. For example, suppose we adopt the rule that M = CT 1/3 where C is a finite constant. If T = 100 and the practitioner decides that M = 29 gives a desired result, then this choice of M could be justified ˆ by using C = 6.2478. Then, in theory, the on the grounds of consistency of practitioner would have to use this value of C should more data become available, but the practitioner is already using all the available data. The arbitrary nature of the choice of M in finite samples has led to the development of data dependent choices of M most notably by Andrews (1991) and Newey and West (1994) in the context of covariance matrix estimation. The basic idea is ˆ (or some asymptotic to choose M to minimize the mean square error (MSE) of ˆ approximation of the MSE). Because the MSE of depends on the serial correlation structure of f(vt , 0 ), the practitioner must choose an “approximate” model for the serial correlation of f(vt , 0 ). Once this “approximate” model is chosen, then the choice of M becomes automatic. However, this approach replaces the choice of M with the choice of an approximate model for the serial correlation in f(vt , 0 ). Because of these and other practical problems inherent in nonparametric estimation of , den Haan and Levin (1997, 1998) recommend using parametric estimators of based on VAR models fit to f(vt , ˆ T ). The parametric approach can attain the same level of generality with regard to the nature of the serial correlation in f(vt , 0 ) provided the lag of the VAR increases at a suitable rate as the sample size increases. Therefore, the choice of M is replaced by the choice of lag length. Again, data dependent methods have been proposed to help make this choice. In finite samples, data dependent lag length methods require the practitioner to choice a maximal lag length. There is no guide on this choice of maximal lag length other than the maximal lag length must increase as T increases but not too fast. So, the choice of lag length is replaced with the choice of maximal lag length.
4. A NEW APPROACH Following Kiefer and Vogelsang (2002a) consider estimating using (4) but with ˆ M=T . This is a specific choice of M that, for M = T. This estimator is denoted by kernels that truncates, uses information about all of the sample autocovariances in the data. For kernels that do not truncate, this choice of bandwidth places relatively large weights on higher order sample autocovariances. It is well known
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that letting M grow at the same rate as T results in an inconsistent estimate of . Thus, the corresponding estimate of V, ˆ M=T W T G T (ˆ T ) Vˆ M=T = [G T (ˆ T ) W T G T (ˆ T )]−1 G T (ˆ T ) W T × [G T (ˆ T ) W T G T (ˆ T )]−1 ,
(5)
is not consistent. However, it is important to keep in mind that a consistent estimate of V is only a sufficient condition for valid asymptotic testing, not a necessary condition. This fact is often overlooked, especially in textbook ˆ M=T delivers an treatments of asymptotic testing. As shown below, use of estimate of V that converges to a random variable that is proportional to V in a useful way and otherwise does not depend on unknown nuisance parameters. Therefore, asymptotically valid testing is still possible except that the distribution theory becomes nonstandard; i.e. t-tests are not asymptotically normal. In a sense, merely appealing to consistency of an estimator of V leads to a less than satisfactory asymptotic approximation for t-tests based on ˆ T . Consistency is best viewed as the minimally required asymptotic property for an estimator. For example, showing that ˆ T is consistent is the natural first step when determining whether it is a useful estimator. But, if one is interested in testing hypotheses about 0 , then consistency of ˆ T is not enough, and an asymptotic normality result for T 1/2 (ˆ T − 0 ) is required. In other words, testing requires at least a first order asymptotic approximation to the finite sample distribution of ˆ T . One should perhaps view consistency as a less than first order asymptotic approximation. When V is replaced with a consistent estimator and V is treated as known asymptotically, a constant is being used to approximate the finite sampling variability of a random variable. In small samples, this approximation can be inaccurate. In other words, the standard asymptotic approximation for the t-test ˆ M=T and a ignores the finite sample variability of the estimate of V. By using first order asymptotic approximation of its finite sampling behavior, it is possible to obtain a more accurate asymptotic approximation for t and F tests regarding 0 .
ˆ M=T 4.1. Asymptotic Distribution of Ω ˆ M=T is derived. When the In this subsection the asymptotic distribution of ˆ model is exactly identified and q = p, then M=T is asymptotically proportional to and it is easy to show that the resulting estimate of V is proportional ˆ M=T is no to V. However, when the model is over-identified and q > p, then 1 longer proportional to asymptotically. This does not pose any problems ˆ M=T W T G T (ˆ T ), does have because the middle term of Vˆ M=T , G T (ˆ T ) W T
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the required asymptotic proportionality to variance nuisance parameters. The following theorem gives the asymptotic result. Theorem 1. Suppose Assumptions 1–4 hold. If k (x) exists for x ∈ [−1, 1] and is continuous, then as T → ∞, 1 1 ∗ ˆ ˆ ˆ ˜ p (r)B ˜ p (s) dr ds∗ . −k (r − s)B G T (T ) W T M=T W T G T (T ) ⇒ 0
0
If k(x) is the Bartlett kernel, then as T → ∞, ˆ M=T W T G T (ˆ T ) ⇒ 2∗ G T (ˆ T ) W T
1
˜ p (r)B ˜ p (r) dr∗ . B
0
) W
ˆ M=T W T G T (ˆ T ) can be used for valid testing is The reason that G T (ˆ T T because it is asymptotically proportional to ∗ ∗ and the random variables 1 1 1 ˜ ˜ ˜ ˜ 0 0 −k (r − s)Bp (r)Bp (s) dr ds and 0 Bp (r)Bp (r) dr do not depend on unknown nuisance parameters.
4.2. Inference Without Truncation In this section it is shown how asymptotically valid (i.e. nuisance parameter free or pivotal) t and F statistics can be constructed using Vˆ M=T . Suppose the hypothesis of interest can be written as H0 : H1 :
r(0 ) = 0 r(0 ) = 0,
where r(0 ) is a m × 1 vector (m ≤ p) of continuously differentiable functions with derivative matrix R(0 ) = ∂r(0 )/∂ with full row rank. Application of the Delta method gives T 1/2 r(ˆ T ) ⇒ −R(0 )V 1/2 B p (1) ≡ N(0, V R ) where V R = R(0 )VR(0 ) . Suppose we use the inconsistent estimator of V R , R(ˆ T )Vˆ M=T R(ˆ T ) , to construct the F statistic F ∗ = Tr(ˆ T ) (R(ˆ T )Vˆ M=T R(ˆ T ) )−1
r(ˆ T ) . m
ˆ M=T is used The only difference between F ∗ and a conventional F test is that instead of a consistent estimator of . In the case where m = 1 a t-statistic can
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be computed as t∗ =
T 1/2 r(ˆ T ) . R(ˆ T )Vˆ M=T R(ˆ T )
A practically relevant t-test is the test of significance of the individual parameters H0 : H1 :
i = 0 i = 0.
In this case the t ∗ statistic can be written as ˆ i t∗ = , se(ˆ i ) ii ii where se(ˆ i ) = T −1 Vˆ M=T and Vˆ M=T is the ith diagonal element of the Vˆ M=T matrix. A theorem which is proved in the appendix establishes the limiting null distributions of t ∗ and F ∗ . Theorem 2. Suppose Assumptions 1–4 hold. If k (x) exists for x ∈ [−1, 1] and is continuous, then as T → ∞,
−1 1 1 ∗ ˜ m (r)B ˜ m (s) dr ds F ⇒ B m (1) −k (r − s)B B m (1)/m, 0
t ∗ ⇒ 1 1 0
0
0
B 1 (1) ˜ 1 (r)B ˜ 1 (s) dr ds −k (r − s)B
.
If k(x) is the Bartlett kernel, then as T → ∞,
−1 1 ˜ m (r)B ˜ m (r) dr F ∗ ⇒ B m (1) 2 B B m (1)/m, 0
B 1 (1) t∗ ⇒ . 1 ˜ 2 2 0 B 1 (r) dr According to the theorem, the asymptotic distributions are free of nuisance parameters although the distributions are nonstandard and depend on the kernel through k (x). Given the kernel, critical values are easily obtained using simulation methods. In the case of the Bartlett kernel for the t ∗ test, analytical critical values are available following Abadir and Paruolo (1997) and Abadir and Paruolo (2002). Asymptotic critical values for t ∗ for a group of popular kernels
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Table 1. Asymptotic Critical Values of t ∗ Statistics. Kernel
90.0%
95.0%
97.5%
99.0%
Bartlett Parzen (a) Parzen (c) Tukey-Hanning QS Daniell Bohman Normal
2.740 2.840 6.122 3.947 5.188 4.822 3.034 3.050
3.764 4.228 9.453 6.207 8.283 7.711 4.530 4.564
4.771 5.671 13.530 9.292 12.374 11.573 6.095 6.157
6.090 8.112 20.663 15.288 20.380 19.180 8.799 9.022
√ Notes: For the Bartlett kernel critical values were obtained by scaling by 1/ 2 the critical values from line 1 of Table 1 in Abadir and Paruolo (1997). See Abadir and Paruolo (2002). For the other kernels the critical values were calculated via simulation methods using i.i.d. N(0, 1) random deviates to approximate the Wiener processes defined in the respective distributions. The integrals were approximated by the normalized sums of 1,000 steps using 50,000 replications. Left tail critical values follow by symmetry around zero. See the Appendix for the kernel formulas.
Table 2. Asymptotic Critical Values for F ∗ Using Bartlett Kernel. m= 90% 95% 97.5% 99% m= 90% 95% 97.5% 99% m= 90% 95% 97.5% 99%
1
2
3
4
5
6
7
8
9
10
14.28 23.14 33.64 51.05
17.99 26.19 35.56 48.74
21.13 29.08 37.88 51.04
24.24 32.42 40.57 52.39
27.81 35.97 44.78 56.92
30.36 38.81 47.94 60.81
33.39 42.08 50.81 62.27
36.08 45.32 54.22 67.14
38.94 48.14 57.47 69.67
41.71 50.75 59.98 72.05
11
12
13
14
15
16
17
18
19
20
44.56 53.70 63.14 74.74
47.27 56.70 65.98 78.80
50.32 60.11 69.46 82.09
52.97 62.83 72.46 85.12
55.71 65.74 75.51 88.86
58.14 68.68 78.09 91.37
60.75 70.59 80.94 94.08
63.35 73.76 83.63 97.41
21
22
23
24
25
26
27
28
65.81 68.64 76.42 79.50 86.20 89.86 99.75 103.2 29
30
70.80 73.41 76.19 78.40 81.21 83.59 85.83 88.11 90.92 93.63 82.00 84.76 87.15 89.67 92.70 95.49 97.57 99.48 102.9 105.8 92.32 94.54 98.06 100.4 103.5 106.6 108.8 110.7 114.6 117.5 105.4 108.0 111.8 114.7 117.6 120.8 123.4 124.5 129.6 132.1
Notes: The critical values were calculated via simulation methods using normalized partial sums of 1,000 i.i.d. N(0, 1) random deviates to approximate the standard Brownian motions in the respective distributions. 50,000 replications were used. m is the number of restrictions being tested.
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are tabulated in Table 1. Asymptotic critical values for F ∗ for the Bartlett kernel for m = 1, 2, . . . , 30 are tabulated in Table 2. Using the truncation lag of M = T when estimating the asymptotic variance delivers an asymptotically valid class of tests even though the asymptotic variance is not consistently estimated. The practical advantage of using M = T is that this choice of truncation lag is specific in finite samples and the first order asymptotic approximation explicitly captures this choice. What remains, though, is the choice of kernel. Using a local asymptotic power analysis in a linear regression model, Kiefer and Vogelsang (2002a) showed that among the kernels given in Table 1, the Bartlett kernel delivers the highest power for the t ∗ statistic.2 Their power analysis easily generalizes to the GMM framework and the result for the Bartlett kernel continues to hold. Details are not provided here for the sake of brevity.
5. LINEAR IV MODELS: CHOICE OF WT AND FINITE SAMPLE PERFORMANCE In this section attention is focused on the special case of linear instrumental variables (IV) models. Beyond serving as an example, this section has two additional goals. The first goal is to explore the choice of the weighting matrix, W T , which is greatly simplified in the linear case because the formula for ˆ T can be written in a closed form. The second goal is to illustrate the finite sample performance of the new tests, and this is also simplified (in terms of computation complexity) in the linear model.
5.1. Linear IV Model Let x t denote a p × 1 vector of regressors. Consider the linear regression y t = x t 0 + u t , where u t is a mean zero error term and x t and u t could be correlated. Suppose that a q × 1 vector of instruments, z t , are available that satisfy the moment conditions E(z t u t ) = 0. For the instruments to be valid we also need the assumption that E(x t z t ) have full row rank equal to q. In terms of the general notation from above, we have vt= (y t , x t , z t ) and f(vt , 0 ) = z t u t = z t (y t − x t 0 ). It follows that g T () = T −1 Tt=1 z t (y t − x t ). Given a weighting matrix, W T , it is easy to
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show that
T
−1 T
T
T ˆ T = z t x t W T z t x t z t x t W T zt yt . t=1
t=1
t=1
(6)
t=1
The asymptotic variance of ˆ T is V = (Q zx W ∞ Q zx )−1 Q zx W ∞ W ∞ Q zx (Q zx W ∞ Q zx )−1 , where Q zx = p lim (T −1 Tt=1 z t x t ) and = limT→∞ VAR (T −1/2 Tt=1 z t u t ). The asymptotic variance matrix can be consistently estimated using ˆ zx )−1 Q ˆ zx W T W ˆ TQ ˆ zx (Q ˆ zx )−1 , ˆ zx W T Q ˆ zx W T Q Vˆ = (Q
(7)
−1 T
ˆ ˆ ˆ zx = T ˆt = where Q t=1 z t x t and is given by (4) using f(vt , T ) = z t u z t (y t − x t ˆ T ). The corresponding variance matrix with M = T is given by
ˆ zx W T Q ˆ zx W T Q ˆ zx )−1 Q ˆ zx W T ˆ M=T W T Q ˆ zx (Q ˆ zx )−1 . Vˆ M=T = (Q
(8)
As shown by Hansen (1982), the weighting matrix that gives the optimal ˆ −1 . In this case, (minimum asymptotic variance) GMM estimator is W T = the asymptotic variance of ˆ T simplifies to V opt = (Q zx −1 Q zx )−1 and the ˆ zx ˆ is ˆ −1 Q ˆ zx )−1 . Because corresponding estimator simplifies to Vˆ opt = (Q ˆ ˆ constructed using T , there can be improvements in the precision of through 0 iteration. Let ˆ T denote an initial estimate of obtained with an arbitrary ˆ ˆ 0T ) denote the estimate of obtained weighting matrix (e.g. W T = I). Let ( 0 ˆ ˆ 0T )−1 we obtain the updated using ˆ T . Now, using the weighting matrix, W T = ( 1 1 ˆ ˆ 1T ) estimate of , ˆ T . Using ˆ T we can then construct a new estimate of , ( 1 −1 ˆ ˆ T ) . This algorithm is iterated until the and a new weighting matrix W T = ( estimate of converges. In what follows, iteration is always used when using ˆ −1 and the corresponding estimate of is denoted by ˆ opt WT = T . If one would like to completely avoid the choice of truncation lag when doing ˆ −1 is required. In the case where inference, a weighting matrix other than W T = the errors and instruments are i.i.d., the optimal weighting matrix simplifies to −1 ˆ −1 WT = Q zz = T
T
−1 z t z t
.
(9)
t=1
Use of this weighting matrix generates the well known generalized IV estimator GIVE (GIVE) which is denoted by ˆ T .
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5.2. Finite Sample Performance Monte Carlo simulations are used to compare and contrast the performance of standard GMM tests with the new tests. The role of the weighting matrix is highlighted in this exercise. The data are driven by the mean zero vector of innovations t t = t t where t is a scalar, t = (2t , 3t ) and t = (1t , 2t , 3t ) . t is assumed to be a Gaussian white noise vector time series with 1 01×3 E(t t ) = I2 , 03×1 I3 where = E(t t ) =
0.5 , 0.5
0.5 = E(t t ) = 0.0 0.5
0.0 0.5 , 0.5
and t = 0 for t ≤ 0. Consider IV estimation of the following simple regression model with ARMA(2, 1) errors: yt = 1 + 2 x2t + 3 x3t + ut ut = 1 ut−1 + 2 ut−2 + t + t−1 ,
(10)
where x it = 0.9x i,t−1 + it ,
i = 2, 3.
There are three instruments, z t = (z 1t , z 2t , z 3t ) available given by zit = 0.9zi,t−1 + it , z3t = 3t .
i = 1, 2,
Given the structure of t , the regressors are correlated with the errors. The instruments z 1t and z 3t are correlated with x 2t while the instruments z 2t and z 3t are correlated with x 3t . Because one instrument is iid and the other two are AR(1) processes, the vector f(vt , ˆ T ) = z t uˆ t contains time series that do not have identical
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serial correlation structure. The instruments are uncorrelated with the regression error as required for IV estimation. The vector of parameters = (1 , 2 , 3 ) is estimated using (6). Three possible ˆ −1 weighting matrices are used. The first is W T = Q zz which is given by (9). This weight matrix does not require a preliminary estimate of nor does it require an ˆ −1 estimate of . The second weight matrix is the optimal weight matrix W T = QS ˆ QS is the quadratic spectral kernel estimator of using the VAR(1) plug-in where data dependent bandwidth proposed by Andrews (1991). The third weight matrix ˆ −1 ˆ is the optimal weight matrix W T = VAR where VAR is the parametric VAR estimator of with the VAR lag length chosen by the AIC criterion. The VAR lag lengths are allowed to be different across equations in the VAR. See den Haan and Levin (1997, 1998) for details. The maximal lag length depends on the sample size and is given in the tables. Both optimal weighting matrices are implemented with up to 20 iterations. The focus is on 2 . The null hypothesis is H 0 : 2 ≤ 0, and the alternative is H 1 : 2 > 0. Because there are three weight matrices, there are three different GIVE ˆ QS ˆ VAR ˆ −1 ˆ −1 (W T = Q estimators of 2 denoted by ˆ 2 zz ), 2 (W T = QS ), and 2 −1
ˆ VAR ). For a given estimator, there are three potential t-statistics. The first (W T = is denoted by t ∗ which is based on standard errors given by (8) using the Bartlett kernel. The second is denoted by t QS which is based on standard errors given by (7) ˆ QS . The third is denoted by t VAR which is based on standard errors given using QS ˆ VAR . For ˆ GIVE by (7) using results for all three t-tests are reported. For ˆ 2 2 VAR results are reported for t ∗ and t QS . For ˆ 2 results are reported for t ∗ and t VAR . The t ∗ and t QS tests were also implemented using VAR(1) prewhitening following ˆ QS Andrews and Monahan (1992). Those tests are denoted by t ∗PW and t PW QS . For 2 , when prewhitening was applied to the tests, it was also applied to the estimate of the GMM weight matrix. To assess the accuracy of the limiting null distributions, data was generated according to (10) with 2 = 0 (1 and 3 were set to zero without loss of generality). Empirical null rejection probabilities were computed using 2,000 replications for the sample sizes T = 25, 50, 100, 200. Asymptotic 5% right tail critical values were used so that the nominal level is 0.05. Results are reported for a range of values for 1 , 2 and . The results are reported in Tables 3–6. Several interesting patterns emerge from the tables. First, in most cases empirical null rejection probabilities are closer to 0.05 for t ∗ than for t QS or t VAR , and in some cases the differences are quite large. GIVE Focusing on ˆ 2 it can be seen that t ∗PW tends to the be the least size distorted test
214
Table 3. Empirical Null Rejection Probabilities in Linear IV Model T = 25, 5% Nominal Level, 2,000 Replications.
1
GIVE ˆ 2
2
QS ˆ 2
VAR ˆ 2
t∗
t ∗PW
t QS
t PW QS
t VAR
t∗
t ∗PW
t QS
t PW QS
t∗
t ∗PW
t VAR
−0.3 0 0.3 0.5 0.7 0.9 0.95 0.99 1.3 1.3 1.3
0 0 0 0 0 0 0 0 −0.8 −0.6 −0.4
0.014 0.016 0.022 0.032 0.050 0.092 0.112 0.127 0.046 0.068 0.125
0.015 0.016 0.020 0.026 0.044 0.079 0.099 0.112 0.032 0.049 0.104
0.017 0.021 0.032 0.049 0.076 0.142 0.166 0.184 0.098 0.121 0.206
0.021 0.029 0.038 0.052 0.079 0.143 0.169 0.185 0.059 0.084 0.168
0.024 0.030 0.036 0.047 0.073 0.132 0.151 0.168 0.054 0.078 0.158
0.017 0.021 0.036 0.046 0.071 0.118 0.136 0.144 0.072 0.095 0.162
0.022 0.027 0.039 0.051 0.067 0.111 0.130 0.139 0.061 0.076 0.144
0.031 0.035 0.054 0.076 0.114 0.184 0.199 0.217 0.169 0.187 0.262
0.040 0.049 0.067 0.084 0.116 0.184 0.202 0.218 0.107 0.132 0.221
0.012 0.018 0.029 0.038 0.055 0.092 0.108 0.123 0.048 0.067 0.122
0.015 0.017 0.027 0.034 0.050 0.093 0.098 0.119 0.040 0.053 0.103
0.045 0.041 0.051 0.067 0.098 0.157 0.184 0.201 0.083 0.105 0.190
0.0
−0.3 0 0.3 0.5 0.7 0.9 0.95 0.99 1.3 1.3 1.3
0 0 0 0 0 0 0 0 −0.8 −0.6 −0.4
0.053 0.066 0.094 0.127 0.176 0.238 0.234 0.219 0.070 0.134 0.230
0.051 0.063 0.085 0.107 0.146 0.205 0.208 0.190 0.049 0.095 0.193
0.068 0.108 0.159 0.200 0.260 0.339 0.336 0.320 0.151 0.239 0.340
0.074 0.102 0.140 0.165 0.223 0.278 0.279 0.260 0.089 0.162 0.265
0.067 0.093 0.122 0.151 0.183 0.241 0.235 0.205 0.086 0.141 0.230
0.071 0.090 0.128 0.165 0.216 0.274 0.271 0.247 0.115 0.173 0.269
0.073 0.089 0.120 0.158 0.185 0.237 0.244 0.222 0.085 0.145 0.228
0.097 0.143 0.202 0.251 0.308 0.372 0.367 0.364 0.235 0.297 0.387
0.113 0.149 0.186 0.218 0.263 0.316 0.314 0.302 0.155 0.224 0.309
0.052 0.075 0.108 0.125 0.158 0.207 0.200 0.183 0.076 0.116 0.189
0.053 0.064 0.090 0.113 0.152 0.183 0.190 0.163 0.058 0.101 0.164
0.095 0.119 0.160 0.192 0.229 0.274 0.273 0.243 0.125 0.190 0.261
TIMOTHY J. VOGELSANG
−0.8
−0.3 0 0.3 0.5 0.7 0.9 0.95 0.99 1.3 1.3 1.3
0 0 0 0 0 0 0 0 −0.8 −0.6 −0.4
0.085 0.098 0.125 0.150 0.189 0.230 0.226 0.204 0.081 0.145 0.216
0.074 0.082 0.100 0.121 0.157 0.195 0.193 0.174 0.055 0.114 0.186
0.141 0.161 0.199 0.245 0.286 0.334 0.329 0.311 0.158 0.239 0.328
0.118 0.121 0.148 0.180 0.221 0.264 0.258 0.243 0.093 0.167 0.252
0.110 0.119 0.135 0.167 0.191 0.229 0.208 0.193 0.090 0.140 0.216
0.121 0.140 0.166 0.191 0.231 0.253 0.252 0.228 0.117 0.183 0.249
0.114 0.118 0.138 0.159 0.193 0.226 0.227 0.204 0.094 0.155 0.217
0.190 0.226 0.256 0.296 0.342 0.364 0.366 0.346 0.241 0.304 0.373
0.165 0.172 0.191 0.223 0.265 0.304 0.298 0.284 0.159 0.232 0.295
0.095 0.097 0.121 0.133 0.166 0.189 0.195 0.173 0.080 0.132 0.190
0.087 0.093 0.105 0.113 0.156 0.172 0.168 0.157 0.059 0.110 0.171
0.145 0.161 0.180 0.208 0.235 0.271 0.257 0.232 0.130 0.190 0.259
Testing in GMM Models Without Truncation
0.8
ˆ GIVE , W T = ˆ −1 ˆ −1 Notes: The data was generated according to model (10) with 1 = 2 = 3 = 0. The GMM weight matrices are W T = Q QS for zz for 2 QS VAR −1 ˆ VAR for ˆ 2 . The null hypothesis is H 0 : 2 ≤ 0 and the alternative hypothesis is H 1 : 2 > 0. t ∗ and t QS were implemented ˆ 2 and W T = without prewhitening while t ∗PW and t PW QS were implemented with VAR(1) prewhitening. t VAR was implemented so that each equation of the ˆ VAR could have different lag lengths. The BIC criterion was used to choose the lag lengths with a maximal lag length VAR used to construct of 3.
215
216
Empirical Null Rejection Probabilities in Linear IV Model T = 50, 5% Nominal Level, 2,000 Replications.
Table 4.
1
GIVE ˆ 2
2
QS ˆ 2
VAR ˆ 2
t∗
t ∗PW
t QS
t PW QS
t VAR
t∗
t ∗PW
t QS
t PW QS
t∗
t ∗PW
t VAR
−0.3 0 0.3 0.5 0.7 0.9 0.95 0.99 1.3 1.3 1.3
0 0 0 0 0 0 0 0 −0.8 −0.6 −0.4
0.015 0.016 0.022 0.030 0.049 0.097 0.127 0.136 0.022 0.049 0.109
0.014 0.017 0.022 0.026 0.041 0.094 0.115 0.121 0.013 0.031 0.087
0.014 0.016 0.019 0.026 0.050 0.138 0.169 0.182 0.040 0.060 0.150
0.016 0.016 0.017 0.025 0.052 0.126 0.156 0.168 0.011 0.029 0.116
0.015 0.015 0.018 0.026 0.052 0.128 0.160 0.171 0.020 0.037 0.115
0.020 0.023 0.028 0.039 0.057 0.114 0.143 0.150 0.038 0.062 0.134
0.021 0.025 0.030 0.041 0.060 0.109 0.132 0.143 0.029 0.048 0.111
0.018 0.020 0.026 0.034 0.061 0.156 0.193 0.202 0.076 0.100 0.185
0.016 0.019 0.024 0.035 0.068 0.141 0.178 0.187 0.035 0.059 0.138
0.014 0.018 0.023 0.034 0.044 0.100 0.120 0.139 0.029 0.044 0.109
0.017 0.019 0.026 0.032 0.048 0.098 0.118 0.128 0.020 0.039 0.097
0.015 0.019 0.020 0.030 0.061 0.139 0.165 0.186 0.032 0.049 0.125
0.0
−0.3 0 0.3 0.5 0.7 0.9 0.95 0.99 1.3 1.3 1.3
0 0 0 0 0 0 0 0 −0.8 −0.6 −0.4
0.048 0.062 0.087 0.106 0.145 0.214 0.218 0.183 0.050 0.094 0.188
0.050 0.061 0.077 0.089 0.117 0.172 0.176 0.145 0.030 0.062 0.150
0.056 0.077 0.110 0.143 0.206 0.293 0.306 0.271 0.093 0.157 0.287
0.058 0.074 0.087 0.109 0.148 0.219 0.226 0.197 0.032 0.065 0.183
0.049 0.072 0.092 0.105 0.131 0.179 0.184 0.151 0.045 0.072 0.156
0.061 0.067 0.100 0.124 0.174 0.243 0.248 0.219 0.069 0.125 0.215
0.067 0.069 0.088 0.102 0.141 0.205 0.206 0.170 0.052 0.088 0.172
0.062 0.086 0.131 0.183 0.242 0.328 0.338 0.316 0.141 0.220 0.324
0.071 0.086 0.106 0.135 0.182 0.248 0.259 0.232 0.056 0.105 0.210
0.049 0.068 0.087 0.102 0.140 0.203 0.194 0.163 0.059 0.108 0.176
0.054 0.062 0.073 0.092 0.118 0.168 0.172 0.141 0.036 0.072 0.141
0.060 0.076 0.105 0.125 0.151 0.201 0.205 0.178 0.060 0.100 0.176
TIMOTHY J. VOGELSANG
−0.8
−0.3 0 0.3 0.5 0.7 0.9 0.95 0.99 1.3 1.3 1.3
0 0 0 0 0 0 0 0 −0.8 −0.6 −0.4
0.077 0.090 0.101 0.122 0.156 0.214 0.219 0.179 0.050 0.101 0.193
0.070 0.073 0.079 0.088 0.117 0.168 0.171 0.135 0.033 0.072 0.148
0.099 0.115 0.149 0.177 0.226 0.304 0.303 0.266 0.098 0.170 0.284
0.074 0.073 0.081 0.098 0.134 0.215 0.214 0.191 0.034 0.081 0.185
0.081 0.088 0.094 0.106 0.134 0.180 0.180 0.144 0.045 0.076 0.157
0.090 0.110 0.122 0.149 0.187 0.240 0.248 0.212 0.069 0.135 0.217
0.075 0.082 0.095 0.112 0.139 0.197 0.199 0.166 0.050 0.104 0.178
0.124 0.152 0.189 0.227 0.267 0.343 0.346 0.324 0.150 0.234 0.335
0.089 0.097 0.108 0.130 0.170 0.248 0.240 0.220 0.064 0.121 0.215
0.078 0.089 0.105 0.125 0.155 0.187 0.191 0.156 0.057 0.107 0.173
0.068 0.069 0.076 0.096 0.124 0.162 0.167 0.134 0.038 0.081 0.141
0.090 0.100 0.110 0.129 0.163 0.200 0.204 0.165 0.057 0.103 0.183
Testing in GMM Models Without Truncation
0.8
ˆ GIVE , W T = ˆ −1 ˆ −1 Notes: The data was generated according to model (10) with 1 = 2 = 3 = 0. The GMM weight matrices are W T = Q QS for zz for 2 QS VAR −1 ˆ VAR for ˆ 2 . The null hypothesis is H 0 : 2 ≤ 0 and the alternative hypothesis is H 1 : 2 > 0. t ∗ and t QS were implemented ˆ 2 and W T = without prewhitening while t ∗PW and t PW QS were implemented with VAR(1) prewhitening. t VAR was implemented so that each equation of the ˆ VAR could have different lag lengths. The BIC criterion was used to choose the lag lengths with a maximal lag length VAR used to construct of 5.
217
218
Table 5.
Empirical Null Rejection Probabilities in Linear IV Model T = 100, 5% Nominal Level, 2,000 Replications. 1
GIVE ˆ 2
2
QS ˆ 2
VAR ˆ 2
t∗
t ∗PW
t QS
t PW QS
t VAR
t∗
t ∗PW
t QS
t PW QS
t∗
t ∗PW
t VAR
−0.3 0 0.3 0.5 0.7 0.9 0.95 0.99 1.3 1.3 1.3
0 0 0 0 0 0 0 0 −0.8 −0.6 −0.4
0.015 0.014 0.023 0.034 0.046 0.105 0.123 0.124 0.017 0.034 0.086
0.017 0.017 0.024 0.033 0.044 0.099 0.120 0.118 0.008 0.027 0.075
0.010 0.011 0.013 0.019 0.040 0.128 0.171 0.183 0.022 0.038 0.120
0.012 0.013 0.014 0.021 0.039 0.120 0.163 0.178 0.002 0.012 0.096
0.019 0.017 0.018 0.022 0.045 0.121 0.162 0.168 0.011 0.028 0.098
0.018 0.022 0.026 0.035 0.050 0.109 0.137 0.138 0.023 0.044 0.105
0.020 0.021 0.029 0.039 0.052 0.103 0.136 0.128 0.017 0.037 0.093
0.011 0.013 0.019 0.024 0.049 0.135 0.178 0.198 0.036 0.058 0.144
0.013 0.015 0.017 0.025 0.047 0.128 0.176 0.190 0.009 0.028 0.110
0.015 0.020 0.025 0.034 0.046 0.097 0.125 0.128 0.024 0.046 0.108
0.021 0.018 0.028 0.036 0.047 0.100 0.119 0.118 0.014 0.033 0.087
0.018 0.021 0.023 0.024 0.049 0.128 0.171 0.175 0.016 0.035 0.108
0.0
−0.3 0 0.3 0.5 0.7 0.9 0.95 0.99 1.3 1.3 1.3
0 0 0 0 0 0 0 0 −0.8 −0.6 −0.4
0.049 0.064 0.079 0.094 0.112 0.162 0.161 0.123 0.045 0.075 0.150
0.051 0.063 0.072 0.084 0.099 0.136 0.137 0.094 0.022 0.053 0.112
0.053 0.066 0.090 0.118 0.152 0.210 0.219 0.186 0.063 0.109 0.204
0.057 0.065 0.075 0.092 0.117 0.155 0.157 0.135 0.013 0.027 0.122
0.059 0.061 0.088 0.088 0.105 0.134 0.127 0.098 0.033 0.063 0.113
0.054 0.068 0.089 0.112 0.149 0.188 0.177 0.156 0.065 0.098 0.182
0.055 0.064 0.080 0.101 0.124 0.153 0.146 0.112 0.041 0.068 0.134
0.061 0.073 0.106 0.140 0.191 0.243 0.252 0.233 0.097 0.149 0.255
0.063 0.068 0.090 0.106 0.138 0.180 0.181 0.153 0.027 0.058 0.142
0.049 0.061 0.079 0.108 0.127 0.168 0.145 0.117 0.054 0.096 0.156
0.050 0.066 0.073 0.093 0.119 0.133 0.125 0.095 0.031 0.062 0.123
0.058 0.067 0.093 0.103 0.124 0.150 0.141 0.108 0.044 0.080 0.138
TIMOTHY J. VOGELSANG
−0.8
−0.3 0 0.3 0.5 0.7 0.9 0.95 0.99 1.3 1.3 1.3
0 0 0 0 0 0 0 0 −0.8 −0.6 −0.4
0.070 0.076 0.088 0.099 0.118 0.161 0.155 0.122 0.053 0.080 0.147
0.062 0.070 0.077 0.080 0.092 0.129 0.125 0.089 0.027 0.048 0.107
0.077 0.091 0.115 0.127 0.163 0.213 0.217 0.186 0.075 0.113 0.203
0.053 0.053 0.061 0.068 0.096 0.134 0.149 0.122 0.016 0.028 0.114
0.074 0.080 0.085 0.094 0.104 0.132 0.128 0.095 0.037 0.069 0.117
0.078 0.098 0.109 0.125 0.150 0.185 0.174 0.154 0.068 0.098 0.179
0.065 0.078 0.089 0.099 0.114 0.150 0.136 0.102 0.046 0.058 0.132
0.086 0.105 0.130 0.168 0.209 0.259 0.260 0.238 0.111 0.154 0.258
0.065 0.068 0.078 0.093 0.116 0.158 0.169 0.145 0.030 0.057 0.143
0.069 0.083 0.100 0.106 0.134 0.161 0.145 0.119 0.061 0.085 0.163
0.065 0.071 0.084 0.091 0.105 0.134 0.123 0.097 0.037 0.058 0.117
0.080 0.088 0.096 0.108 0.122 0.143 0.140 0.103 0.054 0.085 0.135
Testing in GMM Models Without Truncation
0.8
ˆ GIVE , W T = ˆ −1 ˆ −1 Notes: The data was generated according to model (10) with 1 = 2 = 3 = 0. The GMM weight matrices are W T = Q QS for zz for 2 QS VAR −1 ˆ VAR for ˆ 2 . The null hypothesis is H 0 : 2 ≤ 0 and the alternative hypothesis is H 1 : 2 > 0. t ∗ and t QS were implemented ˆ 2 and W T = without prewhitening while t ∗PW and t PW QS were implemented with VAR(1) prewhitening. t VAR was implemented so that each equation of the ˆ VAR could have different lag lengths. The BIC criterion was used to choose the lag lengths with a maximal lag length VAR used to construct of 10.
219
220
Table 6.
Empirical Null Rejection Probabilities in Linear IV Model T = 200, 5% Nominal Level, 2,000 Replications. 1
GIVE ˆ 2
2
QS ˆ 2
VAR ˆ 2
t∗
t ∗PW
t QS
t PW QS
t VAR
t∗
t ∗PW
t QS
t PW QS
t∗
t ∗PW
t VAR
−0.3 0 0.3 0.5 0.7 0.9 0.95 0.99 1.3 1.3 1.3
0 0 0 0 0 0 0 0 −0.8 −0.6 −0.4
0.027 0.029 0.030 0.038 0.045 0.085 0.105 0.108 0.025 0.031 0.064
0.031 0.033 0.031 0.039 0.044 0.084 0.102 0.104 0.016 0.028 0.059
0.015 0.008 0.005 0.013 0.032 0.119 0.167 0.183 0.018 0.029 0.091
0.013 0.006 0.005 0.013 0.033 0.115 0.169 0.203 0.001 0.006 0.067
0.025 0.019 0.015 0.018 0.039 0.123 0.157 0.153 0.012 0.022 0.081
0.024 0.026 0.033 0.042 0.046 0.085 0.115 0.121 0.031 0.035 0.073
0.029 0.027 0.032 0.039 0.045 0.088 0.112 0.109 0.022 0.034 0.069
0.012 0.007 0.007 0.015 0.039 0.126 0.178 0.191 0.024 0.033 0.097
0.010 0.006 0.009 0.017 0.040 0.126 0.177 0.206 0.007 0.013 0.082
0.023 0.028 0.030 0.039 0.046 0.086 0.111 0.117 0.028 0.036 0.071
0.028 0.033 0.035 0.039 0.048 0.085 0.112 0.109 0.019 0.029 0.066
0.025 0.018 0.019 0.021 0.042 0.130 0.162 0.157 0.016 0.028 0.090
0.0
−0.3 0 0.3 0.5 0.7 0.9 0.95 0.99 1.3 1.3 1.3
0 0 0 0 0 0 0 0 −0.8 −0.6 −0.4
0.057 0.062 0.069 0.073 0.080 0.104 0.112 0.104 0.047 0.060 0.093
0.057 0.062 0.065 0.068 0.071 0.087 0.091 0.080 0.029 0.043 0.067
0.050 0.062 0.072 0.089 0.106 0.151 0.159 0.164 0.061 0.086 0.133
0.055 0.060 0.061 0.067 0.078 0.011 0.129 0.124 0.007 0.012 0.063
0.057 0.055 0.069 0.072 0.082 0.108 0.108 0.094 0.042 0.061 0.085
0.052 0.063 0.073 0.076 0.095 0.120 0.124 0.136 0.054 0.078 0.120
0.054 0.064 0.072 0.071 0.083 0.094 0.098 0.094 0.044 0.063 0.085
0.051 0.067 0.080 0.101 0.130 0.172 0.185 0.195 0.074 0.107 0.173
0.056 0.066 0.068 0.080 0.100 0.128 0.140 0.142 0.021 0.033 0.086
0.050 0.063 0.075 0.077 0.085 0.113 0.116 0.113 0.052 0.068 0.109
0.052 0.063 0.067 0.077 0.084 0.099 0.094 0.091 0.037 0.050 0.089
0.055 0.059 0.076 0.083 0.096 0.115 0.120 0.109 0.048 0.068 0.099
TIMOTHY J. VOGELSANG
−0.8
−0.3 0 0.3 0.5 0.7 0.9 0.95 0.99 1.3 1.3 1.3
0 0 0 0 0 0 0 0 −0.8 −0.6 −0.4
0.069 0.066 0.072 0.074 0.077 0.103 0.109 0.105 0.051 0.068 0.089
0.064 0.061 0.061 0.062 0.066 0.083 0.085 0.077 0.033 0.044 0.067
0.064 0.072 0.085 0.098 0.111 0.150 0.162 0.159 0.070 0.093 0.140
0.044 0.041 0.043 0.043 0.048 0.094 0.117 0.124 0.011 0.011 0.063
0.065 0.068 0.074 0.082 0.085 0.103 0.103 0.093 0.046 0.061 0.095
0.075 0.072 0.084 0.088 0.097 0.121 0.122 0.139 0.065 0.082 0.121
0.068 0.065 0.069 0.072 0.077 0.090 0.095 0.092 0.046 0.058 0.082
0.080 0.087 0.102 0.116 0.136 0.183 0.188 0.198 0.085 0.115 0.181
0.053 0.052 0.058 0.065 0.076 0.101 0.131 0.141 0.025 0.035 0.079
0.073 0.074 0.077 0.084 0.090 0.113 0.109 0.113 0.056 0.077 0.105
0.068 0.066 0.076 0.071 0.078 0.093 0.091 0.088 0.042 0.051 0.080
0.070 0.075 0.085 0.098 0.102 0.110 0.115 0.109 0.051 0.068 0.098
Testing in GMM Models Without Truncation
0.8
ˆ GIVE , W T = ˆ −1 ˆ −1 Notes: The data was generated according to model (10) with 1 = 2 = 3 = 0. The GMM weight matrices are W T = Q QS for zz for 2 QS VAR −1 ˆ VAR for ˆ 2 . The null hypothesis is H 0 : 2 ≤ 0 and the alternative hypothesis is H 1 : 2 > 0. t ∗ and t QS were implemented ˆ 2 and W T = without prewhitening while t ∗PW and t PW QS were implemented with VAR(1) prewhitening. t VAR was implemented so that each equation of the ˆ VAR could have different lag lengths. The BIC criterion was used to choose the lag lengths with a maximal lag length VAR used to construct of 15.
221
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TIMOTHY J. VOGELSANG
and t QS tends to be the most size distorted. t VAR usually performs better than t QS or t PW QS . Second, in models with more persistent serial correlation all the statistics tend to over-reject . This fact is well known in the HAC literature. Third, prewhitening usually reduces size distortions even though the data is not AR(1) in most cases. GIVE Fourth, the asymptotic approximations tend to be more accurate for ˆ 2 than QS VAR PW ˆ ˆ for 2 or 2 when the t QS , t QS and t VAR statistics are used. This could occur QS VAR because the weight matrices have more sampling variability for ˆ 2 or ˆ 2 than GIVE ∗ ∗ ˆ for 2 . However, contradicting this logic is the fact that t and t PW are often VAR GIVE QS less distorted with ˆ 2 compared to ˆ 2 although with ˆ 2 they are much ∗ more size distorted. Overall, the least size distorted test is t PW when computed GIVE VAR QS and ˆ 2 . Regardless of the test, ˆ 2 yields tests with the greatest size using ˆ 2 distortions. Given that the asymptotic null approximation appears best for t ∗PW , the obvious question to ask is: does t ∗PW have any power? The answer is yes as the next simulation experiment illustrates. To simulate finite sample power of the tests, data was generated according to (10) for 2 = 0.0, 0.2, 0.4, . . ., 4.8, 5.0 (again, 1 = 3 = 0). Results are reported for T = 50 for two error models. The first model has 1 = 0.9, 2 = 0 while the second model has 1 = 1.3, 2 = −0.8. Power was calculated using finite sample critical values, i.e. power is size adjusted. This was done so that the empirical null rejection probabilities are the same for all statistics thus making power comparisons more meaningful in a theoretical sense. Of course, such size correction cannot be done in practice on actual data, but it serves a useful purpose here for comparing the performance of different tests statistics. As before, 2,000 replications were used. The results are given by Figs 1–8. Figures 1–3 and 5–7 group the tests by estimators of 2 . Figures 4 and 8 give power of t ∗PW across the three estimators of 2 . Focusing on Figs 1–3 and 5–7 we see that, given the estimator of 2 , power is similar for all the tests. Perhaps surprisingly, we see in Fig. 6 that t ∗ and t ∗PW has slightly higher power than the other tests when 2 is small. Figure 4 shows the expected result that using optimal GMM estimates leads to tests with higher (slightly) power. Counter-intuitively, Fig. 8 shows that power can be higher GIVE QS VAR for ˆ 2 than ˆ 2 and ˆ 2 when using t ∗PW . It is not obvious why this occurs although it may have something to do with the sampling variability of the GMM weight matrices. The finite sample results suggest that t ∗PW is a good alternative to standard HAC robust tests. t ∗PW has a more accurate asymptotic null approximation and power is competitive. The simulations also suggest that using “optimal” GMM weighting matrices do not necessarily lead to more powerful tests in finite samples. Clearly, more work is required to fully understand the finite sample differences in size and power across the weight matrices.
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GIVE Fig. 1. Finite Sample Power (Size Adjusted, 5% Level), ˆ 2 Model (11), T = 50, 1 = 0.9, 2 = 0.0.
QS Fig. 2. Finite Sample Power (Size Adjusted, 5% Level), ˆ 2 Model (11), T = 50, 1 = 0.9, 2 = 0.0.
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VAR Fig. 3. Finite Sample Power (Size Adjusted, 5% Level), ˆ 2 Model (11), T = 50, 1 = 0.9, 2 = 0.0.
Fig. 4. Finite Sample Power (Size Adjusted, 5% Level), Model (11), T = 50, 1 = 0.9, 2 = 0.0.
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GIVE Fig. 5. Finite Sample Power (Size Adjusted, 5% Level), ˆ 2 Model (11), T = 50, 1 = 1.3, 2 = −0.8.
QS Fig. 6. Finite Sample Power (Size Adjusted, 5% Level), ˆ 2 Model (11), T = 50, 1 = 1.3, 2 = 0.8.
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VAR Fig. 7. Finite Sample Power (Size Adjusted, 5% Level), ˆ 2 Model (11), T = 50, 1 = 1.3, 2 = −0.8.
Fig. 8. Finite Sample Power (Size Adjusted, 5% Level), Model (11), T = 50, 1 = 1.3, 2 = −0.8.
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6. CONCLUSIONS This paper proposes a new approach to testing in the GMM framework. The new tests are constructed using HAC robust standard errors computed using nonparametric spectral density estimators without truncation. While such standard errors are not consistent, a new asymptotic theory shows that they lead to valid tests nonetheless. In the context of the linear IV model, in simulations the new tests and the associated limiting distribution theory are shown to have better finite sample size, i.e. a more accurate first order asymptotic null approximation, than HAC robust tests based on both nonparametric and parametric consistent estimates of standard errors. Finite sample power of the new tests is shown to be comparable to standard tests. Because use of a truncation lag equal to the sample size is a specific recommendation, it is fully automatic. This approach could potentially lead to a standard of practice (which does not currently exist) for the computation of HAC robust standard errors in GMM models.
NOTES ˆ M=T could be obtained by demeaning f(vt , ˆ T ) 1. Asymptotic proportionality to of ˆ M=T . However, it can be shown that the term G T (ˆ T )W T ˆ M=T W T G T (ˆ T ) before computing ˆ M=T . is computationally equivalent whether or not f(vt , ˆ T ) is demeaned before computing On the other hand, as shown by Hall (2000), demeaning improves the power of tests for over-identified restrictions and is recommended. 2. It is interesting to note that Kiefer and Vogelsang (2002a) showed that the testing ˆ M=T with the approach proposed by Kiefer et al. (2000) is exactly equivalent to using 0.5 Bartlett kernel.
ACKNOWLEDGMENTS Thanks to Alastair Hall for helpful conversations on GMM. Helpful comments were provided by participants at the 2001 Summer Meetings of the Econometrics Society at the University of Maryland and the LSU Advances in Econometrics Conference, November, 2002. I also thank the Center for Analytical Economics at Cornell University. Financial support is gratefully acknowledged from the National Science Foundation grants SES-9818695, SES-0095211 and from the Cornell Institute for Social and Economic Research for a research development grant.
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REFERENCES Abadir, K. M., & Paruolo, P. (1997). Two mixed normal densities from cointegration analysis. Econometrica, 65, 671–680. Abadir, K. M., & Paruolo, P. (2002). Simple robust testing of regression hypotheses: A comment. Econometrica, 70, 2097–2099. Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59, 817–854. Andrews, D. W. K., & Monahan, J. C. (1992). An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica, 60, 953–966. den Haan, W. J., & Levin, A. (1997). A practictioner’s guide to robust covariance matrix estimation. In: G. Maddala & C. Rao (Eds), Handbook of Statistics: Robust Inference (Vol. 15, pp. 291–341). New York: Elsevier. den Haan, W. J., & Levin, A. (1998). Vector Autoregressive Covariance Matrix Estimation. Working Paper, International Finance Division, FED Board of Governors. Gallant, A. (1987). Nonlinear statistical models. New York: Wiley. Gallant, A., & White, H. (1988). A unified theory of estimation and inference for nonlinear dynamic models. New York: Basil Blackwell. Hall, A. (2000). Covariance matrix estimation and the power of the overidentifying restrictions test. Econometrica, 68, 1517–1528. Hansen, B. E. (1992). Consistent covariance matrix estimation for dependent heterogenous processes. Econometrica, 60, 967–972. Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica, 50, 1029–1054. Kiefer, N. M., & Vogelsang, T. J. (2002a). Heteroskedasticity-autocorrelation robust testing using bandwidth equal to sample size. Econometric Theory, 18, 1350–1366. Kiefer, N. M., & Vogelsang, T. J. (2002b). Heteroskedasticity-autocorrelation robust standard errors using the bartlett kernel without truncation. Econometrica, 70, 2093–2095. Kiefer, N. M., Vogelsang, T. J., & Bunzel, H. (2000). Simple robust testing of regression hypotheses. Econometrica, 68, 695–714. Newey, W. K., & McFadden, D. L. (1994). Large sample estimation and hypothesis testing. In: R. Engle & D. L. McFadden (Eds), Handbook of Econometrics (Vol. 4, pp. 2113–2247). Amseterdam, The Netherlands: Elsevier. Newey, W. K., & West, K. D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55, 703–708. Newey, W. K., & West, K. D. (1994). Automatic lag selection in covariance estimation. Review of Economic Studies, 61, 631–654. Phillips, P. C. B., & Durlauf, S. N. (1986). Multiple regression with integrated processes. Review of Economic Studies, 53, 473–496. Robinson, P. (1998). Inference-without smoothing in the presence of nonparametric autocorrelation. Econometrica, 66, 1163–1182. White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48, 817–838. White, H. (1984). Asymptotic theory for econometricians. New York: Academic Press.
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MATHEMATICAL APPENDIX: PROOFS In this appendix proofs are given for the lemma and theorems. Define the function [rT ] [rT ] [rT ] − 1 [rT ] + 1 −k − k −k . D T (r) = T 2 k T T T T Note that D T (−r) = D T (r) because k(−r) = k(r). If k (r) exists, then limT→∞ D T (r) = k (r) by the definition of the second derivative. If k (r) is continuous, then D T (r) converges to k (r) uniformly. Therefore, when k (r) is continuous, we have uniform convergence of D T (r) to k (r). Then, using Assumptions 1–4 the following joint convergence result holds: G [rT] (ˆ T ), G [rT] (¯ [rT],T ), T 1/2 g [rT] (0 ), D T (r) ⇒ (rG 0 , rG 0 , B q (r), k (r)). ˆ M=T and F ∗ can be written To establish Theorems 1 and 2 it is shown that as continuous functions of (G [rT] (ˆ T ), G [rT] (¯ [rT],T ), T 1/2 g [rT] (0 ), D T (r)). The limiting distributions then follow from the continuous mapping theorem. Proof of Lemma 1: Setting t = T, multiplying both sides of (3) by G T (ˆ T ) W T , and using the first order condition G T (ˆ T ) W T g T (ˆ T ) = 0 gives 0 = G T (ˆ T ) W T g T (0 ) + G T (ˆ T ) W T G T (¯ T,T )(ˆ T − 0 ).
(11)
Solving (11) for (ˆ T − 0 ) and scaling by T 1/2 gives T 1/2 (ˆ T − 0 ) = −[G T (ˆ T ) W T G T (¯ T,T )]−1 G T (ˆ T ) W T T 1/2 g T (0 ). The lemma follows because p lim G T (ˆ T ) W T G T (¯ T,T ) = G 0 W ∞ G 0 by Assumptions 3 and 4 and because G T (ˆ T ) W T T 1/2 g T (0 ) ⇒ G 0 W ∞ B q (1) by Assumptions 2, 3, and 4. Proof of Theorem 1: The first step is to derive the limit of G T (ˆ T ) W T T 1/2 g [rT] (ˆ T ). Setting t = [rT] and multiplying both sides of (3) by T 1/2 gives T 1/2 g [rT] (ˆ T ) = T 1/2 g [rT] (0 ) + G [rT] (¯ [rT],T )T 1/2 (ˆ T − 0 ).
(12)
Using Assumptions 2, 3 and 4 and Lemma 1 it follows from the continuous mapping theorem that T 1/2 g [rT] (ˆ T ) ⇒ B q (r) − rG 0 (G 0 W ∞ G 0 )−1 G 0 W ∞ B q (1).
(13)
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Because p lim G T (ˆ T ) W T = G 0 W ∞ by Assumptions 3 and 4, it follows from (12) and (13) that G T (ˆ T ) W T T 1/2 g [rT] (ˆ T ) ⇒ G W ∞ (B q (r) 0
− rG 0 (G 0 W ∞ G 0 )−1 G 0 W ∞ B q (1)) = G 0 W ∞ B q (r) − rG 0 W ∞ B q (1) ˜ p (r). = ∗ (B p (r) − rB p (1)) ≡ ∗ B
(14)
) W
ˆ M=T W T G T (ˆ T ) in terms The second step of the proof is to write G T (ˆ T T of G T (ˆ T ) W T T 1/2 g [rT] (ˆ T ) and D T (r) and then apply the continuous mapping theorem. Let K il = k((i − l)/T). Using algebraic arguments similar to those used by Kiefer and Vogelsang (2002a), it is straightforward to show that ˆ M=T = T −1
T−1 T−1
[(K il − K i,l+1 ) − (K i+1,l − K i+1,l+1 )]Tg i (ˆ T )Tg l (ˆ T )
l=1 i=1
+T
−1
T−1
(K Tl − K T,l+1 )Tg T (ˆ T )Tg l (ˆ T )
l=1
+ T −1
T
f(vi , ˆ T )K iT Tg T (ˆ T ) .
(15)
i=1
Using (15) it directly follows that ˆ M=T W T G T (ˆ T ) = T −1 G T (ˆ T ) W T
T−1 T−1
[(K il − K i,l+1 )
l=1 i=1
− (K i+1,l − K i+1,l+1 )]G T (ˆ T ) W T Tg i (ˆ T )Tg l × (ˆ T ) W T G T (ˆ T ),
(16)
where the second and third terms of (15) vanish because from (2) it follows that G T (ˆ T ) W T Tg T (ˆ T ) = Tg T (ˆ T ) W T G T (ˆ T ) = 0. Using the fact that T 2 [(K il − K i,l+1 ) − (K i+1,l − K i+1,l+1 )] i−l+1 i−l i−l i−l−1 2 = −T k −k − k −k T T T T i−l = −D T , T
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(16) can be expressed as ˆ M=T W T G T (ˆ T ) G T (ˆ T ) W T = T −1
T−1
T −1
T−1 i=1
l=1
− T D T ((i − l)/T)G T (ˆ T ) W T T 1/2 g i (ˆ T )T 1/2 g l (ˆ T ) W T G T (ˆ T ) 1 1 = −T 2 D T (r − s)G T (ˆ T ) W T T 1/2 g [rT] (ˆ T )T 1/2 g [sT] (ˆ T ) W T G T (ˆ T ) dr ds 2
0
0
∗
1 1
⇒
0
˜ p (r)B ˜ p (s) dr ds∗ , −k (r − s)B
0
where weak convergence follows from (14) and the continuous mapping theorem. The proof for the Bartlett kernel is obtained by noting that T[(K il − K i,l+1 ) − (K i+1,l − K i+1,l+1 )] = 2 for i = l and equals 0 otherwise. Therefore it follows that ˆ M=T W T G T (ˆ T ) G T (ˆ T ) W T = T −1
T−1
2G T (ˆ T ) W T T 1/2 g i (ˆ T )T 1/2 g i (ˆ T ) W T G T (ˆ T )
i=1 ∗˜
˜ p (r) dr∗ , ⇒ 2 Bp (r)B which completes the proof of Theorem 1.
Proof of Theorem 2: Once the result for F ∗ is established, the result for t ∗ trivially follows when m = 1. The proof for the Bartlett kernel is essentially the same and is omitted. Applying the delta method to the result in Lemma 1 and using the fact that B q (1) is a vector of independent standard normal random variables gives T 1/2 r(ˆ T ) ⇒ −R(0 )(G 0 W ∞ G 0 )−1 G 0 W ∞ B q (1) ≡ −R(0 )(G 0 W ∞ G 0 )−1 ∗ B p (1) ≡ ∗∗ B m (1),
(17)
where ∗∗ is the matrix square root of R(0 )(G 0 W ∞ G 0 )−1 ∗ ∗ (G 0 W ∞ G 0 )−1 R(0 ) . Using the result in Theorem 1 along with Assumptions 3 and 4, it directly follows that ˆ M=T W T G T (ˆ T ) R(ˆ T )Vˆ M=T R(ˆ T ) = R(ˆ T )[G T (ˆ T ) W T G T (ˆ T )]−1 G T (ˆ T ) W T × [G T (ˆ T ) W T G T (ˆ T )]−1 R(ˆ T ) ⇒ R(0 )(G 0 W ∞ G 0 )−1 ∗
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× 0
≡ ∗∗
˜ p (r)B ˜ p (s) dr ds∗ (G 0 W ∞ G 0 )−1 R(0 ) −k (r − s)B
0
1 1 0
˜ m (r)B ˜ m (s) dr ds∗∗ , −k (r − s)B
(18)
0
where we use the fact that ˜ p (r) = R(0 )(G 0 W ∞ G 0 )−1 ∗ (B p (r) − rB p (1)) R(0 )(G 0 W ∞ G 0 )−1 ∗ B ˜ m (r). ≡ ∗∗ (B m (r) − rB m (1)) = ∗∗ B Using (17) and (18) it directly follows that −1 r(ˆ T )/m F ∗ = Tr(ˆ T ) R(ˆ T )Vˆ M=T R(ˆ T ) −1 1/2 T r(ˆ T )/m = T 1/2 r(ˆ T ) R(ˆ T )Vˆ M=T R(ˆ T ) ∗∗
⇒ ( B m (1))
1
∗∗
0
≡ B m (1)
1 1 0
1
−1
˜ m (r)B ˜ m (s) dr ds∗∗ −k (r −s)B
∗∗ B m (1)/m
0
−1 ˜ m (r)B ˜ m (s) dr ds −k (r − s)B
B m (1)/m,
0
which completes the proof.
FORMULA APPENDIX: KERNELS The formulas for the kernels considered in this paper are as follows: 1 − |x| for |x| ≤ 1, Bartlett; k(x) = 0 otherwise, 1 2 3 1 − 6x + 6|x| for |x| ≤ 2 , Parzen; (a) k(x) = 2(1 − |x|)3 for 21 ≤ |x| ≤ 1 0 otherwise, (1 + x2 )−1 for |x| ≤ 1, Parzen; (c) k(x) = 0 otherwise,
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Tukey-Hanning;
k(x) =
Quadratic Spectral (QS);
1 2 (1 +
233
cos (x))
0 k(x) =
25 122 x 2
for |x| ≤ 1, otherwise, sin (6x/5) 6x − cos 6x/5 5
sin (x) , x (1 − |x|) cos (x) + sin (|x|) Bohman; k(x) = 0 exp(−4.5x2 ) for |x| ≤ 1, Normal; k(x) = 0 otherwise. Daniell;
k(x) =
for|x| ≤ 1, otherwise,
BAYESIAN ANALYSIS OF MISSPECIFIED MODELS WITH FIXED EFFECTS Tiemen Woutersen ABSTRACT One way to control for the heterogeneity in panel data is to allow for time-invariant, individual specific parameters. This fixed effect approach introduces many parameters into the model which causes the “incidental parameter problem”: the maximum likelihood estimator is in general inconsistent. Woutersen (2001) shows how to approximately separate the parameters of interest from the fixed effects using a reparametrization. He then shows how a Bayesian method gives a general solution to the incidental parameter for correctly specified models. This paper extends Woutersen (2001) to misspecified models. Following White (1982), we assume that the expectation of the score of the integrated likelihood is zero at the true values of the parameters. We then derive the conditions under which a Bayesian √ estimator converges at rate N where N is the number of individuals. Under these conditions, we show that the variance-covariance matrix of the Bayesian estimator has the form of White (1982). We illustrate our approach by the dynamic linear model with fixed effects and a duration model with fixed effects.
Maximum Likelihood Estimation of Misspecified Models: Twenty Years Later Advances in Econometrics, Volume 17, 235–249 © 2003 Published by Elsevier Ltd. ISSN: 0731-9053/doi:10.1016/S0731-9053(03)17011-9
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1. INTRODUCTION In applied work, economist rarely have data that can be viewed as being generated by an homogeneous group. That is, firms or individuals differ in observed and unobserved ways. These unobserved differences are usually referred to as heterogeneity and one can control for the heterogeneity in panel data by allowing for time-invariant, individual specific parameters. Accounting for heterogeneity using such individual or fixed effects avoids distributional and independence assumptions (which are usually not supported by economic theory), see Chamberlain (1984, 1985), Heckman et al. (1998) and Arellano and Honor´e (2001). This fixed effect approach introduces many parameters into the model which causes the “incidental parameter problem” of Neyman and Scott (1948): the maximum likelihood estimator is in general inconsistent. Chamberlain (1984), Trognon (2000) and Arellano and Honor´e (2001) review panel data techniques that give good estimators for specific models. Woutersen (2001) derives a general solution that approximately separates the parameters of interest from the fixed effects using a reparametrization. After the reparametrization, the fixed effects are integrated out with respect to a flat prior. This yields a Bayesian estimator for the parameter ˆ that has a low bias, O(T −2 ) where T is the number of observations of interest, , per individual. Moreover, the asymptotic distribution of ˆ has the following form, √
d
NT (ˆ − 0 )→N(0, I()−1 ).
where I() is the information matrix and T ∝ N ␣ where ␣ > 1/3. Thus, the asymptotic variance of ˆ is the same as the asymptotic variance of the infeasible maximum likelihood estimator that uses the true values of the fixed effects. This paper extends the analysis of Woutersen (2001) by allowing for misspecification of the likelihood. Following White (1982), we assume that the expectation of the score is zero at the true values of the parameters. We then derive√the primitive conditions under which the Bayesian estimator converges at rate N. In particular, we assume the “score” of the integrated likelihood to be zero at the true value of the parameter of interest. Under these conditions, we show that the variance-covariance matrix of the Bayesian estimator has the form of White (1982). Lancaster (2000, 2002) does not derive asymptotic variances and another new feature of this paper is that it derives the asymptotic variance of the integrated likelihood in a fixed T, increasing N asymptotics. We illustrate our approach by the dynamic linear model with fixed effects and a duration model with fixed effects. This paper is organized as follows. Section 2 reviews information-orthogonality as a way to separate the nuisance parameters from the parameter of interest. Section 3 discusses the integrated likelihood approach. Section 4 gives the
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conditions for consistency and derives the variance-covariance matrix under misspecification. Section 5 discusses the dynamic linear model and a duration model and Section 6 concludes.
2. INFORMATION-ORTHOGONALITY The presence of individual parameters in the likelihood can inhibit consistent estimation of the parameters of interest, as shown by Neyman and Scott (1948). For example, the dynamic linear model with fixed effects cannot be consistently estimated by maximum likelihood, as shown by Nickell (1981).1 Information-orthogonality reduces the dependence between the parameters of interest and the individual parameters. We introduce more notation so that we can be specific. Suppose we observe N individuals for T periods. Let the log likelihood contribution of the tth spell of individual i be denoted by L it . Summing over the contributions of individual i yields the log likelihood contribution, L i (, i ) = L it (, i ), t
where  is the common parameter and i is the individual specific effect. Suppose that the parameter  is of interest and that the fixed effect i is a nuisance parameter that controls for heterogeneity. We can approximately separate  from = {1 , . . . , N } by using an information-orthogonal parametrization of the quasi likelihood. In particular, information-orthogonality reduces this dependence between  and by having cross derivatives of the quasi log-likelihood being zero in expectation. That is, EL  (0 , 0 ) = 0 i.e.
y max
y min
L  (0 , 0 ) eL(0 ,0 ) dy = 0,
where y denotes the dependent variable, y ∈ [y min , y max ] and {0 , 0 } denote the true value of the parameters. Cox and Reid (1987) and Jeffrey (1961) use this concept and refer to it as “orthogonality.” Lancaster (2000, 2002) applies this orthogonality idea to panel data and Woutersen (2000) gives an overview of orthogonality concepts. Chamberlain (1984) and Arellano and Honor´e (2001) review panel data econometrics in their handbook chapters. All but two of their models can be written in information-orthogonal form.2
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Suppose that a quasi-likelihood is not information-orthogonal. In that case we reparameterize the quasi-likelihood to make it information-orthogonal. Let the individual nuisance parameter that is not information-orthogonal be denoted by f. We can interpret f as a function of  and information-orthogonal , f(, ), and write the log likelihood as L(, f(, )). Differentiating L(, f(, )) with respect to  and yields ∂L(, f(, )) ∂f = L + Lf ∂ ∂ ∂2 L(, f(, )) ∂f ∂f ∂f ∂2 f = L f + L ff + Lf ∂∂ ∂ ∂ ∂ ∂∂ where L f is a score and therefore EL f = 0. Information-orthogonality requires the cross-derivative ∂2 L(, f(, ))/∂∂ to be zero in expectation, i.e. EL  = EL f
∂f ∂f ∂f + EL ff = 0. ∂ ∂ ∂
This implies the following differential equation EL f + EL ff
∂f = 0. ∂
(1)
If Eq. (1) has an analytical solution then f(·) can be written as a function of {, }. If Eq. (1) has an implicit solution, then the Jacobian ∂/∂f can be recovered from the implicit solution. The Jacobian ∂/∂f is all we need for a reparametrization in a Bayesian framework. The general nonlinear model and the single index model have an information-orthogonal parametrization that is implicit, as shown in Woutersen (2001). For the remainder of the paper, we assume information-orthogonality. The “invariance result” of the maximum likelihood estimator implies that reparametrizations do not change the estimates. In particular, an informationorthogonal parametrization would yield the same estimates for  as a parametrization that is not information-orthogonal. However, the integrating out method does not have this invariance property and this paper shows that informationorthogonality can yield moment functions that are robust against incidental parameters, even under misspecification.
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3. THE INTEGRATED LIKELIHOOD After ensuring information-orthogonality, we integrate out the fixed effects and use the mode of the integrated likelihood as an estimator. That is, ˆ = arg max L I () 
where L I () =
ln
eL
i (,)
di .
i
Misspecification has been, so far, not considered in combination with the integrated likelihood approach as is apparent from the overviews of Gelman et al. (1995) and Berger et al. (1999). The point of this paper, however, is to consider misspecification. In particular, L i (, ) does not need to be a fully specified likelihood. It is sufficient that we specify, as an approximation, a density for y it that is conditional on x it and i . The likelihoodcontribution L it (, ) is the logarithm of this conditional density and L i (, ) = t L it (, ). In particular, the distribution of the fixed effects is left unrestricted. Thus, in this set-up we can think of the Data Generating Process as follows. First, the fixed effects, f 1 , . . . , f N , are generated from an unknown and unrestricted distribution. As a second step, x 11 , . . . , x N1 is generated from another unknown distribution that can depend on f 1 , . . . , f N . Then y 11 , . . . , y N1 is generated by a conditional distribution3 that is approximated by the econometrician. For period t = 2, the distribution of x 12 , . . . , x N2 can depend on f 1 , . . . , f N , x 1 , . . . , x N . Alternatively, x it can allowed to be endogenous in which case the econometrician specifies a density for y it that is conditional on x i,t−1 and f i .
4. ASSUMPTIONS AND THEOREM In this section, we consider estimation while allowing for misspecification of the model. The clearest approach seems to impose the assumptions directly on the integrated likelihood function. White (1982, 1993) assumes that the expectation of the score is zero at the true value of the parameter. Similarly, we assume that the score of the integrated likelihood has expectation zero at the truth. Assumption 1. (i) Let {x i , y i } be i.i.d. and (ii) let EL i,I  = 0 for every i. j,I
This assumption implies, by independence across individuals, that EL i,I  L = 0 for i = j. Note that the regressor x i = {x i1 , x i2 , . . . , x iT } and dependent variable
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y i = {y i1 , y i2 , . . . , y iT } are not required to be stationary and that x i is not required to be exogenous. Assumption 2. (i)  ∈ where which is compact or (ii) L I () L I () is concave in . This is a regularity condition that is often assumed. Assumption 3. (i) EL I () = 0 is uniquely solved for  = 0 ; (ii) L I () is continuous at each  ∈ with probability one; and (iii) E sup∈ ||L I ()|| < ∞. Information-orthogonality, EL  (0 , 0 ) = 0, does not imply EL I () = 0 but the stronger condition L  (, ) = 0 does. However, imposing this stronger condition excludes many interesting models. Thus, it could be that EL  (0 , 0 ) = 0 is not a necessary condition for EL I () = 0 but we do not know examples for which EL I () = 0 and EL  (0 , 0 ) = 0. We therefore recommend to first reparameterize the model so that EL  (0 , 0 ) = 0 and, as a second step, check Assumptions 1–3. Assumption 4. (i) 0 ∈ interior( ); (ii) L I () is continuously differentiable in a neighborhood N of 0 ; (iii) EL I () is continuous at 0 and p
sup∈N ||EL I () − EL I ()||→0; and (iv) EL I (0 ) is nonsingular. Theorem 1. Suppose ˆ = arg min {(L I ()/NT )(L I ()/NT)}. Let Assumptions 1–4 hold. Let N → ∞ while T is fixed. Then √ NT (ˆ − 0 ) → N(0, ) where =
−1
1 EL I ( ) NT  0
Proof: See Appendix A.
1 E{(L I (0 ))(L I (0 )) } NT
−1
1 EL I ( ) NT  0
.
The theorem shows that the integrated likelihood as a convenient tool to derive moments that are robust against incidental parameters as well as robust against misspecification of the parametric error term.
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241
5. EXAMPLES In this section we discuss two examples that illustrate the integrated likelihood approach.
5.1. Dynamic Linear Model Consider the dynamic linear model with fixed effects, y it = y i,t−1  + f i + it for
Eis it = 0
where for
Eit = 0, E2it < ∞
s = t
and
t = 1, . . . , T.
This model is perhaps the simplest model that nests both state dependence and heterogeneity as alternative explanations for the variation in the values of y it across agents. As such, the dynamic linear model is popular in the development and growth literature. For a discussion and further motivation of this model, see Kiviet (1995), Hahn, Hausman and Kuersteiner (2001), Arellano and Honor´e (2001) as well as the references therein. Lancaster (2002) suggests the following informationorthogonal parametrization, f i = y i0 (1 − ) + i e−b()
where
b() =
T 1 T−t t . T t t=1
However, Lancaster (2002) does not derive the asymptotic variance of the integrated likelihood estimator. Woutersen (2001) shows that, under normality of it , the integrated likelihood estimator is adaptive for an asymptotic with T ∝ N ␣ and ␣ > 1/3. That is, the asymptotic variance does not depend on knowledge of in this asymptotic. We now consider the case where the normality of it fails to hold and only assume normality in order to derive the integrated likelihood estimator. Note that 1 EL i = 2 E t {y t−1 − y 0 − b () e−b() } = 0 t EL i = −
b () e−b() E t = 0. 2 t
The log likelihood contribution has the following form, 1 1 L i = − ln(2 ) − 2 (˜yt − y˜ t−1  − e−b() )2 where 2 2 t
y˜ t = y t − y 0 .
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Integrating with respect to gives the integrated likelihood contribution, 1 2 −b() )2 L i,I e−(1/2 ) t (˜yt −˜yt−1 − e = √ d e 2 1 2 2 = √ eb() e−(1/2 ) t (y t −y t−1 −f) df 2 1 b()−(1/22 ) (y t −y t−1 )2 2 2 t e−(T/2 ){f −2f(y t −y t−1 )} df = √ e 2 ∝ eb()−(1/2
2 ){(T/2)(y
t −y t−1 )
2 + (y −y 2 t−1 ) } t t
,
does not depend on f. Taking where we omit the subscript i and ∂/∂f = logarithms and differentiating with respect to  yields 1 = b () + (y − y i,t−1 )y i,t−1 − T(y it − y i,t−1 )y i,t−1 L i,I  2 t it eb()
L i,I  = b () −
1 2 1 y i,t−1 + 2 Ty i,t−1 2 2 t
where b() = 1/T Tt=1 (T − t/t)t , b() = 1/T Tt=1 (T − t)t−1 , b() = 1/T Tt=1 (T − t)(t − 1)t−2 . Note that EL i,I  /NT = 0 for any N, T and that the mode of L I ()/NT is a consistent estimator for  for N increasing. Analogue to the quasi-maximum likelihood estimator of White (1982), the asymptotic variance has the form of Theorem 1, = [(1/NT)EL I ]−1 [(1/NT)E{(L I )(L I ) }][(1/NT)EL I ]−1 . The author views the integrated likelihood as a convenient way to derive moments that can be robust against misspecification of the parametric error term. In particular, the parametric assumptions on the error term are irrelevant for the models with additive error terms that are discussed in Arellano and Honor´e (2001).
5.2. Duration Model with Time-Varying Individual Effects Consider a duration model in which the hazard depends on an individual effect f i , a spell-specific effect u is and observable regressors x is . In particular, consider the following hazard, is (t) = ef i +x is +u is .
(2)
where the subscript i refers an individual and the subscript s refers to a spell of that individual. This hazard depends on two unobservable stochasts, f i and u is .
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243
In particular, the individual specific effect f i can depend on the regressors x is . We avoid distributional assumption on the spell-specific effect u is but we assume that u is is independent of x is and Ee−u is < ∞. Thus, the hazard of Eq. (2) is a generalization of the fixed effect hazard model with regressors where the hazard is ef i +x is  . Chamberlain (1984) developed an estimator for the last model and Van den Berg (2001) gives a current review of duration models. A common criticism of the model with hazard ef i +x is  is that it assumes that variations in the hazard can all be explained by variations in the regressor x is . In other words, the unobservable effect is constant over time, see Van den Berg (2001) for this argument. Equation (2) extends this model by allowing for a spell-specific effect u it . As an approximation of the model of (2) we consider is = ei +x is  where s x is = 0. This hazard implies a log likelihood and the normalization, x = 0, ensures that the log likelihood is information-orthogonal. In particular, s is L i (, i ) = Ti − ei ex is  t is , L i (, i ) = −ei
s
x is ex is  t is ,
s
and L ii (, i ) = −ei
x is ex is  t is .
s
Note that ex is 0 t is is exponentially distributed with mean e−(0,i +u is ) . This implies, EL i (0 , i,0 ) = −E e0,i x is e−(0,i +u is ) = −E x is e−u is = 0 s
s
since s x is = 0 and u is is independent of x is . Similarly, EL i (0 , i,0 ) = 0. i Integrating i with respect to the likelihood gives (T) Ti − s ex is +i t is i,I Li di = ln x  T . L = ln e di = ln ei e { s e is t is } see Appendix B for details. Thus, L i,I  T
=
x is  t is s x is e x is  t e is s
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and x is  t is s x is e = i x  is NT N t is se 2 x is  I L  x e t is − { s x is ex is  t is }2 x  2 = i s is NT N { s e is t is } L I
In Appendix C, it is shown that (1/NT)EL I = 0 for any N and any T ≥ 2. Thus, the mode of L I ()/NT is a consistent estimator for  for N increasing. Moreover, the asymptotic variance has the form of Theorem 1, = [(1/NT)EL I ]−1 [(1/NT)E{(L I )(L I ) }][(1/NT)EL I ]−1 . 5.2.1. Simulation Let the data be generated by the following hazard model, is (t) = ef i +x is +u is .
(3)
This hazard implies that the expected duration, conditional on f i , x is , and u is equals4 1/ef i +x is +u is , i.e. E(t is |f i , x is , u is ) = 1/ef i +x is +u is . Let the exponent of the individual effect, ef i , have a unit exponential distribution and let the individual spell effect, u is , be normally distributed with mean zero and variance 2u . Suppose that we observe a group of N individuals and that we observe an unemployment spell before and after treatment, that is x i1 = 0 for all i and x i2 = 1 for all i. Heckman, Ichimura, Smith and Todd (1998) discuss the estimation of treatment effect models and conclude that the fixed effect model performs very well. This simulation study extends the fixed effect duration model by allowing for an spell specific effect u is , i = 1, . . . , N and s = 1, 2. In particular, the model of Eq. (3) also extends both Chamberlain (1985) and Ridder and Woutersen (2003) by allowing for both random and fixed effects. We first assume that the treatment has no effect on the hazard out of unemployment, that is,  = 0. We then assume that the hazard out of unemployment increases by factor 2.7, That is,  = 1 and e = e ≈ 2.7. The estimator developed in this subsection is denoted by “integrated likelihood estimator.” A naive Bayes estimator that just integrated out the fixed effects and then uses the posterior mode is denoted by “naive Bayes estimator.” We use flat priors for all parameters and base inference on the posterior mode after integrating out the fixed effects f i , i = 1, . . . , N. The model is misspecified in the sense that the individual spell effect, u is , is ignored.
Bayesian Analysis of Misspecified Models with Fixed Effects
Bias ( = 0)
RMSE ( = 0)
245
Bias ( = 1)
RMSE ( = 1)
Integrated likelihood estimator 2u = 21 −0.0008 2 u = 1 0.0145 2 u = 2 −0.0008
0.1334 0.1451 0.1790
0.0039 −0.0010 −0.0042
0.1298 0.1467 0.1826
Naive bayes estimator 2u = 21 1.1346 2 u = 1 1.2197 2 u = 2 1.3739
1.1424 1.2288 1.3856
1.1308 1.2188 1.3674
1.1394 1.2285 1.3795
Note that the two estimators use the same likelihood and priors. However, the “info-ortho Bayes estimator” separates the nuisance parameter from the parameter of interest before integrating out f i , i = 1, . . . , N. As a consequence, the bias is much lower, by about factor 8, for the “integrated likelihood estimator.” Note that, for both estimators, the Root Mean Squared Error (RMSE) is increasing in 2u and that the bias of the “naive Bayes estimator” does not strongly depend on the value of . We conclude that separating the nuisance parameter from the parameter of interest works well for this misspecified model.
6. CONCLUSION This paper extends the integrated likelihood estimator to misspecified models. Using information-orthogonality, we approximately separate the nuisance parameter from the parameter of interest. We use a Bayesian techniques since reparametrization of a nuisance parameter only requires an expression of the Jacobian in a Bayesian framework. Under the condition that the score of the integrated likelihood has expectation zero at the truth, we show that the variancecovariance matrix of the Bayesian estimator has the form of White (1982). Thus, information-orthogonality combined with the integrated likelihood is a promising approach which solves the incidental parameter problem of Neyman and Scott (1948) for a class of misspecified models. We illustrate our approach by two misspecified models with individual effects. In the dynamic linear model, we allow the error term to be non-normal and in the hazard model we allow the individual effect to change over time.
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NOTES 1. The dynamic linear model assumes that y it = y i,t−1  + f i + it and we discuss this model in Section 5.1. 2. The transformation model of Abrevaya (1998) and one discrete choice model by Honor’e and Kyriazidou (2000) are not information-orthogonal. Both models require infinite support for the regressor, can be estimated using a sign function and will be discussed in a separate paper that deals with “information-orthogonality” of sign functions. 3. That is, conditional on f 1 , . . . , f N and x 1 , . . . , x N . 4. Note that t is is exponentially distributed if we condition on f i , x is , and u is .
ACKNOWLEDGMENTS I gratefully acknowlegde stimulating suggestions from Tony Lancaster. The Social Science and Humanities Research Council of Canada provided financial support. All errors are mine.
REFERENCES Abrevaya, J. (1998). Leapfrog estimation of a fixed-effects model with unknown transformation of the dependent variable. Unpublished manuscript, Graduate School of Business, University of Chicago. Arellano, M., & Honor´e, B. E. (2001). Panel data models: Some recent developments. In: J. Heckman & E. Leamer (Eds), Handbook of Econometrics (Vol. 5). Amsterdam: North-Holland. Berger, J. O., Liseo, B., & Wolpert, R. L. (1999). Integrated likelihood methods for eliminating nuisance parameters. Statistical Science, 14, 1–28. Chamberlain, G. (1984). Panel data. In: Z. Griliches & M. D. Intriligator (Eds), Handbook of Econometrics (Vol. 2). Amsterdam: North-Holland. Chamberlain, G. (1985). Heterogeneity, omitted variable bias, and duration dependence. In: J. J. Heckman & B. Singer (Eds), Longitudinal Analysis of Labor Market Data. Cambridge: Cambridge University Press. Cox, D. R., & Reid, N. (1987). Parameter orthogonality and approximate conditional inference (with discussion). Journal of the Royal Statistical Society, Series B, 49, 1–39. Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (1995). Bayesian data analysis. New York: Chapman & Hall. Hahn, J., Hausman, J., & Kuersteiner, G. (2001). Bias corrected instrumental variables estimation for dynamic panel models with fixed effects. MIT Working Paper. Heckman, J., Ichimura, H., Smith, J., & Todd, P. (1998). Characterizing selection bias using experimental data. Econometrica, 66, 1017–1098. Honor´e, B. E., & Kyriazidou, E. (2000). Panel data discrete choice models with lagged dependent variables. Econometrica, 68, 839–874. Jeffreys, H. (1961). Theory of probability (3rd ed.). Oxford: Clarendon Press.
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Kiviet (1995). On bias, inconsistency and efficiency of various estimators in dynamic panel data models. Journal of Econometrics, 68, 53–78. Lancaster, T. (2000). The incidental parameters since 1948. Journal of Econometrics, 95, 391–413. Lancaster, T. (2002). Orthogonal parameters and panel data. Review of Economic Studies (forthcoming). Newey, W. K., & McFadden, D. (1994). Large sample estimation and hypothesis testing. In: R. F. Engle & D. MacFadden (Eds), Handbook of Econometrics (Vol. 4). Amsterdam: North-Holland. Neyman, J., & Scott, E. L. (1948). Consistent estimation from partially consistent observations. Econometrica, 16, 1–32. Nickell, S. (1981). Biases in dynamic models with fixed effects. Econometrica, 49, 1417–1426. Ridder, G., & Woutersen, T. M. (2003). The singularity of the efficiency bound of the mixed proportional hazard model. Econometrica (forthcoming). Trognon, A. (2000). Panel data econometrics: A successful past and a promising future. Working Paper, Genes (INSEE). Van den Berg, G. T. (2001). Duration models: Specification, identification, and multiple duration. In: J. Heckman & E. Leamer (Eds), Handbook of Econometrics (Vol. 5). Amsterdam: North-Holland. White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50, 1–25. White, H. (1993). Estimation, inference and specification analysis. New York: Cambridge University Press. Woutersen, T. M. (2000). Consistent estimation and orthogonality. Working Paper, Department of Economics, University of Western Ontario. Woutersen, T. M. (2001). Robustness against incidental parameters and mixing distributions. Working Paper, Department of Economics, University of Western Ontario.
APPENDIX A THEOREM 1 To be shown
where
√
−1
1 EL I ( ) = NT  0
N(ˆ − 0 ) → N(0, )
1 E{(L I (0 ))(L I (0 )) } NT
−1
1 EL I ( ) NT  0
.
Proof: Let Assumptions 1, 2(i), and 3 hold. Then all the conditions of Newey and McFadden (1994, Theorem 2.6) are satisfied and consistency follows. Assuming that, in addition Assumption 4 holds then the assumptions of Newey and McFadden (1994, Theorem 3.2) are satisfied and asymptotic normality follows where the identity matrix is used a the weighting matrix. Instead of assuming that the parameter space is compact as in 2(ii) we can assume that we assume that 0 is an element of the interior of a convex set and
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L I () is concave for all i as in Assumption 2(ii) and 4(i). All the requirements of Newey and McFadden (1994, Theorem 2.7) are satisfied and consistency of the integrated likelihood estimator follows. Asymptotic normality is again implied by Newey and McFadden (1994, Theorem 3.2).
APPENDIX B DURATION EXAMPLE, INTEGRATED LIKELIHOOD To be shown, L
i,I
= ln
i − eT i e
s
ex is +i t is
di = ln
{
(T) x is  t }T is se
where (·) denotes the Gamma function Proof: Define vi = ei . x  i 1 is L i,I = ln eL dvi = ln vT−1 e−vi s e t is dvi i vi x is  T T−1 −vi s ex is  t is { s e t is } vi e (T) = ln dvi . (T) { s ex is  t is }T x  Note that ({ s ex is  t is }T vT−1 e−vi s e is t is )/(T) is a gamma density with parami eters T and s ex is  t is and that this density integrates to one. The result follows. Q.E.D.
APPENDIX C DURATION EXAMPLE, SCORE To be shown, 1 EL I = 0 NT 
where
x is 0 t is s x is e = i x  is 0 NT N t is se L I
for any N and any T ≥ 2. Proof: x is 0 t x is 0 +0 t 1 is is s x is e s x is e i I x  =E i EL  = E x is 0 +0 t is 0 t NT N N e e is is s s
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249
Note that ex is 0 +0 t is is exponentially with mean e−u is . Also note that x distributed x  + + is is 0 0 0 0 the expectation of e t is / s e t is does not depend on x is . Thus, define i = E(ex is 0 +0 t is )/ s ex is 0 +0 t is . This yields 1 i I x is = E i i x is = 0 EL  = E NT N s N s since s x is = 0. Q.E.D.
