Nonlinear Financial Econometrics: Markov Switching Models, Persistence and Nonlinear Cointegration

Nonlinear Financial Econometrics: Markov Switching Models, Persistence and Nonlinear Cointegration GREGORIOU-3: “FM” —...

Author: Greg N. Gregoriou | Razvan Pascalau

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Nonlinear Financial Econometrics: Markov Switching Models, Persistence and Nonlinear Cointegration

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Also by Greg N. Gregoriou and Razvan Pascalau FINANCIAL ECONOMETRICS MODELING: Derivatives Pricing, Hedge Funds and Term Structure Models FINANCIAL ECONOMETRICS MODELING: Market Microstructure, Factor Models and Financial Risk Measures NONLINEAR FINANCIAL ECONOMETRICS: Forecasting Models, Computational and Bayesian Models

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Nonlinear Financial Econometrics: Markov Switching Models, Persistence and Nonlinear Cointegration Edited by

Greg N. Gregoriou Professor of Finance, State University of New York (Plattsburgh) Research Associate EDHEC Business School, Nice, France and

Razvan Pascalau Assistant Professor of Economics, State University of New York (Plattsburgh)

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Selection and editorial matter © Greg N. Gregoriou and Razvan Pascalau 2011 Individual chapters © respective contributors 2011 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No portion of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6-10 Kirby Street, London EC1N 8TS. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The authors have asserted their rights to be identified as the authors of this work in accordance with the Copyright, Designs and Patents Act 1988. First published in 2011 by PALGRAVE MACMILLAN Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number 785998, of Houndmills, Basingstoke, Hampshire RG21 6XS. Palgrave Macmillan in the US is a division of St Martin’s Press LLC, 175 Fifth Avenue, New York, NY 10010. Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world. Palgrave® and Macmillan® are registered trademarks in the United States, the United Kingdom, Europe and other countries. ISBN: 978–0–230–28364–0 hardback This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. Logging, pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin. A catalogue record for this book is available from the British Library. A catalog record for this book is available from the Library of Congress. 10 9 8 7 6 5 4 3 2 1 20 19 18 17 16 15 14 13 12 11 Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne

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Contents List of Tables

vii

List of Figures

ix

Acknowledgments

x

About the Editors

xi

Notes on Contributors

xii

Chapter Abstracts

xvi

Part I

Markov Switching Models

1 Valuing Equity when Discounted Cash Flows are Markov Jeremy Berkowitz

3

2 Markov Switching Mean-Variance Frontier Dynamics: Theory and International Evidence Massimo Guidolin and Federica Ria

21

3 A Markov Regime-Switching Model of Stock Return Volatility: Evidence from Chinese Markets Thomas C. Chiang, Zhuo Qiao and Wing-Keung Wong

49

Part II

Persistence and Nonlinear Cointegration

4 Nonlinear Persistence and Copersistence Christian Gourieroux and Joann Jasiak 5 Fractionally Integrated Models for Volatility: A Review Dean Fantazzini 6 An Explanation for Persistence in Share Prices and their Associated Returns Derek Bond and Kenneth A. Dyson

77 104

124

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vi Contents

7 Nonlinear Shift Contagion Modeling: Further Evidence from High Frequency Stock Data Mohamed El Hedi Arouri, Fredj Jawadi, Wael Louhichi and Duc Khuong Nguyen 8 Sparse-Patterned Wavelet Neural Networks and Their Applications to Stock Market Forecasting Jack Penm and R.D. Terrell 9 Nonlinear Cointegration and Nonlinear Error-Correction Models: Theory and Empirical Applications for Oil and Stock Markets Mohamed El Hedi Arouri, Fredj Jawadi and Duc Khuong Nguyen Index

143

161

171

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Tables 1.1 Present value models of equity valuation 1.2 Summary statistics: Price-dividend ratios, dividend growth and stock returns 2.1 Summary statistics for international stock returns 2.2 Model selection statistics 2.3 Estimates of two-state Markov switching model 2.4 Summary statistics for recursive, optimal mean-variance portfolio weights 3.1 Descriptive statistics for stock returns of the stock markets of China 3.2 Estimates of the AR (1)-GARCH model 3.3 Estimates of the Markov switching AR (1)-GARCH model 3.4 The summary statistics for GARCH and RS-GARCH models 3.5 One-week-ahead forecast errors of GARCH and RS-GARCH models 3.6 Analyses of volatility linkages among four segmented stock markets 4.1 Persistence decay patterns for Gaussian processes 4.2 Persistence decay patterns of a beta mixture of Gaussian processes 4.3 Estimation of d (true value d = 0.45) 4.4 Estimation of d (true value d = 0.3) 5.1 Volatility models: Diagnostic test statistics for the SP500 log-returns 5.2 Volatility models: Diagnostic test statistics for the NASDAQ log-returns 5.3 Volatility models: Diagnostic test statistics for the DOW-JONES log-returns 6.1 Fractional analysis of price series 6.2 Fractional analysis: Returns series 6.3 Traditional analysis returns series 6.4 Smith’s modified GPH analysis: Price series 6.5 Fractional and modified GPH analysis: Returns 6.6 Perron-Qu td (a, c1 , b, c2 ) Tests 6.7 Qu’s W tests 7.1 Correlation matrix 7.2 Descriptive statistics of intraday returns

10 13 32 33 36 44 56 57 58 59 60 68 85 86 91 92 116 118 120 134 135 136 137 138 139 139 151 151

vii

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viii Tables

7.3 8.1 9.1 9.2 9.3 9.4 9.5

LSTEC-GARCH estimation results Forecasting performance comparison Descriptive statistics and normality test Linear cointegration tests Results of mixing tests Linearity tests NECM estimation results

155 169 184 185 186 186 187

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Figures 1.1 Histograms of the price-dividend, dividend growth and stock returns 1.2 Postwar S&P500 index prices and the price-dividend ratio 1.3 Stock prices and Gordon (1962) model-implied values 1.4 Actual and Markov model-implied price-dividend ratios 1.5 Stock prices and Markov model-implied values 2.1 Smoothed (full-sample) probabilities from two-state Markov switching Model 2.2 Single-state vs. Markov switching mean-variance frontiers 3.1 Stock indices of Chinese segmented stock market 3.2 AR (1)-RS-GARCH (1, 1) estimation for SHA 3.3 AR (1)-RS-GARCH (1, 1) estimation for SZA 3.4 AR (1)-RS-GARCH (1, 1) estimation for SHB 3.5 AR (1)-RS-GARCH (1, 1) estimation for SZB 4.1 Autocorrelogram of Hj (X) 4.2 Transformed Autocorrelogram of Hj (X) 4.3 Nonlinear Autocorrelogram of Hj (X) 4.4 Extremes of Hermite Polynomials, j = 1, 2, 3, 4 5.1 Stock index price levels 7.1 Stock price dynamics (Subsample 1) 7.2 Stock price dynamics (Subsample 2) 7.3 Transition function 7.4 Time-varying transition function 8.1 A wavelet decomposition tree 8.2 A neural network with a single hidden-layer for a time-series system with input vectors y(t), u(t) and w(t) 9.1 Dynamic distribution of oil and stock returns 9.2 Rational polynomial function 9.3 Three-regime logistic transition function 9.4 Time-variation of the three-regime logistic transition function

14 15 16 17 18 37 41 55 63 64 65 66 90 90 91 92 113 150 150 156 156 163 166 184 188 189 190

ix

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Acknowledgments We thank Lisa von Fircks and Renee Takken at Palgrave and the team at Newgen Imaging Systems. We also thank a handful of anonymous referees for the selection of papers for this book. Neither the contributors nor the publisher is held responsible for the contents of each chapter. The contents of each chapter remain the sole responsibility of each author.

x

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About the Editors Greg N. Gregoriou has published 40 books, over 55 refereed publications in peer-reviewed journals and 20 book chapters since his arrival at SUNY (Plattsburgh) in August 2003. Professor Gregoriou’s books have been published by John Wiley & Sons, McGraw-Hill, ElsevierButterworth/Heinemann, Taylor and Francis/CRC Press, PalgraveMacMillan and Risk books. His articles have appeared in the Journal of Portfolio Management, Journal of Futures Markets, European Journal of Operational Research, Annals of Operations Research, Computers and Operations Research, etc. Professor Gregoriou is co-editor and editorial board member for the Journal of Derivatives and Hedge Funds, as well as editorial board member for the Journal of Wealth Management, Journal of Risk Management in Financial Institutions, IEB International Journal of Finance, Market Integrity and Brazilian Business Review. A native of Montreal, Professor Gregoriou obtained his joint Ph.D. at the University of Quebec at Montreal in Finance which merges the resources of Montreal’s four major universities (University of Quebec at Montreal, McGill University, Concordia University and HEC-Montreal). Professor Gregoriou’s interests focus on hedge funds, funds of hedge funds and managed futures. He is also a member of the Curriculum Committee of the Chartered Alternative Investment Analyst Association (CAIA). He is also Research Associate at the EDHEC Business School in Nice, France. Razvan Pascalau joined the School of Business and Economics at SUNY Plattsburgh in Fall 2008. He graduated with a Ph.D. in Economics and MSc in Finance from the University of Alabama. He also holds an MSc in Financial and Foreign Exchange Markets from the Doctoral School of Finance and Banking in Bucharest, Romania. In 2004, he worked full time for the Ministry of Finance in Romania as a Counselor of European Integration. Professor Pascalau’s primary field of interest is (applied) Time Series Econometrics with an emphasis on modeling nonlinear structures in macro and financial data. Research interests also include topics related to Financial Risk Management, International Finance, and Managerial Finance/Economics. He has published in Applied Economic Letters and the Journal of Wealth Management.

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Notes on Contributors Jeremy Berkowitz is an Associate Professor of Finance at the University of Houston’s Bauer College of Business. Prior to this position, he was a Professor at the University of California at Irvine and an Economist at Federal Reserve Board. His research interests include risk management and financial econometrics. Dr. Berkowitz has published his research in the Journal of Finance, Review of Economic Studies, Rand Journal of Economics, Review of Economics and Statistics, Journal of Econometrics, Management Science, Journal of Fixed Income, Journal of Risk, Journal of Law and Economics and Econometric Reviews. He has presented his work at conferences and university seminars in Canada, Europe and across the United States. Derek Bond is a former principal economist at the Northern Ireland Treasury and Director of the Northern Ireland Regional Research laboratory. He is currently a Senior Lecturer in Financial Econometrics at the University of Ulster. He is also past President of the International Statistical Institute’s standing committee on regional and urban statistics. He has published over 50 academic papers. Thomas C. Chiang is the Marshall M. Austin Professor of Finance at Drexel University. He is the author of numerous articles in refereed journals and two books. His recent research interests have included financial contagion, international finance, asset pricing, and financial econometrics. His articles have appeared in the Journal of International Money and Finance; Quantitative Finance; Journal of Money, Credit and Banking; PacificBasin Finance Journal and Journal of Financial Research among others. Dr. Chiang received his Ph.D. from Pennsylvania State University, with a concentration in financial economics and econometrics. Kenneth A. Dyson is currently a Lecturer in Finance at the University of Ulster and formerly worked as a Research Officer in the Department of Accounting and Finance at the University of Essex. Mohamed El Hedi Arouri is currently an Associate Professor of Finance at the University of Orleans, France and a Researcher at EDHEC Business

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Notes on Contributors xiii

School. He holds a Master’s in Economics and a Ph.D. in Finance from the University of Paris X Nanterre. His research focuses on the cost of capital, stock market integration, and international portfolio choice. He has published articles in refereed journals such as International Journal of Business and Finance Research, Frontiers of Finance and Economics, Annals of Economics and Statistics, Finance, and Economics Bulletin. Dean Fantazzini is Associate Professor in Econometrics and Finance at the Moscow School of Economic, Moscow State University. He graduated with honours from the Department of Economics at the University of Bologna, Italy in 1999. He obtained a Master’s in Financial and Insurance Investments at the Department of Statistics – University of Bologna, Italy in 2000 and a Ph.D. in Economics in 2006 at the Department of Economics and Quantitative Methods, University of Pavia, Italy. Before joining the Moscow School of Economics, he was research fellow at the Chair for Economics and Econometrics, University of Konstanz, Germany and at the Department of Statistics and Applied Economics, University of Pavia, Italy. Specialist in time series analysis, financial econometrics, multivariate dependence in finance and economics. The author has to his credit more than 20 publications, including three monographs. Christian Gourieroux is Professor at the Department of Economics, University of Toronto, Canada and INSEE, France, and Director of Research at the Center for Research in Economics and Statistics (CREST). Massimo Guidolin Ph.D., University of California is a Chair Professor of Finance at Manchester Business School. He has also served as an Asst. Vice-President and Senior Policy Consultant (Financial Markets) within the U.S. Federal Reserve system (St. Louis FED). Since December 2007 he has been co-director of the Center for Analysis of Investment Risk, at Manchester Business School. His research focuses on predictability and non-linear dynamics in financial returns, with applications to portfolio management, and sources and dynamics of volatility and higher-order moments in equilibrium asset pricing models. His research has been published in the American Economic Review, Journal of Financial Economics, Review of Financial Studies, Journal of Business, and Journal of Econometrics among others. Joann Jasiak is Associate Professor in the Department of Economics, York University, Canada. She is the author of several articles published

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xiv Notes on Contributors

in scientific journals and books, such as Financial Econometrics (with C. Gourieux, Princeton University Press, 2001) and The Econometrics of Individual Risk (with C. Gourieux, Princeton University Press, 2006). Fredj Jawadi is currently an Associate Professor at the University of Evry Val d’Essonne and researcher at Amiens School of Management and EconomiX in France. He holds a Master in Econometrics and a Ph.D. in financial econometrics from the University of Paris X Nanterre, France. His research topics cover modeling asset price dynamics, nonlinear econometrics, international finance and financial integration in developed and emerging countries. He has published in Journal of Risk and Insurance, Applied Financial Economics, Finance and Economics Bulletin. He is also co-author of The Dynamics of Emerging Stock Markets (Springer, 2010). Duc Khuong Nguyen is Professor of Finance and Head of the Department of Economics, Finance and Law at ISC Paris School of Management (France). He holds an MSc and a Ph.D. in Finance from the University of Grenoble II (France). His principal research areas concern emerging markets finance, market efficiency, volatility modeling and risk management in international capital markets. His most recent articles are published in Review of Accounting and Finance, American Journal of Finance and Accounting, Economics Bulletin, European Journal of Economics, Finance and Administrative Sciences, and Bank and Markets. Waël Louhichi is currently an Associate Professor of Finance at Rennes 1 University. He was previously an Assistant Professor at Amiens School of Management. He obtained a Ph.D. from both Perpignan University (France) and FUCaM (Belgium). He is a researcher at CREM Rennes (UMR 6211 CNRS). His main area of research is market microstructure and has published several articles in national and international journals, including Banque & Marchés, Review of Accounting and Finance, and Management Decision. Jack Penm is currently a researcher at Australian National University (ANU). He obtained a Ph.D. in electrical engineering from University of Pittsburgh, USA, and a Ph.D. in finance from ANU. He is an author/coauthor of more than 80 papers published in various journals.

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Notes on Contributors xv

Zhuo Qiao received his Ph.D. in Economics from the National University of Singapore. He is currently an Assistant Professor at University of Macau. His research areas include financial econometrics and international finance. His papers have been published in Journal of International Financial Markets, Institutions & Money, Journal of Multinational Financial Management, Global Finance Journal and Economics Letters. Federica Ria is a research affiliate with the Center for Analysis of Investment Risk, at Manchester Business School. She received her Master’s from the University of Insubria, Italy. She is also a junior administrative officer with Katarsis Capital Advisors Sa., in Switzerland. Her research focuses on the econometrics of Markov switching processes and their applications in finance. R.D. Terrell is a financial econometrician, and officer in the general division of the Order of Australia. He served as Vice-Chancellor of the ANU from 1994 to 2000. He has also held visiting appointments at the London School of Economics, the Wharton School, University of Pennsylvania, and the Econometrics Program, Princeton University. He has published a number of books and research monographs and around eighty research papers in leading journals. Wing-Keung Wong is Professor of Economics at the Department of Economics and Institute for Computational Mathematics, Hong Kong Baptist University. He obtained his Ph.D. at the University of WisconsinMadison in 1989, majoring in Finance and Statistics. His research papers have appeared in Economic Theory, Contemporary Accounting Research, Mathematical Finance, Journal of Financial Markets, Journal of Business and Economic Statistics and European Journal of Operational Research among others.

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Abstracts 1 Valuing Equity when Discounted Cash-Flows are Markov Jeremy Berkowitz We derive new methods for valuing equity and a wide variety of other assets by assuming that both dividend payments and discount rates are Markov. Our approach does not require any particular model of the dynamics of dividend growth and is general enough to include most existing models of the term structure of interest rates. Dividend growth can be stationary or explosive, linear or nonlinear and can be modeled in discrete time or continuous time. We describe the relationship of our approach to existing methods of valuing stocks and present an application to S&P 500 index prices.

2 Markov Switching Mean-Variance Efficient Frontier Dynamics: Theory and International Evidence Massimo Guidolin and Federica Ria It is well known that regime switching models are able to capture the presence of rich non-linear patterns in the joint distribution of asset returns. After reviewing key concepts and technical issues related to specifying, estimating, and using multivariate Markov switching models in financial applications, in this chapter we map the presence of regimes in means, variances, and covariances of asset returns into explicit dynamics of the Markowitz mean-variance frontier. In particular, we show both theoretically and through an application to international equity portfolio diversification that substantial differences exist between bull and bear, regime-specific frontiers, both in statistical and in economic terms. Using Morgan Stanley Capital International (MSCI) investable indices, we characterize mean-variance frontiers and optimal portfolio strategies in bull periods, in bear periods, and in periods where high uncertainty exists on the nature of the current regime.

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Abstracts xvii

3 A Markov Regime-Switching Model of Stock Return Volatility: Evidence from Chinese Markets Thomas C. Chiang, Zhuo Qiao and Wing-Keung Wong This chapter presents a regime switching GARCH model (RS-GARCH) to examine the volatile behavior and volatility linkages among the four major segmented Chinese stock indices. We find evidence of a regime shift in the volatility of the four markets, and the RS-GARCH model appears to outperform the single regime GARCH model. The evidence suggests that B-share markets are more volatile and shift more frequently between high- and low-volatility states. B-share markets are found to be more sensitive to international shocks, while A-share markets seem immune to international spillovers of volatility. Also, volatility linkages among the four segmented markets are regime-dependent.

4 Nonlinear Persistence and Copersistence Christian Gourieroux and Joann Jasiak This chapter examines relationships between various forms of persistence in nonlinear transformations of stationary and nonstationary processes. We introduce the concept of persistence space that is used to define the degrees of persistence in univariate and multivariate time series. For illustration, we examine and compare persistence in a fractionally integrated process and in a beta mixture of AR(1) processes. We also propose the concept of persistence by trajectory that allows us to define nonlinear cointegration and discuss identification problems that arise in this context.

5 Fractionally Integrated Models for Volatility: A Review Dean Fantazzani Many empirical studies have showed the extreme degree of persistence of shocks to the conditional variance process, whose effects can take a considerable time to decay. Therefore the distinction between stationary and unit root processes seems to be far too restrictive. Indeed, the propagation of shocks in a stationary process occurs at an exponential rate of decay (so that it only captures the short-memory), while for an unit root process the persistence of shocks is infinite. Fractional models have been recently proposed in the financial literature to fill the gap between short and complete persistence. The main motivation to use these kinds

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xviii Abstracts

of models is that the propagation of shocks occurs at a slow hyperbolic rate of decay, as opposed to the exponential decay associated with the stationary and invertible ARMA class of processes, or the infinite persistence resulting from non-stationary processes. We review the main developments of this growing field of research, trying to highlight the main advantages and disadvantages of the approaches proposed so far.

6 An Explanation for Persistence in Share Prices and their Associated Returns Derek Bond and Kenneth A. Dyson The question of whether the returns on shares exhibit long memory has attracted considerable attention. Recently several studies have also raised the possibility that some share prices could exhibit long memory. As well as exploring the financial implications of such findings – namely: potential, exploitable, profitable inefficiencies in the market – it is also important to consider whether the apparent long memory is caused by nonlinearity. This chapter does this using tests based on the semiparametric estimator of the fractional integration parameter‘d’ suggested by Smith (2005), Shimotsu (2006), Perron and Qu (2008).

7 Nonlinear Shift Contagion Modeling: Further Evidence from High Frequency Stock Data Mohamed El Hedi Arouri, Fredj Jawadi, Waël Louhichi and Duc Khuong Nguyen This chapter investigates the contagion hypothesis for the French and German stock markets using a combination of a Switching Transition Error Correction model and a Generalized Autoregressive Conditional Heteroscedasticity (STEC-GARCH) model. The main advantage of this double nonlinear error-correction modeling is to specify a time-varying process that apprehends the dynamic evolution of the contagion and reproduces its speed, its extreme regimes as well as its intermediate states, by taking into account the possible linkages between these markets. More importantly, these techniques capture two kinds of nonlinearity: nonlinearity in the mean and nonlinearity in the variance. Applying this modeling on the intraday data of the CAC40 and DAX100 indices over the pre-crisis period (2004–2006) and the post-crisis period (2007–2009), our results indicate significant shift contagion between

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Abstracts xix

studied markets. There is also evidence of nonlinear time-varying error correcting-mechanism toward the long-run equilibrium.

8 Sparse-Patterned Wavelet Neural Networks and Their Applications to Stock Market Forecasting Jack Penm and R.D. Terrell Wavelet neural networks combine the theories of wavelet analysis and neural networks. This Chapter proposes construction approaches to develop sparse-patterned wavelet neural networks, which demonstrate the ‘presence and absence’ restrictions on the coefficients of a subset time-series system. To demonstrate the effectiveness of the proposed nonlinear approaches, the developed sparse-patterned wavelet neural networks are applied to stock market forecasting.

9 Nonlinear Cointegration and Nonlinear Error Correction Models: Theory and Empirical Applications for Oil and Stock Markets Mohamed El Hedi Arouri, Fredj Jawadi and Duc Khuong Nguyen This chapter aims to present the recent developments of nonlinear cointegration and nonlinear error correction models (nonlinear cointegration regressions, attractors, mixing tests, nonlinear cointegration tests, threshold cointegration models and nonlinear ECM) that were introduced by Escribano and Mira (2002) [Journal of Time Series Analysis, Vol. 23, 509–522] and to discuss their applications in the field of empirical finance and energy studies. We also provide evidence regarding their superiority over useful linear cointegration tools. After pointing out the limitations of linear cointegration models at the theoretical level, we apply both linear and nonlinear cointegration techniques to reproduce the short- and long-term empirical linkages between oil and world stock markets over the last three decades. Our findings show several intriguing facts. Indeed, while linear modeling models fail to apprehend the significant relationship between oil and stock markets and, rather, conclude with their segmentation, the hypothesis of financial and oil market integration is not rejected regarding nonlinear cointegration models. More interestingly, this cointegration relationship between oil and world stock markets yields an on-going process that is partially activated, only per regime when oil price deviations against the world market are away from the equilibrium and exceed some threshold.

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Part I Markov Switching Models

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1 Valuing Equity when Discounted Cash Flows are Markov Jeremy Berkowitz

1.1 Introduction In the absence of arbitrage opportunities, the value of a claim that is expected to pay a stream of payments dt is equal to its discounted expected cash flow. This result is a consequence of the martingale representation established in Harrison and Kreps (1979) and Harrison, Pliska (1981). Yet for valuing stocks, no formal methods of implementing the present value model are in widespread use. A great deal of attention has been focused on valuing individual equities cross-sectionally from a small set of factors. The CAPM, the Consumption-based CAPM and the APT are all models in which asset price risk and hence return are driven by a small number of state variables. These models have been subject to countless empirical investigations and have generally been rejected for US equity and bond returns (e.g., Hansen and Singleton (1982), Gibbons, Ross and Shanken (1989), Hansen and Jagannathan (1991), Ferson and Harvey (1992)). In the present article, we avoid specification of preferences and hence of the stochastic discount factor. Our starting point is to present a value model in which the value of a stock is a claim on the firm’s future earnings. We then show that if the discounted cash flows are Markovian, the present value formula collapses to a convenient closed-form expression. Derivatives have long been valued by assuming a process for the underlying claim. For options, the underlying stock value is typically assumed to be a geometric Brownian motion or similar process. Our approach is analogous in that we are assuming the underlying dividends are Markov. The geometric Brownian motion is Markov and therefore a special case but the approach holds far more generally. 3


4 Jeremy Berkowitz

When the process is Markov, the conditional expectation is linear in the conditioning value. It is this property that is particularly useful for valuing stocks. It implies that conditional expectations have the semigroup property so that forecasts at any horizon can be written as a simple function of the one-step ahead forecast. Therefore, in this class of models, the one-step ahead forecast is sufficient to calculate all necessary forecasts. Since the value of a stock is the expected value of an infinite sum, this “chain-rule” property can be used to collapse infinitely many expectations into a convenient closed-form expression. The result is a formula extending the Gordon (1962) model to any stream of discounted dividends that are Markov. From an empirical perspective, the Markov property seems reasonably general. Markov processes include a wide range of common models such as continuous time diffusions and jump-diffusions and finite-ordered autoregressive models (linear or nonlinear) in discrete time. As a simple example, we fit a simple first-order vector autoregression to the Shiller (1989) long-term stock return monthly data set. Even such a simple dynamic model is able to generate stock prices that exhibit substantial volatility and track actual prices extremely well. Our results go part of the way to address Campbell and Shiller (1987, 1989) and Barsky and Delong (1993) who find that the volatility of observed prices is too high to be consistent with present value models. Indeed, we are able to construct models with arbitrarily high volatility of the SDF. This is because our approach is partial equilibrium in the sense that the discount rate process can be anything (Markov). For example, there is no restriction that the SDF be related to macroeconomic factors. Lastly, an advantage to making the Markov assumption is that it makes available the extensive tools developed in Duffie and Singleton (1993). They develop methods for estimating the parameters of dynamic models in which the state-vector is a Markov process. Hansen and Scheinkman (1995) and Stanton (1997) develop tools for estimating continuous-time Markov processes. Durlauf (1991) and Berkowitz (2001) develop tests of the null hypothesis that a process has the martingale property. The chapter is organized as follows. Section 1.2 describes the mathematical framework and presents the martingale representation results. The application to stock pricing is discussed in section 1.3. In section 1.4 we study the performance of a vector autoregression in valuing S&P500 index prices. Section 1.5 concludes.


Valuing Equity when Discounted Cash Flows are Markov 5

1.2 A Markov martingale representation We begin by making the standard assumption that the financial variables of interest are random variables. Assumption A1. The process {xt } is a set with t ∈ [0, ∞] of real-valued random variables adapted with respect to a filtration t which has the usual interpretation as the time t information set. The crucial restriction on the behavior of dividends and on the term structure of the interest rate will be that they have the Markov property. Formally, consider the following definition. Definition. The process {xt } is Markov if it has a probability distribution with the property that pr xt+1 |t = pr xt+1 |xt (1.1) for all t. The initial value x0 has a distribution P0 . Additionally, we make explicit that the conditional expectations are assumed to exist. This assumption will be maintained throughout the remainder of the chapter. Assumption A2. For all t, xt is square integrable with respect to the conditional probability pr xt+1 |xt . Given these two technical regularity conditions, any Markov process can be written as a martingale (e.g., Stroock and Varadhan (1972)). This is convenient because the present value formula is nothing more than an infinite set of expected values. The following proposition presents a formal statement of the result. Proposition 1 The process {xt } is Markov if and only if xt = P1−T E xt+T |xt for t ≤ T where P1 is the one-step ahead transition probability. Proof. First, we show that the Markov property implies the existence of a martingale representation. From assumption A1, the {xt } form a set of measurable functions and assumption A2 ensures compactness. Since the process is integrable, it is equivalent to E xt+1 |xt = yp1 (y|xt )dy (1.2)


6 Jeremy Berkowitz

where p1 (y|xt ) = pr xt+1 = y|xt is the one-step ahead transition probability, conditional probability of xt . Since the process is Markov, the transition probabilities obey the Chapman-Kolmogorov equation, ω pr xs+T |xt = pr xs+T |xω (1.3) s pr xs |xt dω for s > 0, T > 0. This defines a contraction semigroup of linear operators Ps g(xt ) = g(ω)pr xω (1.4) s |xt dω where g(·) is any integrable test function (e.g., Ethier and Kurtz (1986, p.160)). Taking g(xt ) = xt and s = 1, ωpr xω dω = P1 xt |x (1.5) t t+1 the integral is linear in the conditioning value xt . Since (1.6) is the condi tional expectation, we conclude that E xt+1 |xt = P1 xt where the vector P1 does not depend on xt . This immediately leads to the martingale property since E xt+1 |xt = Pxt where P = p(y)dy. Therefore E xt+1 − Pxt |xt = 0 which allows us to write the asset value xt = P −1 E xt+1 |xt . Now we show the converse. Assume that there exists a P such that the asset can be written as a martingale, xt = EP xt+1 . The martingale property is preserved by taking arbitrary functions, E h(xt+1 )|t = E h(xt+1 )|xt

(1.6)

where h : R → R is any Borel measurable, integrable function (e.g., Norris (1997, Thm 4.1.3)). So h(xt ) = EP h(xt+1 )

(1.7)

for arbitrary h which is equivalent to pr(xt+1 |t ) = pr(xt+1 |xt ) where the probability is under P. Lastly, these results extend to any horizon T > 1 by induction. QED. Therefore the Markov property is actually equivalent to the existence of a martingale representation. The relationship holds regardless of whether the process is defined in discrete or continuous time. In this sense, riskneutral valuation is much more general than the stochastic calculus and Ito’s lemma.



The reasoning may be stated as follows. When the process is Markov, the conditional expectations obey the Chapman-Kolmogorov equation, Ps+T (, xt ) = Ps (, y)Ps (y, xt )dy (1.8) for s > 0, T > 0 where Ps (, xt ) = pr(xt+s ∈ |xt ) is the conditional probability. This says that transition probabilities over any time span s + t can be represented as the inner product of the probability of a transition to an intermediate point sand the probability of subsequent transitions from s to T . It is this property (the semi-group property) that guarantees the conditional expectation operator has a linear representation. As a conse quence, E xt+1 |xt = Pxt for some P. The martingale representation then follows immediately since E[xt+T − P T xt ] = 0 and so xt = P −T E[xt+T |xt ]. An equivalent way to write the Markov property is that conditional expectations depend only on today’s value E h(xt+1 )|t = E h(xt+1 )|xt (1.9) for any arbitrary integrable function h. This is extremely convenient because if xt is an asset price, then the value of a derivative is given by h(xt ). Corollary 1. If the process xt is Markov, then any asset or wealth process h(xt ) is a martingale, h(xt ) = P −T E h(xt+T )|xt , if h is any continuous function. Relation to Harrison-Pliska (1981) Define an arbitrage opportunity as the property that Xt = 0, E [XT |t ] > 0

(1.10)

for some t ≤ T . No arbitrage opportunities are possible if Xt = 0 ⇒ Et [XT ] = 0 so the standard no arbitrage condition can be written Xt = 0 ⇔ Et [XT ] = 0.

(1.11)

An asset with zero value at time t must have identically zero payoffs. In their seminal chapter, Harrison and Pliska (1981) show that the no arbitrage condition is equivalent to the existence of a martingale representation. In other words, the two properties: 1. no arbitrage and, 2. martingale representation are equivalent. Our proposition adds a condition: 3. Markov, as another equivalent property. This reveals a somewhat


8 Jeremy Berkowitz

surprising link between a lack of arbitrage opportunities and the Markov quality. To appreciate the direct connection between no arbitrage on the one hand, and the Markov property consider the following corollary from proposition 1. Corollary 2. A process Xt is Markov if and only if t+T

h(Xt+T ) = h(Xt ) +

(P − I)h(Xs )ds

(1.12)

t

for arbitrary bounded function h. ∂ h(X ) = Ah(X ), where A is the infinitesiTaking the limit as T → 0, ∂t t t mal generator of the process (when it exists). So the Backward equation follows from the Markov property. Suppose, for example, the process is a diffusion. Then the generator is known: Ah(Xt ) = µt

σ 2 ∂2 ∂ h(Xt ) + t h(Xt ). ∂Xt 2 ∂X 2 t

The PDE can be derived directly from the Markov property of the diffusion and its infinitesimal operator. Alternatively, the PDE can be viewed as a result of no arbitrage which in turn is equivalent to the Markov property. But note the following distinction. The martingale representation expresses the asset price as an expected payoff with respect to a “riskneutral” density. This density is not necessarily equal to the objective physical density. On the other hand, both the Markov property and the no arbitrage condition are statements about asset values under the objective, physical probabilities.

1.3 Application to valuing stocks In this section we show that if discounted dividends are Markov, the present value formula can be written as a geometric series expansion. The result is a generalized Gordon (1962) model that is valid for any Markov process. Suppose that dividends and the term structure of interest rates are Markov in the following sense.



Assumption B1. The stochastic discount factor mt = m(r t ) is a continuous function of a k-vector set of state variables, r t where (dt , rt ) is Markov. The vector r t contains the state vectors that describe the evolution of the discount rate. Assumption B2. For all t, the expected discounted dividends are welldefined, Et mt+k dt+k < ∞.

Let At = Et

dt+1 mt+1 dt mt

be the scalar, one-step ahead growth rate of

discounted dividends. From corollary 1, Et [mt+T dt+T ] = AT t mt dt for all forecast horizons T > 0. Plugging this into the present value, pt = E t

∞

mt+k dt+k = mt dt (I + At + A2t + · · ·) = (I − At )−1 dt

(1.13)

k=0

using a standard result from operator theory (e.g., Reed and Simon (1980, 191)). d This expression of course requires that Et mt,t+1 dt+1 < 1 so that the t geometric expansion is well-defined. Economically, this means that dividends must grow more slowly than the discount rate (on average). Since observed prices are finite, one might say we infer that this holds. We summarize the result as follows. Proposition 2. If there are no arbitrage opportunities, the value of an E d mt+1 −1 asset with payments {dt } is given by pt = 1 − t t+1 dt . dt mt An equivalent statement of the result is −1

gt+1 pt = 1 − Et dt (1.14) 1 + rt+1 where gt +1 = dt +1 /dt is the dividend growth rate and rt +1 is the time t to time t + 1 discount rate. This is the natural generalization of the Gordon (1962) model to any situation in which dividends and discount rate are Markov. The present value model expresses the stock price as an average of an infinite set of future payments. In the Markov world, long-run average behavior is determined entirely by the one-step ahead conditional distribution. This is what permits collapsing the infinite set of conditional expectations operators into a function of the one-step ahead conditional expectation.


10 Jeremy Berkowitz

Another way to view this result is as an analog to the “chain rule of forecasting”. This principle is the observation that in order to make k-step ahead forecasts, one must implicitly say something about the path that will be taken in steps one through k − 1. When the process is Markov, conditional expectations have the semi-group property and are thus geometric in the forecast horizon. 1.3.1 Dividend and discount rate models Our framework can be applied to arbitrarily complicated dividend, discount rate models. This is because all that is required are the one-step ahead transitions. Given the one-step ahead distribution, the stock value can be calculated exactly. Table 1.1 summarizes the present value expressions implied by various common models as indicated in column 1. Column 2 shows the present value given each model’s restrictions on the SDF and/or the dividend growth process. Column 3 reports the same present value formula simplified by using the semi-group property or

Table 1.1 Present value models of equity valuation Model CAPM

Present value expression Et

C-CAPM with CARA utility

Et

APT (factor models)

Et

Gordon model

Et

Markov model

Et

∞

a + bRm,t+k dt+k

k=0 ∞ c

t+k ct

k=0

−γ

dt+k

a + bFt dt+k

∞

1 d + g)k k (1 k=0 (1+R) ∞

−1 d dt 1 − Et a + bRm,t+1 t+1 d t

∞ k=0

Chain-rule formula

1

k k=0 (1+R ) t+j

dt

d

dt 1 − Et t+1 dt

ct+1 ct

−γ −1

−1 d dt 1 − Et a + bFt t+1 d t

dt

1+g r−g

−1 1+g dt 1 − Et 1+r t+1 t+1

j=1

Notes: Table presents the present value expression under various restrictions on the stochastic discount factor and/or dividend growth. The chain-rule formula refers to the same model when discounted dividends are Markov. For the CAPM model (a, b) are constants and Rm,t+k is the return on the market portfolio. C-CAPM denotes the consumption-based capital asset pricing model with time-separable constant absolute risk aversion utility. APT valuation refers to any linear factor model in which Ft are the factors.



“chain-rule” of forecasting that is applicable if discounted dividends are Markov. The bottom row of Table 1.1 indicates that the Markov model cannot be implemented until we specify a process for the discount rate and dividend growth. For example, we might model r t using the model of Cox, Ingersoll and Ross (1985). The Cox, Ingersoll and Ross (CIR) model is widely used for pricing zero coupon bonds and derivatives. The approach is based on the following stochastic process for the short rate: √ drt = a(µ − rt )dt + σ rt dz The model exhibits mean reversion of the interest rate, rt . The coefficient a is the speed of mean reversion, µ is the long run average rate and σ is the volatility. The CIR model implies that Et rt+1 = αµ + (1 − α)rt .

(1.15)

Given parameter estimates (µ, σ ), the one period discount is easily calculated from the current rate. Similarly, there is a good deal of flexibility in modeling the dynamic behavior of dividends. In the discrete time framework, we might assume that dividends are first-order autoregressive, dt = β0 + β1 dt−1 + εt

(1.16)

where εt ∼ iid(0, σ ). The persistence, β1 , does not need to be less than one – the dividend payout process can be stationary or nonstationary. Barsky and Delong (1993), for example, model dividends as moving average of order one in differences: dt = εt +

t−1

(1 − θ )εt−1 + g0

i=1

with θ in the range of .96. In this case, the generalized Gordon model (1.18) no longer holds but is used as an approximation.1 Our suggestion is quite similar in spirit but we require the dividend process be Markov. Given specifications for dividend growth and the discount rate, the Markov present value collapses to

pt = 1 −

−1 gˆt+1 dt 1 + rˆt+1

(1.17)

where the hats denote one-step ahead conditional expectations.


12 Jeremy Berkowitz

1.4 S&P500 Index prices and the equity premium puzzle Empirical investigations of the present value formula applied to US equity returns have been largely negative. In particular, actual stock returns appear to be far more volatile than volatility in dividends and discount rates would justify. Campbell and Shiller (1987, 1989) and Barsky and Delong (1993) find that the fundamental price implied by various present value models commoves with the actual price. However, the volatility of the “fundamental price” implied by the model is not high enough to match actual price volatility. It has long been argued that the volatility of observed prices is too high to be consistent with reasonable models of the stochastic discount factor. Formal testing of the consumption-based asset pricing model leads to convincing rejections of the CCAPM (e.g., Hansen and Singleton (1983)). The calibration exercises of Mehra and Prescott (1985) suggest implausibly high risk aversion or an equity premium puzzle. In a similar vein, Cochrane and Hansen (1992) and Cecchetti, Lam and Mark (1994) show that the CCAPM implies a stochastic discount factor that cannot attain the minimum volatility bounds of HansenJagannathan (1991). Again, this is equivalent to stating that the data imply an implausibly high risk aversion coefficient. Yet, on the positive side, one can easily construct a model with arbitrarily high volatility. Regardless of whether a particular parameterization of the CCAPM can generate sufficient volatility in asset prices, an infinite number of model can. In the representative CCAPM with constant −γ absolute risk aversion, mt,t+1 = Ct+1 /Ct so the volatility of the SDF is driven entirely by that of the consumption growth rate. The parameter γ is fixed and serves only to control the degree to which volatility in consumption translates into asset price volatility. More generally, the SDF is the derivative that transforms the martingale-equivalent density into the true physical density (the RadonNikodym derivative). As such, it can display a wide array of Markovian dynamics, at least in theory. A natural question is whether a simple first-order Markov model can improve on existing methods. If so, it argues against the finding that present value models based on expected dividends cannot replicate actual stock volatility. We take monthly S&P500 price, dividend and interest rate data from Robert Shiller’s long-term stock return data set.2 All series are


Valuing Equity when Discounted Cash Flows are Markov 13 Table 1.2 Summary statistics: Price-dividend ratios, dividend growth and stock returns January 1871 to December 2008 Standard Excess 1st 99th Mean deviation Skewness kurtosis percentile percentile Price-dividend 14.54 ratio Dividend 0.29 growth rate Stock return 8.58 Interest rate 4.80

7.52

2.24

5.48

6.35

44.89

1.12

−1.23

9.63

−3.75

3.39

5.04 2.75

1.49 1.18

17.53 3.39

−4.57 0.56

19.85 14.60

monthly and extend from January 1871 through December 2008 for a total of 1,656 observations. Stock price data are monthly averages of daily closing prices. The nominal interest rate series is the prime commercial chapter rate. Further detail on the dataset is available in Shiller (1989). Table 1.2 reports various summary statistics. The long-term average (net) annual return on the index is about 8.6 percent. Since the average short term commercial chapter rate is about 4.8 percent, the average equity premium is 3.8 percent. The price-dividend ratio, pt /dt and dividend growth rate dt+1 /dt − 1. Table reports summary statistics for S&P500 price, dividend, earnings and returns data. All series are monthly from January 1871 through December 2008 for a total of 1,656 observations and are taken from Robert Shiller’s S&P500 data set. The data consists of monthly stock price, dividends, and earnings data and the consumer price index (to allow conversion to real values). Stock price data are monthly averages of daily closing prices. The nominal interest rate series is the prime commercial chapter rate. Both ratios, indeed all four variables, are extremely fat-tailed and the tail percentages reveal substantial variation. Histograms are plotted in Figure 1.1 to give a sense of the sample distributions. Figure 1.2 plots the log index price (solid line) over the more familiar period, January 1947 to December 2008. The dashed line is the log pricedividend ratio times five over the same period. We multiply by five to make the two series visible on the same scale. The graph clearly confirms the observation that movements in the overall price level coincide with swings in the price-dividend ratio.


14 Jeremy Berkowitz Price-dividend ratio

300

Dividend growth rate

500

250

400

200 300 150 200 100 100

50 0 0

10

20

30

40

50

Stock returns

500

0 0.9

400

300

300

200

200

100

100

0.8

0.9

1

1.1

1.2

1.3 1.4

0 0.7

1

1.05

C-P Rate

500

400

0 0.7

0.95

0.8

0.9

1

1.1

1.2

1.3 1.4

Figure 1.1 Histograms of the price-dividend, dividend growth and stock returns Notes: Histograms of monthly S&P500 index price-dividend ratios, dividend growth rates and S&P500 stock returns. All series are taken from Robert Shiller’s long term monthly stock dataset available at http://www.econ.yale.edu/˜shiller.

In Figure 1.3, the log of the monthly S&P500 is plotted along with the “fundamental” price implied by the Gordon (1962) model in which the dividend growth rate and the expected rate of return are set equal to their sample averages over the post-war period. As discussed above, the fundamental price commoves with the actual price. However, the volatility of the fundamental price is not high enough to match actual price volatility. As a Markov alternative, consider a simple vector autoregression in the dividend growth gt = dt /dt−1 and the discount rate gt−p gt gt−1 = B1 + · · · + Bp + εt (1.18) rt rt−1 rt−p



7.5

price level price-dividend ratio

7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1950

1960

1970

1980

1990

2000

2008

Figure 1.2 Postwar S&P500 index prices and the price-dividend ratio Notes: The solid line is the log of monthly S&P500 stock prices from January 1947 to December 2008. The dashed line is the log of the dividend-price ratio (separate scale).

where rt is the stock return from t-1 to t and p is the lag order. From proposition 2, the stock price is given by

pt = 1 − E t

−1 dt+1 dt (1 + rt+1 )dt

(1.19)

and plugging in the vector autoregression this becomes

   1 −  

e1 A 1 + e2 A

gt rt

−1     gt  rt

dt

(1.20)

where e1 and e2 are vectors equal to zero except for the first element of e1 and the p+1st element of e2 which equal one. The formula (1.24) is


16 Jeremy Berkowitz

7.5

S&P500 Gordon

7 6.5 6 5.5 5 4.5 4 3.5 3 2.5

1950

1960

1970

1980

1990

2000

2008

Figure 1.3 Stock prices and Gordon (1962) model-implied values Notes: The solid line is the log of the monthly S&P500 stock prices from January 1947 to December 2008. The dashed line is the fundamental price implied by a constant dividend-price ratio equal to the sample average. The dashed line is the VAR-implied price.

closely analogous to Campbell and Shiller (1987) but does not require logs or linear approximations. The coefficients in the first-order vector autoregression are estimated via least-squares and the model-implied price-dividend ratio is plotted in Figure 1.4. The solid line is the actual ratio and the dashed is the model predicted. Clearly, a first-order autoregressive model is capable of matching the dynamics in the price-dividend ratio. In fact, the model implied ratio is more volatile than the actual price. Figure 1.5 shows the log fundamental price (1.24) along with the actual S&P500 log price level. As would be expected in light of the volatility in the price-dividend ratio, the Markov model price tracks the actual price quite well but with excessive volatility.



5.5 Actual Markov

5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

1950

1960

1970

1980

1990

2000

2008

Figure 1.4 Actual and Markov model-implied price-dividend ratios Notes: The solid line is the log of the monthly price-dividend ratio from January 1947 to December 2008. The dashed line is the price-dividend ratio implied by the first-order autoregressive model.

This is entirely plausible in a partial equilibrium approach such as ours. The price-dividend ratio, and hence the stock price, is driven by one-step ahead expectations. If the expectations themselves are sufficiently volatile, then the rational price will also be quite volatile – even if dividend growth is smooth.

1.5 Conclusion This chapter is motivated by a desire to find a formal framework for valuing stocks from the underlying payments. It turns out that the required framework is very general and extends to any Markov process. In fact, the approach can be applied to other assets besides stocks. It may be reasonable for example to model currencies and commodities as simple


18 Jeremy Berkowitz

S&P500 Markov

8

7

6

5

4

3

2 1950

1960

1970

1980

1990

2000

2008

Figure 1.5 Stock prices and Markov model-implied values Notes: The solid line is the log of the monthly S&P500 stock prices from January 1947 to December 2008. The dashed line is the fundamental price implied by the VAR(1) Markov model in levels. See text for details.

Markov processes. It would be also interesting to assess whether it is possible to derive implications for optimal dividend payout policy from the Markov assumption. The generalized Gordon model we have derived is a moment condition. This suggests testing the present value model directly with moment-based tests such as the GMM J-statistic. A rejection of the moment condition implies there is a systematic departure of the price from its value. For example, the equity premium puzzle can be derived directly without appeal to logs or Normality. Lastly, a further advantage to the Markov assumption is that it makes available the tools developed in Duffie and Singleton (1993) for estimation. An extensive set of results are derived for estimating and conducting inference on the unknown parameters. Since this is precisely the assumption we make on the dividend and discount rate process, it is immediately applicable as an estimation tool.



Notes Acknowledgements: I gratefully acknowledge helpful discussions with Peter Christoffersen, Tom George and Bong-Soo Lee. Any remaining errors or inaccuracies are mine. 1. The reason the Barsky-Delong model is only approximate is that the MA model is not Markov. 2. The data are available on the web at http://www.econ.yale.edu/∼shiller/

References Barsky, R. B. and DeLong, B. J. (1993) “Why Does the Stock Market Fluctuate?” The Quarterly Journal of Economics, 108(2),: 291–311. Berkowitz, J. (2001) “Generalized Spectral Estimation of the Consumption-Based Asset Pricing Model,” The Journal of Econometrics, 104(2): 269–288. Campbell, J. and Shiller, R. (1987) “Cointegration Tests of Present Value Models,” Journal of Political Economy, 95(5): 1062–1087. Campbell, J. and Shiller, R. (1989) “The Dividend-Price Ratio and Expectations of Future Dividends and Discount Factors,” Review of Financial Studies, 1(3): 195–227. Cecchetti, S.G., Lam, P. and N. Mark (1994) “Testing Volatility Restrictions on Intertemporal Marginal Rates of Substitution Implied by Euler Equations,” Journal of Finance, 49(1): 123–152. Cochrane, J. and Hansen, L. P. (1992) “Asset Pricing Explorations for Macroeconomics,” O. Blanchard and S., (Eds), 1992 NBER Macroeconomics Annual, 115–165. Cox, J.C., Ingersoll, J. E. and S.A. Ross. (1985) “A Theory of the Term Structure of Interest Rates,” Econometrica 53(2): 385–408. Duffie, D. and Singleton, K. (1993) “Simulated Moments Estimation of Markov Models of Asset Prices,” Econometrica, 61(4), 929–952. Durlauf, S. N. (1991) “Spectral Based Testing of the Martingale Hypothesis,” Journal of Econometrics, 50(3): 355001E376. Ethier, S.N. and Kurtz, T. G. (1986) Markov Processes: Characterization and Convergence, John Wiley: New York, NY. Ferson, W. and Harvey, C. R. (1992) “Seasonality and Consumption-Based Asset Pricing,” Journal of Finance, 47(2): 511–552. Gibbons, M. R., Ross, S. A. and J. Shanken (1989) “A Test of the Efficiency of a Given Portfolio,” Econometrica, 57(5): 1121–1152. Gordon, M. (1962) The Investment, Financing, and Valuation of the Corporation. Irwin: Homewood, IL. Hansen, L.P. and Jagannathan, R. (1991) “Implications of Security Market Data for Models of Dynamic Economies,” Journal of Political Economy, 99(2): 225–262. Hansen, L. P. and Scheinkman, J. A. (1995) “Back to the Future: Generating Moment Implications for Continuous Time Markov Processes,” Econometrica, 63(4): 767–804. Hansen, L. P. and Singleton, K. J. (1983) “Stochastic Consumption, Risk Aversion and the Temporal Behavior of Asset Returns,” Journal of Political Economy, 91(2): 249–265.


20 Jeremy Berkowitz Harrison, M. and Kreps, D. (1979) “Martingales and Arbitrage in Multiperiod Securities Markets,” Journal of Economic Theory, 20(1): 381–408. Harrison, M. and Pliska, S. (1981) “Martingales and Stochastic Integrals in the Theory of Continuous Trading,” Stochastic Processes and their Applications, 11(3): 215–260. Norris, J. R. (1997) Markov Chains, Cambridge University Press: Cambridge, UK. Reed, S. M. and Simon, B. (1997) Functional Analysis, Academic Press: New York. Shiller, R.J. (1989) Market Volatility, MIT Press: Cambridge, MA. Stanton, R. (1997) “A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk,” Journal of Finance, 52(5): 1973–2002. Stroock, D.W. and Varadhan, S. R. (1972) “On the Support of Diffusion Processes with Applications to the Strong Maximum Principle,” Proc. of Sixth Berkeley Symposium on Mathematics Statistics and Probability III, 333–359.


2 Markov Switching Mean-Variance Frontier Dynamics: Theory and International Evidence Massimo Guidolin and Federica Ria

2.1 Introduction One class of models that has gained growing attention in the financial econometrics and asset pricing literatures relies on multivariate extensions of the seminal work by Hamilton (1989) in macroeconomics and by Turner et al. (1989) in financial economics on the presence of Markov regimes in many important time series, including asset returns. This chapter provides a short primer to the structure, estimation issues, and potential applications of multivariate Markov switching models. An illustrative application to an international equity portfolio diversification problem is provided with reference to standard MSCI indices. Although our brief review of methods for and technical issues with multivariate Markov switching models competes with a number of alternative reference articles and books, our chapter also addresses one question that has been receiving growing attention in the empirical finance literature:1 How important and economically valuable can it be for an investor to use information on the current and predicted state of the capital markets when planning her optimal diversification strategies? Clearly, if international equity returns are disconnected from the underlying economic regimes and as such tend to be display statistical properties (such as expected returns, variances, and covariances) that are not predictable over time, then the answer is trivial: Because such states either fail to exist or are irrelevant to the risk-reward opportunities offered by the equity markets, a rational investor may safely ignore the issue. This means that the now classical, Markowitz-style mean-variance recipes offered in most textbooks would be correct and the investor ought to 21


22 Massimo Guidolin and Federica Ria

choose optimal portfolio weights on the basis of simple, naive historical estimates of means, variance, and covariances. If. on the contrary, such regimes exist and may be identified, estimated, and predicted, then it is an open question whether an investor should take notice of them, and go through the relatively sophisticated econometric techniques – the subject of Section 2.2 in this chapter – required by her acknowledging this state-dependence. The issue of whether non-linear predictability patterns may be exploitable for dynamic asset allocation purposes is an important and much researched one. There is an early applied portfolio management literature that has noticed that making portfolio choices state-dependent may deliver important ex ante performance improvements. For instance, Clarke and de Silva (1998) note that no static mix to be applied to standard mean-variance portfolios can be used to achieve a point along a state-dependent efficient frontier. The more efficient and desirable riskreward combinations on the state-dependent frontier can be achieved only by systematically altering portfolio allocations in response to changes in the investment opportunities as the economy switches back and forth among different states. Chow et al. (1999) implicitly introduce the idea that even scarcely persistent, infrequent regimes may play an important role in portfolio choice, especially when it comes to estimating risk. They think about the issue by distinguishing time-measured observations from event-measured observations. They propose to estimate risk parameters not from time-measured but instead from event-measured data and conjecture that such an alternative approach may provide a better representation of the density from which portfolio returns may be drawn during turbulent market periods. More recently, the literature has shifted towards writing and solving portfolio problems under the assumption of statistical frameworks in which asset returns follow a switching dynamics. Ramchand and Susmel (1998) examine the relationship between correlation and variance in a regime-switching ARCH model estimated on weekly stock returns data for the US and a few other major markets. They find that correlations between US and other world markets are 2 to 3.5 times higher when the US market is in a high variance state. Ang and Bekaert (2002a) consider bivariate and trivariate regime switching models that capture asymmetric correlations in volatile and stable markets and characterize a US investor’s optimal asset allocation under constant relative risk aversion. Guidolin and Timmermann (2008b) use an international portfolio diversification application to propose a new tractable approach to solving asset allocation problems under Markov switching with a large number of assets.


Markov Switching Mean-Variance Frontier Dynamics 23

2.2 Multivariate Markov switching models Suppose that the n × 1 random vector yt follows a k-regime Markov switching (MS) VAR(p) process with heteroskedastic component, compactly MSIAH(k,p): yt = µSt +

p

Aj,St yt−j + St εt

(2.1)

j=1

with εt ∼ NID(0, In ). St is a latent state variable driving all the matrices of parameters appearing in (2.1). µSt is an n × 1 vector that collects the n regime-dependent intercepts, while the n × n matrix St represents the factor applicable to state St in a state-dependent Choleski factorization of the variance covariance matrix of the variables of interest, St . Obviously, a non-diagonal St makes the n variables simultaneously cross-correlated. For instance, in Guidolin and Ono (2006) n is broken down into n1 asset returns and n2 macroeconomic predictors, with n1 + n2 = n. Then a non-diagonal St captures simultaneous comovements between asset returns and macro factors, while dynamic (lagged) linkages both across different asset markets and between financial markets and macroeconomic influences are captured by the VAR(p) component. We assume the absence of roots outside the unit circle, thus making the process stationary. In fact, conditionally on the unobservable state St , (2.1) defines a standard Gaussian reduced form VAR(p) model. On the other hand, when k > 1, alternative hidden states are possible and they will influence both the conditional mean and the volatility/correlation structures characterizing the multivariate process in (2.1), St = 1, 2, . . . , k ∀t. These unobservable states are generated by a discrete-state, homogeneous, irreducible and ergodic first-order Markov chain:2 Pr(St = j|{Sj }t−1 , {yj }t−1 ) = Pr St = j|St−1 = i = pij , j=1 j=1 where pij is the generic [i, j] element of the k × k transition matrix P. Ergodicity implies the existence of a stationary vector of probabilities ξ satisfying ξ = P ξ. Irreducibility implies that ξ > 0, meaning that all unobservable states are possible. In practice, P is unknown and hence ξ can be at most estimated given knowledge on P extracted from the information set t = {yj }tj=1 . For simplicity, we will denote as ξ such an estimated vector of ergodic state probabilities. When n is large, model (2.1) implies the estimation of a large number of parameters, k[n + pn2 + n(n + 1)/2 + (k − 1)]. For instance, for k = 2, n = 8,



and p = 1, this implies the estimation of 2 × [8 + 82 + 4 × 9 + 1] = 218 parameters!3 Model (2.1) nests a number of simpler models in which either some of the parameter matrices are not needed or some of these matrices are independent of the regime. These simpler models may greatly reduce the number of parameters to be estimated. Among them, the financial econometrics literature (see, e.g., Ang and Bekaert, 2002a, and Guidolin and Nicodano, 2009) has devoted special attention to MSIH(k) models, yt = µSt + St εt ,

(2.2)

in which p = 0, and to MSIA(k, p) homoskedastic models, yt = µSt +

p

Aj,St yt−j + εt ,

j=1

in which the covariance matrix is constant. Of course, a limit case of (2.1) is obtained when k = 1: yt = µ +

p

Aj yt−j + εt .

(2.3)

j=1

This is a standard multivariate Gaussian VAR(p) model, a benchmark in a large portion of the existing empirical macroeconomics and finance literature. 2.2.1 Estimation and inference The first step towards estimation and prediction of a MSIAH model is to put the model in state-space form. Collect the information on the time t realization of the Markov chain in a random vector ξt ≡ [I(St = 1) I(St = 2) . . . I(St = k)]’, where I(St = i) is an indicator variable. In practice the sample realizations of ξt will always consist of unit vectors ei characterized by a 1 in the i-th position and by zeros everywhere else. Another property is that E[ξt |ξt−1 ] = P ξt−1 . The state-space form is composed of two equations: yt = Xt ξt ⊗ ιn + ∗ ξt ⊗ In εt ξt+1 = Fξ t + ut+1

(measurement equation)

(transition equation)

(2.4)

where Xt is a n × (np + 1) vector of predetermined variables with structure ] ⊗ ι , is a (np + 1) × nk matrix collecting the conditional [1yt−1 . . . yt−p n



mean parameters,    =  

µ1 A11 .. . Ap1

··· ··· .. . ···

µk A1k .. . Apk

   ,  

∗ is a n × nk matrix collecting all the possible k square root (Choleski decomposition) factors [ 1 2 . . . k ] such that ∀t, ∗ ξt ⊗ In ξt ⊗ In ( ∗ ) = St , the St -regime covariance matrix of the asset return innovations εt . Moreover, εt ∼ NID(0, In ), and in the transition equation ut+1 is a zero-mean discrete random vector that can be shown to be a martingale difference sequence. Also, the elements of ut+1 are uncorrelated with εt+1 as well as ξt−j , εt−j , yt−j , and Xt−j ∀j ≥ 0. To the dynamic state-space system (2.4) operational, assume that the multivariate process (2.1) started with a random draw from the unconditional probability distribution defined by the vector of state probabilities ξ. Finally, from the definition of transition probability matrix, it follows that since E[ut+1 |ξt ] = 0 by assumption, then E[ξt+1 |ξt ] = Fξ t implies that F corresponds to the transposed transition probability matrix P . Multivariate Markov switching models are estimated by maximum likelihood. In particular, estimation and inferences are based on the EM (Expectation-Maximization) algorithm proposed by Dempster et al. (1977) and Hamilton (1989), a filter that allows the iterative calculation of the one-step ahead forecast of the state vector ξt+1|t given the information set t and the construction of the log-likelihood function of the data. Because the MLE first-order conditions all depend on the smoothed probabilities ξˆ t|T ≡ Pr ξt |T ; θ, ρ (i.e., the state probabilities estimated on the basis of the full sample of data), they all present a high degree of non-linearity in the parameters, γ ≡ [θρ] . As a result, these first-order conditions have to be solved numerically, although convenient iterative methods exist. In fact, the expectation and maximization steps can be used in iterative fashion until convergence, see Guidolin and Ono (2006) for details. As for the properties of the ML estimators, under standard regularity conditions (such as identifiability, stability and the fact that the true parameter vector does not fall on the boundaries) Hamilton (1989, 1993) has proven consistency and asymptotic normality of γ˜ = [θ˜ ρ] ˜ :

√ d T (γ˜ − γ) → N 0, Ia (γ)−1 ,



where Ia (γ) is the asymptotic information matrix,

p(yt |γ) ∂ 2 ln T t=1 −1 . Ia (γ) ≡ lim −T E ∂γ∂γ T →∞ Although other choices exist – i.e., either to use the conditional scores or a numerical evaluation of the second partial derivative of the loglikelihood function – in applications it has become typical to employ a White-style “sandwich” sample estimator of Ia (γ) which yields the estimate ˜ γ) ˜ 1 (γ)) ˜ −1 I2 (γ)] ˜ ˜ = T −1 [I2 (γ)(I V ar( where ˜ = T −1 I1 (γ)

T

˜ [ht (γ)] ˜ ht (γ) ˜ = [ht (γ)]

t=1

˜ = −T −1 I2 (γ)

˜ ∂ ln p(yt |t−1 ; γ) ∂γ

T ˜ ∂ 2 ln p(yt |t−1 ; γ) . ∂γ∂γ

t=1

As a consequence, and with one important exception, standard inferential procedures are available to test statistical hypothesis. In particular, call ϕ : Rq → Rr a (smooth) function that imposes q − r restrictions on the q-dimensional parameter vector θ. We want to test H0 : ϕ(γ) = 0 vs. H1 : ϕ(γ) = 0 under the assumption that under both hypothesis the number of regimes k is identical. Define θ˜ r as the restricted estimator, obtained under the null hypothesis. Lagrange Multiplier (LM) tests are undoubtedly the preferred tests as they only require the estimation of the restricted model. While the scores of an unrestricted model, t t ∂diagητ ((γ))Fτ (γ) ˜ ≡ ˜ = st (γ) hτ (γ) ξˆ τ |t , ∂γ γ=γ˜ τ =1

τ =1

have zero mean vector by construction, the scores of the restricted model obtained by MLE and imposing ϕ(θ) = 0 can be used to obtain the standard test statistic: d ˜ θ˜r )]−1 ST (θ˜r ) → χr2 LMT ≡ ST (θ˜r ) [V ar(

and θ˜ r denotes the restricted estimator. For where r = rank ∂ϕ(θ) ∂θ

instance, a test of the hypothesis of homoskedasticity (H0 : vech(i ) = vech(M )i = 1, 2, . . . , M) implies r = (k − 1) n(n+1) restrictions and can be 2



formulated as a linear restriction on the matrix ∗ . As an alternative, the Likelihood Ratio (LR) test might be employed: d LR ≡ 2 ln L(θ˜ ) − ln L(θ˜r ) → χr2 . This test requires the estimation of both the restricted and the unrestricted models, which for n high enough may be quite cumbersome and require a host of diagnostic checks on the performance of the EM algorithm in locating a truly global maximum of the likelihood function. Finally, standard t and F statistics can be calculated in the form of a Wald test. The exception to standard inferential procedures concerns the number of non-zero rows of the transition matrix P, i.e., the number of regimes k. In this case, even under the assumption of asymptotic normality of ˜ standard testing procedures suffer from non-standard the estimator γ, asymptotic distributions of the likelihood ratio test statistic due to the existence of nuisance parameters under the null hypothesis. We specifically discuss this problem in Section 2.2.2. 2.2.2 Model selection and diagnostic checks In the absence of Markov switching dynamics in the matrices of autoregressive coefficients and in the covariance matrix of the vector process – i.e., for simple MSI(k, 0) and MSI(k)-VAR(p) processes – it is possible to show that general multivariate Markov switching models possess a standard VARMA representation that helps define a somewhat precise mapping between non-linear Markov switching processes and their linear counterparts. In particular, under a few regularity conditions, (2.1) possesses a VARMA(k + p − 1, k − 1) representation, where k + p − 1 is the autoregressive order and k − 1 is the moving average order. These results give a useful starting point in a simple-to-general specification approach: 1. A researcher may start out by conducting a standard Box-Jenkins’ style model selection procedure applied to the class of VARMA models. The reason is that given the existence of VARMA(a, b) representations for MS processes, it is then possible to solve a simple bivariate system of linear equations to recover k and p from the selected values for a and b. 2. Given such a p∗ , the focus shifts on the number of regimes k. Krolzig (1997) has suggested the analysis of each component of the vector yt in isolation to detect the appropriate number of regimes for each of them, say ki for yit , i = 1, 2, . . . , n. In this case the (V)ARMA equivalence



results can be fully exploited. For each time series, the best fitting ARMA model could be selected using Box-Jenkins or any other ARMA specification criteria. Call {ki∗ }ni=1 the sequence of resulting number of states for each univariate variable under study. 3. Given {ki∗ }ni=1 , the total number of regimes characterizing the multi variate process might be in principle as high as ni=1 ki∗ if the regimes are not simultaneously perfectly correlated with each other, i.e., if it does not occur that at least a subset of variables are governed by the same hidden Markov chain. This latter hypothesis is testable using standard inferential procedures. As illustrated in our application in Section 2.3 as well as in a number of chapters in the recent literature, an alternative set of methods to perform data-driven model selection relies on information criteria, such the Schwartz, Hannan-Quinn, and Akaike criteria (see e.g., Sin and White, 1996, for evidence on information criteria performance in non-linear models). Interestingly, very few chapters have addressed the issue of the small-sample and asymptotic performance of these information criteria specifically for the case of Markov switching models. Once a restricted set of MSIAH models has been estimated, the need of further improvements could arise as the result of diagnostic checks. Although the EM algorithm naturally delivers estimates of the parameters γ˜ besides the smoothed sequence of probability distributions {ξˆ t|T }T therefore lead to define the (smoothed) residuals t=1 and would

as u˜ t ≡ yt − Xt ξˆ t|T . These are not well suited to the use in diagnostic checks as they are full-sample random statistics and hence they structurally overestimate the explanatory power of the MS model. On the contrary the one-step predictions errors

e˜ t|t−1 ≡ yt − Xt Fξˆ t−1|t−1 , are limited information statistics (being based on filtered probabilities) and uncorrelated with the information set t−1 because E[yt |t−1 ] = ˆ ξˆ t−1|t−1 and therefore form a martingale difference sequence Xt BF E[˜et|t−1 |t−1 ] = 0. Therefore standard tests of this hypothesis (such as Portmanteau tests of no serial correlation) could be used. In the presence of Markov switching heteroskedastic components, researchers in empirical finance (e.g., Kim and Nelson, 1999) have also suggested to check whether the smoothed, standardized residuals contain any residual ARCH effects. Standard LM-type as well as Ljiung-Box tests can be applied. This is a way to check whether Markov switching covariances may be sufficient to capture most of the dynamics in volatility, else



explicit ARCH-type modeling (even of Markov switching type, as in Hamilton and Susmel, 1994, or Guidolin, 2009) may be required. Another important type of diagnostic check concerns the number of regimes k. The problem is that under any number of regimes smaller than k there are a few structural parameters of the unrestricted model – the elements of the transition probability matrix associated with the rows that correspond to disappearing states – that can take any values without influencing the resulting likelihood function. We say that these parameters become a nuisance to the estimation. The result is that the presence of these nuisance parameters gives the likelihood surface so many degrees of freedom that computationally one can never reject the null that the non-negative values of those parameters were purely due to sampling variation. Different alternative ways have been proposed to develop sound inferential procedures concerning the number of regimes in multivariate Markov switching models. Hansen (1992) proposes to see the likelihood as a function of the unknown nuisance parameters so that the asymptotic distribution is generated in each case numerically from a grid of transition and regime-dependent nuisance parameters. The test statistic is LWT ≤ sup LWT (ρ) ρ

where the right hand side converges in distribution to a function of a Brownian bridge. In most of the cases a closed form expression cannot be found and the bound must be calculated by simulation and becomes data-dependent. In addition, Davies (1977) bounds the LR test but avoids the problem of estimating the nuisance parameters and derives instead an upper bound for the significance level of the LR test under nuisance parameters:

√

x 1 −1 Pr (LR > x) ≤ Pr χ12 > x + 2x exp − . 2 2 The bound holds if the likelihood has a single peak. A related test is proposed by Wolfe (1971) and applied by Turner et al. (1989). The modified LR test is: LRWolfe = −

2 d ˜ − ln L(γ˜ r )] → χr2 (T − 3) [ln L(γ) T

where γ˜ r is obtained under the null of simple multivariate normality and r = k(k − 1) because in the absence of regime switching there are k(k − 1) which cannot be estimated.



Finally, common sense suggests that correct specification of a Markov switching model should give smoothed probability distributions {ξˆ t|T }T t=1 that consistently signal switching among states with only limited periods in which the associated distribution is flatly spread out over the entire support and uncertainty dominates. Regime Classification Measures have been popularized as a way to assess whether the number of regimes k is adequate. In simple two-regime frameworks, the early work by Hamilton (1988) offered a rather intuitive regime classification measure: RCM1 = 100

T k k2 Pr St = j|y1 , y2 , . . . , yT ; γ˜ , T t=1 j=1

i.e., a corrected sample average of the products of the smoothed state probabilities. Clearly, when a switching model offers precise indications on the nature of the regime at each time t, for at least one value of j = 1, . . . , k, Pr St = j|y1 , y2 , . . . , yT ; γ˜ ∼ = 1 so that k Pr St = j|y1 , y2 , . . . , yT ; γ˜ ∼ = 0 because most probabilities are zero. j=1

Therefore, a good switching model will imply RCM1 ∼ = 0.4

2.3 One application: Regimes in international stock returns In this Section we report one illustrative example of how multivariate Markov switching models may be used to capture the key dynamic features of large-scale, complicated financial phenomena and how they can be used to support sophisticated financial decision making. For reasons of space, we illustrate only the key points of our model specification search and of its asset management implications. Although the data are particular to this chapter, methods and qualitative results may be considered a special case (extension) of results in related chapters by Guidolin and Na (2008) and Guidolin and Nicodano (2009) to which a Reader is referred for additional details. Our empirical analysis is also related to Ang and Bekaert (2004), who solve a large-scale international mean-variance portfolio choice problem in which a version of the zerobeta CAPM is assumed ex-ante and the world market portfolio follows a two-state Markov switching model in its mean. 2.3.1 Data We examine MSCI international monthly equity index data for the sample 1988:01–2008:08, for a total of 248 observations. In this application,



we investigate the regime switching properties of the five major developed country/area value-weighted indices published by MSCI, i.e., (in order of declining capitalization), North America (US and Canada), Japan, Europe ex-UK, Pacific ex-Japan, and United Kingdom. The five indices are all expressed in US dollars. This means that we adopt the point of view of a US investor that is considering un-hedged international portfolio diversification decisions. The 1988–2008 sample period is a plausible estimation interval of time for many investors that could be interesting in performing econometric analysis on relatively recent equity data, while the fact that the sample extends well into 2008 allows us to reach conclusions that are likely to be robust to the recent turmoil in international equity markets. Table 2.1 reports standard descriptive statistics. The table gives only one surprising indication: over a 20-year time span, the Japanese stock market has given on average a slightly negative mean return (−0.6 percent in annualized term) and therefore a negative Sharpe ratio. Otherwise, means, volatilities and realized Sharpe ratios (computed using as a reference the US 1-month T-bill rate) are all within expected ranges. Pacific, continental European, and North American stocks all displayed significant deviations from a single-state IID Gaussian benchmark, as evidenced by the statistically significant Jarque-Bera statistics. All the three indices are characterized by negative skewness and kurtosis in excess of the Gaussian benchmark (three). Also Japanese stock returns display some excess kurtosis, but this is not sufficient to bring the Jarque-Bera statistic above the 5 percent significance threshold. Although none of the indices appears to be predictable based on its correlation structure, all the five indices present evidence of conditional heteroskedasticity, as shown by the Ljung-Box portmanteau tests applied on squared returns.

2.3.2 Model Selection Table 2.2 shows the results of a few model selection criteria applied to our 5 × 1 vector of international stock returns. Clearly, we have estimated a range of alternative MSIAH models, including simple singlestate (i.e., Gaussian IID and Gaussian VAR) ones. Significantly, with a total of 1,240 observations, one encounters difficulties at obtaining reliable estimates of richly parameterized models in which the number of parameters exceeds 100 so that the saturation ratio (i.e., the number of observations available to estimate each parameter, on average) is below ten. This is the case of the MSIAH(3,1), MSIA(4,1), and MSIAH(4,1) models.


GREGORIOU-3: “CHAP02” — 2010/11/10 — 21:37 — PAGE 32 — #12 4.9275

0.766∗

0.756∗∗ 3.924

4.3971

6.3103

−0.051

0.707

5.4283

0.746∗

0.103

0.080

0.084

−0.064

0.072

Sharpe ratio

** = significant at 1%; * = significant at 5%.

North America

United Kingdom

Europe ex-UK

Japan

Pacific ex-Japan

St. Dev.

Mean

1.093

0.673

1.103

−0.272

0.927

Median

10.4

14.1

−10.9 −14.3

13.8

21.7

15.3

Max.

−15.6

−21.6

−23.1

Min.

Table 2.1 Summary statistics for international stock returns

−0.441∗

0.038

−0.542∗

0.101

−0.530∗

Skewness

3.714

3.178

4.059

3.696

4.685∗

Kurtosis

13.309∗∗

0.389

23.73∗∗

5.43

40.94∗∗

Jarrque-Bera

7.77

11.23

15.48

10.94

18.62

LB(12)

34.88∗∗

57.88∗∗

25.22∗

52.51∗

28.59∗∗

LB(12)squares

GREGORIOU-3: “CHAP02” — 2010/11/10 — 21:37 — PAGE 33 — #13 −18.321 −18.862 −18.285

−18.540 −18.857

Four-state models 2345.95 129.780 0.000 −17.874 −18.272 2430.31 298.485 0.000 −17.554 −18.333 No converge achieved(too many parameters)

MSI(4,0) MSIH(4,0) MSIA(4,1)

−18.234 −18.138

Three-state models 2307.85 53.570 0.000 −17.811 −18.116 2405.03 259.939 0.000 −17.825 −18.485 2369.16 168.350 0.000 −16.708 −17.650 No converge achieved (too many parameters)

−18.120 −17.880

MSI(3,0) MSIH(3,0) MSIA(3,1) MSIAH(3,1)

−17.949 −17.848 −17.794 −17.148 −16.975

−17.951 −17.498 −18.332 −18.643 −18.218 −18.242 −18.282

0.000 0.000 0.000 0.000 0.000

_ _ −18.178 −18.284 −17.907 −17.802 −17.756

38.180 81.037 52.804 89.839 129.718

_ _

Two-state models 2300.15 2321.58 2307.46 2329.91 2349.85

AIC

MSI(2,0) MSIH(2,0) MSH(2,0) MSIA(2,1) MSIAH(2,1)

HQ

Single state models 2281.06 2284.99

BIC

MSIA(1,0) MSIA(1,1)

Davies’ approx. p-value

Model(k,p)

LR Statistic

Loglikelihood

Table 2.2 Model selection statistics

47 92 147

36 66 111 141

27 42 37 77 92

20 45

Number of parameters

1240 1240 1235

1240 1240 1235 1235

1240 1240 1240 1235 1235

1240 1235

26.4 13.5 8.4

34.4 18.8 11.1 8.8

45.9 29.5 33.5 16.0 13.4

62.0 27.4

VAR:56.54 (0.244)

H:42.86 (0.002) I:28.24 (0.000) VAR:59.52 (0.168) H:39.88 (0.001)

Number Saturation of obs. ratio Tests


The table 2.2 shows that the Davies’ approximate p-value for a test of the null of k = 1 regimes vs. the alternative hypothesis of k ≥ 2 (the specific value for k depends then on each of the models considered) is always zero for all Markov switching frameworks considered, independently on their specific structure in terms of choice of p and of whether variance and covariances ought to be allowed to be a function of the Markov state. The associated LR statistics are in fact so large that the same conclusion is likely to emerge regardless of the nuisance parameter correction applied to compute the p-values in the table. For instance, Wolfe’s test statistic delivers p-values which are also zero. Therefore, it is clear that the data seem to require the specification of Markov switching dynamics, which is consistent with earlier findings by Ramchand and Susmel (1998), Ang and Bekaert (2002a), and Guidolin and Timmermann (2008b). Next, we employ three information criteria to select among multi-state regime switching models. For each of three criteria (Bayesian-Schwartz, Hannan-Quinn, and Akaike), Table 2.2 boldfaces the three best models. Here, one has to recall that by construction (they reward fit as represented by the negative of the average log-likelihood function and penalize over-parameterization by adding a positive term), information criteria illustrate an increasingly good trade-off between fit and parsimony as their values decline. As one would expect in the light of their relative penalties for model size, the BIC tends to favor small models and in fact, a single-state Gaussian IID model enters the set of best models. On the contrary, AIC shows a bias in favor of relatively large models like the MSIH(4,0) model which enters the AIC’s best-three set in spite of its rather modest saturation ratio of 13.5.5 Hannan-Quinn is usually in an intermediate position when compared to BIC and AIC, although in our application it yields indications which are identical to AIC. However, in spite of these differences, the tables also stress the existence of the one, and only one, model that receives good scores from all the criteria: a parsimonious (42 parameters for a saturation ratio of 29.5) MSIH(2,0), i.e., a model with two regimes, regime-dependent covariance matrices, and no vector autoregressive component.6 In fact, the very last column of Table 2.2 also proceeds to test – using standard likelihood ratio tests – whether any expansion over the MSIH(2,0) may be required by the data. In particular, the null of a MSIH(2,0) against the alternative of a richer MSIAH(2,1) with p = 1 can be rejected with a p-value of 0.244. At the same time, the null of a simpler MSI(2,0) model vs. the MSIH(2,0) (i.e., no regime-switching heteroskedasticity) is rejected with a p-value of 0.002; the null of a MSH(2,0) model vs. the MSIH(2,0) (i.e., no regime switching in conditional means)



is rejected with a p-value of zero. We have also tried to use Krolzig’s (1997) VARMA-MSIA mapping method: because we find that a VARMA(2,1) seems to be required by the 5×1 vector at hand, we obtain that k +p−1 = 2 and k − 1 = 1. Solving for k and p, this gives k∗ = 2 and p∗ = 1. However, both LR tests and information criteria advise us to select instead k = 2 and p = 0, augmented by a regime-dependent heteroskedastic component. 2.3.3 A two-state model Table 2.3 reports estimates of the two-state model and, as a benchmark, of a matching single-state model – in this case a simple Gaussian IID model that implies constant means, variance, and covariances (means and variances are the same as in Table 2.1). Starting with the singlestate model, it is clear that all pairs of stock indices are characterized by positive and highly statistically significant correlation coefficients, ranging between 0.44 and 0.74. Although these correlations are high, not even 0.74 would be able to deprive a mean-variance investor from considerable gains from portfolio diversification. The second panel of Table 2.3 reports instead the estimates of the Markov switching model. The first regime is a bear state with low (negative, and statistically significant in three cases out of five) mean returns, high volatility, and high (above-average, represented by the single-state panel of the table) correlations.7 In this state, expected returns are negative and statistically significant in Japan, the UK, and the US; volatilities are high and ranging between 17 percent (Japan) and 27 percent (Pacific ex-Japan) per year; correlations exceed their simple Gaussian IID counterparts and average 0.67 with many spikes in excess of 0.7: when stock prices are hit by negative shocks, they all tend to fall together. The bear state has however only moderate persistence – once in the bear regime, market tend to stay in this state for 3 months – and characterizes 38 percent of the data in the long run (equivalently, 0.38 is the ergodic probability). Figure 2.1 shows that the most prolonged bear periods may be identified with the Spring-Summer of 1998 (the Asian flu), the dotcom market crash of 2000–2001, and a few months between late 2007 and the Summer of 2008. The second state is a bull regime characterized by positive (for all indices but one) mean returns, low volatilities, and moderate (belowaverage) correlations, in some cases not significantly different from zero. This regime displays considerable persistence – once in this regime, international stock markets tend to display this dynamics for 5 months on average; as a result, 62 percent of any long sample ought to be generated by this bull state. Figure 2.1 shows (by taking the complement to one of


36 Massimo Guidolin and Federica Ria Table 2.3 Estimates of two-state markov switching model Panel A - SINGLE STATE MODEL Pacific EX JP 1. Mean returns 2. Correlations/ Volatilities Pacific EX JP JP Europe EX UK UK North America

0.746*

Japan

−0.051

5.428** 0.444** 0.592** 0.621** 0.601**

6.310** 0.462** 0.480** 0.368**

Europe EX UK

UK

North America

0.766*

0.707*

0.756**

4.928** 0.744** 0.669**

4.397 0.664**

3.924**

Panel B - TWO-STATE MODEL

Pacific EX JP 1. Mean returns Bear/High Volatility State Bull/low Volatility State

−1.762

−2.781**

Europe EX UK −1.864

UK

−2.001*

North America −1.523*

1.003

1.781**

1.752**

1.635**

2. Correlations/ Volatilities Bear/High Volatility State Pacific EX JP 7.684** JP 0.676** Europe EX UK 0.560** UK 0.628** North America 0.602**

4.831** 0.715** 0.681** 0.625**

6.404** 0.748** 0.739**

4.238** 0.674**

4.577**

Bull/low Volatility State Pacific EX JP 3.811** JP 0.309** Europe EX UK 0.541** UK 0.589** North America 0.523**

6.478** 0.308** 0.351** 0.187

3.740** 0.713** 0.523**

3.979** 0.583**

3.223**

3. Transition probabilities Bear/High Volatility State Bull/low Volatility State

1.714**

JP

Bear/High Volatility State

Bull/low Volatility State

0.674**

0.326

0.201

0.799

Panel C - MARKOV CHAIN PROPERTIES, TWO-STATE MODEL

Ergodic Probs

Bear 0.381

Bull 0.619

Avg. dur.

Bear 3.07

Bull 4.98

** = significant at 1% size or lower; * = significant at 5% size.


Markov Switching Mean-Variance Frontier Dynamics 37 Bear/high volatility state 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 2008

2007

2006

2005

2004

2003

2002

2001

2000

1999

1998

1997

1996

1995

1994

1993

1992

1991

1990

1989

1988

0.0

Figure 2.1 Smoothed (full-sample) probabilities from two-state Markov switching Model

the plotted probabilities) that long stretches of time – well in excess of 5 months, in fact – such as 1989–1990, 1991–1992, and 2003–2007 are characterized as draws from the bull regime. We perform diagnostic checks based on the one-step predictions errors. A number of alternative Portmanteau statistics and tests all indicate that the one-step errors are approximately martingale difference sequences. In particular, there is only weak evidence of residual ARCH effects in the prediction errors, which seems to be a rather common finding in similar applications. 2.3.4 Time-varying mean-variance frontiers One of the goals of this Section is to illustrate how multivariate Markov switching models may be put at work. We start doing this by computing regime-specific Markov switching mean-variance frontiers (MSMVF, for short). MSMVFs are a simple generalizations of standard textbook, Markowitz-style efficient frontiers to the case in which the (predicted) moments – in particular, means, variances, and covariances – of returns of assets in the choice menu are time-varying and driven by the realization of a Markov chain within a MSIAH process. Besides the econometric estimates of a Markov switching model – in our case to be identified with those presented and commented in Section 3.3 – two basic ingredients inform the construction of a MSMVF: how to go from MSIAH parameter estimates to predictions of means, variances, and covariances of portfolio returns; the investment horizon for which the MSMVF is to be built.8



Before presenting some illustrative results based on the MSIH(2,0) estimates, we describe how forecasts of future moments may be computed in a Markov switching framework. To get some intuition on the factors that determine the predictions of means and variances of asset returns under a MSIH process for returns, consider first the case with a single risky asset (n = 1), yt = µSt + σS2 εt , t where εt ∼ N(0, 1). Call ξˆ t|t the 2 × 1 vector of (filtered) probabilities of being in each of the two alternative regimes, based only upon the information available up to time t, ξˆ t|t ≡ [Pr(St = 1|Ft ) Pr(St = 2|Ft )] . Notice that the (row) vector of time t + 1 predicted probabilities of the two states can be computed as: ˆ ξˆ t+1|t = [ξˆ1,t|t pˆ 11 + ξˆ2,t|t (1 − pˆ 22 )ξˆ1,t|t (1 − pˆ 11 ) + ξˆ2,t|t pˆ 22 ] = ξˆ t|t P,

ˆ The predicted mean for where pˆ ij is the row-i, column-j element of, P. period t + 1 is then Et [yt+1 ] = ξˆ1,t|t pˆ 11 µˆ 1 + ξˆ1,t|t (1 − pˆ 11 )µˆ 2 + (1 − ξˆ1,t|t )(1 − pˆ 22 )µˆ 1 + (1 − ξˆ1,t|t )pˆ 22 µˆ 2 . In general, and extending this result to the case of a T -step ahead fore ˆ ∗ is a matrix that stacks in ˆ ∗ , where µ cast, we have that Et [rt+T ] = ξˆ t Pˆ T µ each row the regime-dependent mean return estimates for each asset. In the case of a generic number n of assets, this expression easily general ˆ ∗ where now µˆ ∗ is 2 × n and Et [yt+T ] yields a izes to: Et [yt+T ] = ξˆ t Pˆ T µ 1×n vector of predicted means. Next, consider the prediction of the variance of one asset return in period t + 1 conditional on the information set Ft : Vart [yt+1 ] = ξˆ1,t|t pˆ 11 E[(µˆ 1 − Et [yt+1 ] + σˆ 1 εt+1 )2 ] + ξˆ1,t|t (1 − pˆ 11 )E[(µˆ 2 − Et [yt+1 ] + σˆ 2 εt+1 )2 ] + (1 − ξˆ1,t|t )(1 − pˆ 22 )E[(µˆ 1 − Et [yt+1 ]σˆ 1 εt+1 )2 ] + (1 − ξˆ1,t|t )pˆ 22 E[(µˆ 2 − Et [yt+1 ] + σˆ 2 εt+1 )2 ]. Once again, this is easily extended to the conditional variance of returns in period t + T : Vart [yt+T ] = ξˆ t|t Pˆ T

(µˆ 1 − Et [yt+T ])2 + σˆ 12 (µˆ 2 − Et [yt+T ])2 + σˆ 22

.



The implication is that unless µˆ 1 = µˆ 2 = Et [yt+T ] (i.e., unless there is no regime switching in the conditional mean function), variance will exceed the level computed as a predicted state-probability weighted combination of the regime-specific variance estimates. These results are easy to generalize to the case of multiple assets, n ≥ 2, which is obviously the relevant case in an asset allocation perspective. Consider a portfolio of assets summarized by the nx1 vector of percentage portfolio weights at time t, ωt . Under a simple MSIH(2,0) switching p p model, the return on the portfolio, rt+1 , is rt+1 = ωt yt+1 = ωt µSt+1 + ωt St εt+1 . The expected portfolio return next period is then simply p ˆ ∗ ωt . The variance of portfolio returns can Et [r ] = ω Et [yt+T ] = ξˆ Pˆ T µ t

t

t+1

be written in the following form p p Et (r −µ )2 = ω (ξˆ1,t+T |t E[(yt+T − Et [yt+T ])(yt+T − Et [yt+T ]) t+T

t+T

t

|St+T = 1] + ξˆ2,t+T |t E[(yt+T − Et [yt+T ])(yt+T − Et [yt+T ]) |St+T = 2])ωt, where the n × n matrix of squared return deviations from the mean in state St+T is given by E[(yt+T − Et [yt+T ])(yt+T − Et [yt+T ]) |St+T ] = St+T + 

2



1 ] 1 ] 1 ] µ1S µ1S − Et [yt+T − Et [yt+T − Et [yt+T ··· µ1S  t+T

t+T

t+T   2 2 ] N N ]   × µ × µ − E [y − E [y t t t+T t+T St+T St+T     



2  2  2 ] 2 2 ] 2 2 ]  µ  − E [y − E [y · · · µ − E [y µ t t+T t t+T t t+T  St+T  St+T St+T

  N N  × µ1 − E [y 1 ] × µS − Et [yt+T ]  t t+T  . St+T t+T       . . . .   . . . . .   . . .   

2 

 N  N ] N ] N − Et [yt+T µN − Et [yt+T µN  µS  St+T − Et [yt+T ]  t+T  St+T 1 ] × µ2 2 ] × µ1S − Et [yt+T − E [y t t+T S t+T

t+T

The second term, which shows the deviations of the state-specific conditional means from its overall expectation in each state and for each asset, does not arise in single-state models. This term could be potentially important for portfolio allocation purposes. Again, the implication is that unless µi1 = µi2 i = 1, 2, . . . , n, the conditional variance ˆ1+ of the T -periods ahead portfolio returns is not simply ωt (ξˆ1,t+T |t Ω ˆ 2 )ωt , involving instead a complex matrix reflecting cross ˆξ2,t+T |t Ω



products of deviations of the conditional means from the unconditional means for the assets. A MSMVF reflects the basic intuition of Markowitz: investors decide optimal portfolios on the basis of the trade-off between portfolio risks and expected returns; risk should be measured by the variance of portfolio returns. Moreover, for any given level of expected return, a rational, risk-averse investor will choose the portfolio with minimum variance from amongst the set of all possible portfolios. In particular, suppose that the goal of our investor is to select a portfolio comprised of the n risky T T T assets, in the form of a vector of portfolio weights ωT t ≡ [ω1t ω2t . . . ωnt ] , n such that i=1 = (ωT t ) ιn = 1. Unless we shall state otherwise, notice T that ωit < 0 is admissible. The investor has a risk-return trade-off goal over T periods; therefore she cares for optimizing the trade-off between the portfolio expected return and variance over the interval [t, t +T ]. The investor’s problem may be formulated as a constrained minimization: p

min Vart [rt:t+T ; ωT t ] ωT t

p

s.t. (i) µ = Et [rt:t+T ; ωT t ];

(ii) (ωT t ) ιN = 1,

(2.5) p p p p p where rt:t+T ≡ rt+1 + rt+2 + · · · + rt+T = T τ =1 rt+τ is the total portfolio return between time t and T ≥ 1, µ is the considered mean portfolio goal, p p T and the notations Et [rt:t+T ; ωT t ] and Vart [rt:t+T ; ωt ] stress that the (predicted) time t expectation and variance depend on the selected portfolio weights ωT t in the ways discussed early on. This is a simple quadratic optimization problem that can be solved with the method of Lagrange ˆT multipliers. The resulting ω t (µ) will be a straightforward function of ˆ as well as ˆ ∗ , P, ˆ ∗, the basic Markov switching parameter matrices, µ the (filtered) state probabilities collected in the vector ξˆt . Since ξˆt ∈ Ft and the perceived state probabilities change over time, the result is that ∀µ the variance-minimizing weights become themselves a function of either the current state St – if known – or at least of the vector of state probabilities ξˆt . Consider now solving the program in (2.5) for all possible, different choices of µ ∈ (−1, +∞). This delivers a set of variance minimizing ˆT ˆT weights {ω t (µ); µ ∈ (−1, +∞)}. In correspondence to each vector ω t (µ) it is then possible to compute the associated portfolio expected return and p ∗ ˆT risk (where in fact Et [rt:t+T ; ω t (µ )] = µ by construction). The set of all ˆT possible mean-variance combinations induced by {ω t (µ); µ ∈ (−1, +∞)} is the mean-variance frontier. Since in the presence of Markov switchˆT ing dynamics all optimal variance-minimizing portfolios in {ω t (µ); µ ∈


Markov Switching Mean-Variance Frontier Dynamics 41 Markov Switching M-V frontier 1-month ahead

0.14 0.12

Expected return

0.10 0.08 0.06 0.04 0.02 0.00 –0.02 –0.04 –0.06 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44 0.48 Standard deviation Single state

Bear state

Bull state

Ergodic probs

Markov Switching M-V frontier 24-month ahead

0.10 0.08

Expected return

0.06 0.04 0.02 0.00 –0.02 –0.04 –0.06 0.00

0.04

0.08

0.12

0.16

0.20

0.24

0.28

0.32

0.36 0.40

0.44

0.48

Standard deviation Single state

Bear state

Bull state

Ergodic probs

Figure 2.2 Single-state vs. Markov switching mean-variance frontiers

(−1, +∞)} will generally depend on the state (or the perception of the state, as captured by ξˆt ) the resulting MSMVF will be state-dependent. Figure 2.2 shows three sets of MSMVFs, each corresponding to a different choice of the horizon T , i.e., 1-, 3-, and 24-month ahead. Within each set, four different mean-variance frontiers are plotted. Three of them are MSMVFs and correspond to three alternative and key configurations



of the (filtered) state vector ξˆt , i.e., when an investor has knowledge of the current regime being bear, being bull, or when an investor ignores the nature of the current regime and simply assigned to each of the two states a probability equal to their long-run, ergodic probabilities. The latter case corresponds to a plausible situation of ignorance on the nature of the current state. A fourth frontier is provided as a benchmark and corresponds to the (time invariant) frontier an investor derives from a Gaussian IID model as in top panel of Table 2.3. Clearly, modeling Markov regimes has massive effects when the investment horizon is short: the bull and bear MSMVFs are substantially different, with the bull MSMVF implying a substantially better risk-return trade-off. For instance, if an investor were to expect a portfolio return of µ = 0.015 (i.e., a rather aggressive 18 percent per year), in the bull regime she would be able to reach this goal bearing a rather moderate risk of 8.7 percent per year; however, the same expected return target in a bear regime would force the investor to accept a much higher risk of 27.3 percent per year. Interestingly, already for T = 1 month, the ergodic and single-state (IID) MSMVFs are very hard to tell apart from the Figure. Importantly, this does not have to happen as Markov switching models have to produce ergodic densities for the variables that are neither Gaussian nor even approximately similar to single-state models (see Guidolin and Timmermann 2006b, for related comments). However, in this illustration, this turns out to be the case. The bottom panel of Figure 2.2 shows that provided an investor has a sufficiently long horizon, the bull, bear, and ergodic MSMVFs all come to coincide. This is to be expected as the longer the horizon, the higher is the chance that the predicted state probabilities used to compute predicted means, variances, and covariances will come to essentially coincide with the model-implied ergodic state probabilities. Since the ergodic MSMVF clearly cannot depend on T , for a sufficiently long horizon it also happens that bull, bear, and ergodic MSMVFs all converge to the single-state MSMVF. A plot for the case T = 3 months (unreported) falls in-between the top and bottom panels, even though T = 3 has been selected to show that Markov switching effects are not short-lived and will potentially affect optimal portfolio choices for horizons that are plausible in practice.

2.3.5 Portfolio implications As a last illustrative step, we have also computed portfolio weights using simple mean-variance preferences. One can interpret such an exercise as



equivalent to computing the MSMVFs in Section 2.3.4 and then select˜T ing an optimal vector of weights ω t after super-imposing on the plots in Figure 2.2 some standard sets of mean-variance indifference curves. Similarly, to Guidolin and Na (2008), consider an investor with the simple objective: p T ] − λ Var [r p T] max Vt:t+T (ωT ) ≡ E [r ; ω ; ω t t:t+T t t t:t+T t t T 2 ωt

s.t. (ωT t ) ιN = 1,

(2.6) where λ > 0 is the coefficient of risk aversion characterizing the investor’s preferences. At time t the investor maximizes the expected utility objective (2.6) by implementing a simple buy-and-hold strategy (no dynamic rebalancing) in which ωT t is selected at time t and held up to time T . In particular, we have proceeded to compute optimal international diversification weights (among developed markets) on the basis of the MSIH(2,0) estimates of Section 2.3.3 and setting λ to (locally) match the behavior of an investor with constant coefficient of relative risk aversion of 5.9 We have performed this exercise recursively between 1998:01 and 2008:08. Table 2.4 reports a number of summary statistics for these recursive sets of (128) portfolio weights for the cases T = 1 and 12 months and for the MSIH(2,0) and the single-state model. Table 2.4 reports means, medians, standard deviations and the 10 percent empirical confidence bands for recursive portfolio weights.10 Table 2.4 shows important – although never dramatic – differences between single- and Markov switching recursive portfolio weights. For instance, focusing on the case in which short sales are admitted, while a single-state model implies that Europe ex-UK for T = 1 should receive an average weight of 27 percent (25 percent using the median), the two-state MSIH model yields an average weight of 62 percent (66 percent using the median); similarly, the weight to UK stocks is 38 percent (equal to the median) vs. 59 percent (55 percent using the median). The fact that European stock weights are inflated by Markov switching dynamics is compensated by lower weights assigned to US stocks. A few other facts emerge from Table 2.4. As one would expect, MSIH weights are much more volatile than single-state weights are, and this difference is particularly strong in the T = 1 case. This should be expected because an investor that uses the Markov switching framework will actively try to time the international markets’ bull and bear regimes and to tailor her optimal risk-return trade-off on the basis of the underlying dynamics.


GREGORIOU-3: “CHAP02” — 2010/11/10 — 21:37 — PAGE 44 — #24 0.199 0.013 0.175 0.029 0.585

0.137 0.000 0.300 0.079 0.485

Pacific ex-Japan Japan Europe ex-UK United Kingdom North America


Mean

0.148 0.000 0.358 0.078 0.384

0.190 0.010 0.169 0.024 0.602

Median

0.053 0.000 0.233 0.076 0.328

0.051 0.012 0.029 0.026 0.082

Std. Dev

Mean

No short sales

5% Upper

Median

0.273 0.029 0.214 0.070 0.686 0.059 0.008 0.165 0.022 0.746

0.057 0.005 0.158 0.018 0.751

0.057 0.000 0.050 0.000 0.042

0.194 0.000 0.650 0.189 0.877

0.055 0.000 0.067 0.000 0.878

0.060 0.000 0.068 0.000 0.873

Two-State Model, Mean-Variance Preferences

0.139 0.000 0.140 0.000 0.449

Single-State Model, Mean-Variance Preferences

5% Lower

10% Confidence bands

T = 1 month

0.029 0.000 0.022 0.000 0.036

0.017 0.009 0.026 0.015 0.037

Std. Dev

0.014 0.000 0.041 0.000 0.832

0.040 0.000 0.135 0.004 0.699

5% Lower

0.093 0.000 0.098 0.000 0.933

0.082 0.021 0.207 0.041 0.789

5% Upper

10% Confidence bands

T = 12 months

Table 2.4 Summary statistics for recursive, optimal mean-variance portfolio weights (Sample Period: 1998:01–2008:08)

GREGORIOU-3: “CHAP02” — 2010/11/10 — 21:37 — PAGE 45 — #25 0.425 −1.506 0.265 0.384 1.431

0.487 −1.618 0.615 0.587 0.928


Pacific ex-Japan Japan Europe ex-UK United Kingdom North America 0.476 −1.387 0.664 0.546 0.730

0.434 −1.408 0.250 0.379 1.370

0.285 1.175 0.463 0.348 0.970

0.093 0.488 0.089 0.059 0.589

0.549 −0.898 0.395 0.470 2.252 0.248 −0.742 0.420 0.355 0.718

0.252 −0.740 0.416 0.358 0.729

0.085 −3.235 0.035 0.083 0.008

0.907 −0.142 1.283 0.976 2.647 0.310 −1.414 0.229 0.427 1.424

0.305 −1.378 0.222 0.418 1.296

Two-State Model, Mean-Variance Preferences

0.314 −2.164 0.128 0.312 0.751

Single-State Model, Mean-Variance Preferences

Short Sales admitted

0.181 0.817 0.157 0.238 0.331

0.039 0.099 0.042 0.048 0.134

0.058 −2.559 0.028 0.088 1.050

0.196 −0.854 0.372 0.291 0.545

0.583 −0.270 0.476 0.778 1.877

0.299 −0.622 0.475 0.413 0.893


2.4 Conclusion This chapter has provided a primer on the main statistical and technical issues characterizing multivariate Markov switching models. An application to international portfolio diversification has illustrated the potential effects of basing a standard but key financial decision making exercise on a model from the Markov switching class. Of course, a number of important aspects of the application of Markov switching methods to strategic asset allocation, international diversification, and risk management have failed to play a role in our chapter, such as the interaction between regime shifts and hedging demands when an investor frequently (continuously) rebalances her portfolio (see Guidolin and Timmermann, 2008a), the role of preferences in which either predictions of conditional moments higher than mean and variance (see Guidolin and Timmermann, 2008b) or the entire predictive density for all portfolios/assets enter the portfolio problem, such as in the power utility (constant relative risk aversion) case (see Guidolin and Timmermann, 2008a), the role of transaction costs in reducing the out-of-sample, realized payoffs from portfolio strategies that exploit the presence of regime shifts (see Guidolin and Na, 2008).

Notes 1. For instance, a reader is invited to consult Fruhwirth-Schnatter (2006), Kim and Nelson (1999), or Krolzig (1997). 2. The assumption of a first-order Markov process is not restrictive, since a higher order Markov chain can always be re-parameterized as a higher dimensional first-order Markov chain, i.e., substitutability exists between the order of the Markov chain driving St and the number of regimes k. 3. This is the sense in which Marron and Wand (1992) conclude that mixtures of normal distributions provide a flexible family that can be used to approximate many distributions. Mixtures of normals can also be viewed as a non-parametric approach to modeling the return distribution if the number of states, k, is allowed to grow with the sample size. 4. When applied to models such that k > 2, RCM1 has one disadvantage: a model can imply an enormous degree of uncertainty on the current regime, but still ˜ ∼ have kj=1 Pr(St = j|y1 , y2 , . . . , yT ; γ) = 0 for most values of t. As a result, it is rather common to witness that as k exceeds 2, almost all switching models (good and bad) will imply values of RCM1 that decline towards 0. Guidolin (2009) proposes a few alternative measures that correct this problem. 5. In the non-linear literature, all models with saturation ratios below 20 are normally regarded with suspicion. Table 2 has boldfaced all models with a saturation ratio of at least 20.


Markov Switching Mean-Variance Frontier Dynamics 47 6. The MSIH(2,0) model also yields a good RCM1 of 14.4 and a Guidolin’s (2009) RCM2 of 16.9 which are relatively low. For instance, a MSIH(3,0) model returns a RCM2 of 93.2, which is largely disappointing. 7. Japan is an exception to these cross-regime volatility patterns. 8. Notice that in standard unconditional (single-state) mean-variance analysis, the investment horizon makes no difference because the forecast of future means, variances, and covariances are the currently estimable means, variances, and covariances. 9. See Guidolin and Timmermann (2008b) for details. One can show that Et [u(Wt+T )] ∼ = κ0 (γ ) + κ1 (γ )Et [Wt+T ] + κ2 (γ )Vart [Wt+T ] ∝ Et [Wt+T ] − λVart [Wt+T ] where κ2 (η) ≡ − 14 ην −(1+η) [2 + 2(η + 1) + (η + 1)(η + 2)] < 0, i.e., λ is interpretable as a non-linear function of η. 10. The 10 percent lower bound is the fifth percentile and the 10 percent upper bound is the ninety-fifth percentile of the empirical distribution of portfolio weights. In the table, we have boldfaced 10 percent confidence bands that fail to include zero, i.e., indices for which an exercise gives a clear indication as to the sign of the average commitment to the portfolio.

References Ang, A. and Bekaert, G. (2002a) “International Asset Allocation with Regime Shifts,” Review of Financial Studies, 15(4): 1137–1187. Ang, A. and Bekaert, G. (2004) “How Regimes Affect Asset Allocation,” Financial Analysts Journal, 60(2): 86–99. Chow, G., E. Jacquier, M. Kitzman and K. Lowry. (1999) “Optimal Portfolios in Good Times and Bad,” Financial Analysts Journal, 55(3): 65–73. Clarke, R. and de Silva, H. (1998) “State-Dependent Asset Allocation,” Journal of Portfolio Management, 24(2): 57–64. Davies, R. (1977) “Hypothesis Testing When a Nuisance Parameter is Present Only Under the Alternative,” Biometrika, 64(2): 247–254. Dempster, A. P., M. Schatzoff, and N. Wermuth. (1977) “A Simulation Study of Alternatives to Ordinary Least Squares,” Journal of the American Statistical Association, 72(357): 77–91. Fruhwirth-Schnatter, S. (2006) Finite Mixture and Markov Switching Models, NY: Springer. Guidolin, M. (2009) “Detecting and Exploiting Regime Switching ARCH Dynamics in US Stock and Bond Returns,” in G. Gregoriou (Ed.), Stock Market Volatility, Chapman Hall: London. Guidolin, M. and Na, C. F. (2008) “The Economic and Statistical Value of Forecast Combinations Under Regime Switching: An Application to Predictable US Returns,” in M. Wohar and D. Rapach (Eds.), Frontiers of Economics and Globalization, Volume 3, Forecasting in the Presence of Structural Breaks and Model Uncertainty, Emerald Publishing Ltd: NY. Guidolin, M. and Nicodano, G. (2009) “Small Caps in International Equity Portfolios: The Effects of Variance Risk,” Annals of Finance, 5(1): 15–48.


48 Massimo Guidolin and Federica Ria Guidolin, M. and Ono, S. (2006) “Are the Dynamic Linkages Between the Macroeconomy and Asset Prices Time-Varying?” Journal of Economics and Business, 58(5–6): 480–518. Guidolin, M. and Timmermann, A. (2006a) “An Econometric Model of Nonlinear Dynamics in the Joint Distribution of Stock and Bond Returns,” Journal of Applied Econometrics, 21(1): 1–22. Guidolin, M. and Timmermann, A. (2006b) “Term Structure of Risk under Alternative Econometric Specifications,” Journal of Econometrics, 131(1–2): 285–308. Guidolin, M. and Timmermann, A. (2008a) “Size and Value Anomalies under Regime Shifts,” Journal of Financial Econometrics, 6(1): 1–48. Guidolin, M. and Timmermann, A. (2008b) “International Asset Allocation under Regime Switching, Skew and Kurtosis Preferences,” Review of Financial Studies, 21(2): 889–935. Hamilton, J. (1988) “Rational Expectations Econometric Analysis of Changes in Regime,” Journal of Economic Dynamics and Control, 12(2–3): 385–413. Hamilton, J. (1989) “A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle,” Econometrica, 57(2): 357–384. Hamilton, J. (1993) “Estimation, Inference, and Forecasting of Time Series Subject to Changes in Regime,” in G. Maddala, C. Rao and H. Vinod, Handbook of Statistics, vol. 11, Amsterdam: North Holland. Hamilton, J. and Susmel, R. (1994) “Autoregressive Conditional Heteroskedasticity and Changes in Regime,” Journal of Econometrics, 64(1–2): 307–333. Hansen, B. (1992) “The Likelihood Ratio Test Under Non-Standard Conditions: Testing the Markov Switching Model of GNP,” Journal of Applied Econometrics, 7(S): S61–S82. Kim, C. J. (1994) “Dynamic Linear Models with Markov-Switching,” Journal of Econometrics, 60(1–2): 1–22. Kim, C. J. and Nelson, C. R. (1999) State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications, Cambridge, MA: MIT Press. Krolzig, H. M. (1997) Markov-Switching Vector Autoregressions: Modeling, Statistical Inference, and Application to Business Cycle Analysis, Berlin: Springer-Verlag. Marron, J. S. and Wand, M. P. (1992) “Exact Mean Integrated Squared Error,” Annals of Statistics, 20(2): 712–736. Ramchand, L. and Susmel, R. (1998) “Volatility and Cross Correlation across Major Stock Markets,” Journal of Empirical Finance, 5(4): 397–416. Sin, C. Y. and White, H. (1996) “Information Criteria for Selecting Possibly Misspecified Parametric Models,” Journal of Econometrics, 71(1–2): 207–225. Turner, C., Startz R. and C. Nelson. (1989) “A Markov Model of Heteroskedasticity, Risk, and Learning in the Stock Market,” Journal of Financial Economics, 25(1): 3–22.


3 A Markov Regime-Switching Model of Stock Return Volatility: Evidence from Chinese Markets Thomas C. Chiang, Zhuo Qiao and Wing-Keung Wong

3.1 Introduction As a mechanism for the development of the Chinese stock markets, issues of Chinese stocks are mainly divided into A-shares (SHA and SZA) and B-shares (SHB and SZB); both A-shares and B-shares are listed on the Shanghai Stock Exchange (SHSE) and the Shenzhen Stock Exchange (SZSE) of mainland China.1 The Chinese government also allows some companies to issue H, red-chip, N, and S shares in accordance with different listing locations and investors. Among these types of shares, H, red-chip, N, and S shares are traded on the Hong Kong Stock Exchange (HKSE), the New York Stock Exchange (NYSE), and the Singapore Stock Exchange (SSE). There are reasons for issuing different types of stocks in Chinese markets. First, the traditional economic units were believed to lack the capacity to compete with modern corporate power. To insulate these units from the impact of external shocks, the A-share markets were artificially separated from B-share markets.2 Second, issuances of a variety of stocks are designed to cater to the needs of different financial environments that will help Chinese businesses to raise capital in order to facilitate their functioning. Thus, the existence of dual economic characteristics, accompanied by the restriction of foreign currency conversion, government regulations, and information asymmetry between domestic and foreign investors, presents the possibility of arbitrage and creates some market segmentation among different Chinese stock exchanges (Sun and Tong, 2000).

49


50 Thomas C. Chiang, Zhuo Qiao and Wing-Keung Wong

Researchers in international finance (Frankel and Schmukler, 2000; Yang, 2003) recognize that the issue of market segmentation is closely tied to information asymmetry. Given the fact that rational B-share investors have relatively less knowledge about Chinese corporate structure and market fundamentals, they are unwilling to pay the same prices as the well-informed domestic investors do. Asymmetric information thus implies a discount on B-shares. Bailey (1994) analyzes eight Chineseshare stocks from March 1992 to March 1993 and finds a substantial discount in B-share prices relative to A-share prices. Su (1998) investigates 47 stocks from 1993 through 1996 and finds that the average daily discount of B-shares relative to A-shares is about 62.2 percent. By examining a sample consisting of 68 firms issuing both A-share and B-share stocks, Chen et al. (2001) find that the average B-share discount on the SHSE was about 66.2 percent from 1992 to 1997 and that on the SZSE was about 52.4 percent.3 A separate line of research has been advanced by examining the linkage between Chinese stock markets and international stock markets or the linkages among four segmented markets. Chakravarty et al. (1998) report the bivariate return correlations among the A- and B-share indices, as well as the Hong Kong, Japanese, and US market indices. Their results suggest that the Chinese market is still isolated, even after the introduction of B-shares. Moreover, the prices of B-shares are sensitive only to A-share prices and have little relationship to the foreign markets. In addition, there is some evidence of two-way information flows between A-share and B-share markets. However, evidence shows that A-share returns are more likely to lead B-share returns, on average, rather than the reverse causation. Recently, a number of studies have focused on the linkages between Chinese stock markets and foreign markets. In particular, Lin and Wu (2003) find that Chinese stock markets have a weak linkage with Hong Kong, Taiwan, and US markets. However, Lean and Wong (2004) report that after the Asian financial crisis, the Chinese stock markets have been more highly cointegrated with the major world markets, including the US, the UK, and Japan, and three regional counterparts, Taiwan, Hong Kong, and Singapore. By extending the research work from return linkages to volatility spillovers, Brooks and Ragunathan (2003) examine the Chinese stock volatility and find no evidence of volatility spillover from B-share prices to A-share prices or vice versa. Using the daily open and close price data from 1994 to 2001, Wang and Firth (2004) further examine stock returns and the transmission of volatilities across Greater China’s four markets,


Markov Regime-Switching Model of Stock Return Volatility 51

i.e., SHSZ, SZSE, HKSE, and the TSE (Taipei Stock Exchange), and three developed international markets, Tokyo, London, and New York. They find no evidence that Mainland China’s two stock markets are affected by contemporaneous or delayed “bad news.” Recently, Qiao et al. (2008) study the volatility spillover effect of the Chinese markets and find that there are bi-directional volatility spillovers between the two A-share markets and the two B-share markets and between the two B-share markets and the Hong Kong market. However, only unidirectional volatility spillover effects from the two A-share markets to the Hong Kong market are present. Among the alternative markets, they find that the two A-share markets are most influential in both mean and volatility spillover effects. Notice that the evidence on the stock return relationship between Aand B-share markets or their linkages with foreign markets is useful, since this information can be used to justify market efficiency or to construct an optimal, internationally diversifiable portfolio. The evidence of volatility spillover is also meaningful, since it provides information about checking for risk shifting. Despite the investment/financial significance of the stability of the stock return correlations and cross-market volatility covariances, very few attempts have been made to investigate volatility changes across regimes and markets in Chinese stock indices. This stability issue is particularly relevant to China, since the stock markets over recent years have experienced a sequence of policy innovations, financial market reform, “Asian disease,” “Russian crisis,” and the global financial crises triggered by a dramatic rise in mortgage delinquencies and foreclosure in the United States. All these shocks are likely to have a significant impact on return correlations and volatility covariances as is evident from Karolyi and Stulz’s study (1996). To provide more insight into the volatility characteristics and evaluate how external shocks are affecting Chinese stocks, it is crucial to distinguish between the high-volatility state and the low-volatility state, since market behavior is expected to be different in different states. This motivates us to adopt the regime-switching GARCH (RS-GARCH) model, which allows stochastic regime shifts in both the conditional mean and conditional volatility, to analyze the volatility behavior in Chinese stock markets. More important, this model has the capacity to deal with abrupt changes; the by-product, estimates of the “smoothed probability,” offers us a very powerful tool for studying the volatility of switching behaviors in each of the segmented stock markets.4 The remainder of this chapter is organized as follows. Section 3.2 discusses the RS-GARCH model specification and estimation procedure.



Section 3.3 presents the data used and their corresponding descriptive statistics. Section 3.4 provides empirical results of the RS-GARCH model and a discussion. Section 3.5 investigates whether volatility linkages among the four segmented markets are regime-dependent. Section 3.6 contains concluding remarks.

3.2 Regime-switching GARCH model 3.2.1 Structure of the model The observed financial time series often undergo alternating periods of calm and turbulence with clusters of volatility. To model this phenomenon, a GARCH type model has been widely used (Engle, 1982; Bollerslev, 1986; Nelson, 1991; Bollerslev et al. 1992). One of the most problematic features of these models is the excessive persistence imputed to the volatility of financial variables (Chou, 1998). This suggests that shocks to the conditional variance that occurred in the distant past continue to have non-trivial effects on the current value. This lasting persistence indeed is the reason attributed to the relatively poor forecasts for GARCH models. A number of researchers have observed that the high persistence of GARCH models and hence the poor forecasting performance might be related to structural changes in the data generating process. Specifically, Diebold (1986) and Lamoureux and Lastrapes (1990) argue that the usually high persistence in GARCH models is caused by the structural changes in the volatility process during the estimation period. Schwert and Seguin (1990), Nelson (1991), Engle and Mustafa (1992) further point out that GARCH models may be limited in their ability to model the volatility of financial time series if structural breaks, caused by certain dramatic events such as the stock market crash in 1987 and Asian crises in 1997, are present. This means that to obtain more robust estimates of conditional volatility would require a more general class of GARCH models, allowing for regime shifts as part of the data-generating process.5 The regime switching GARCH models introduced by Hamilton (1990), Hamilton and Susmel (1994), Cai (1994), and Gray (1996) help us to address this issue. These models allow the conditional volatility process to switch between a discrete number of states, with the transitions between states governed by a hidden and finite order Markov chain. In addition, the transition probability of the Markov process determines the probability of volatility switching, thus indicating the expected duration of each regime. To examine the behavior of the conditional volatility of



return of the four segmented Chinese stock markets by incorporating these dynamic features, in this chapter, we apply a Regime-Switching GARCH model (RS-GARCH) by first employing a regime-switching model proposed by Gray (1996) as a mixture-of-distributions representation such that:  N[µ1t , h1t ] with probability p1t Rt |t−1 (3.1) N[µ , h ] with probability (1 − p ) 2t 2t 1t where Rt = 100(ln P t − ln Pt−1 ) is the stock return for each series and Pt is the price index on the time t, µit and hit are the conditional mean and conditional variance at time t, respectively. Both µit and hit are allowed to switch between two regimes; for instance, regime probability p1t = Pr(st = 1|t−1 ) is the probability for Regime 1 conditional on a past information set available up to time t − 1. To construct the model, we further specify conditional mean by using an AR (1) process due to partial-price adjustment, limit-price policy, the existence of feedback trading, or other forms of market frictions (Kim and Rogers, 1995; Koutmos, 1998; Antoniou et al. 2005). The conditional estimate volatility is thereafter assumed to evolve by a GARCH (1, 1) process as popularized by Bollerslev et al. (1992): Ri,t = φi0 + φi1 Ri,t−1 + εit where εit t−1 ∼ N(0, hit ) with State i = 1, 2 2 +b h hit = ai0 + ai1 εt−1 i1 t−1

(3.2)

(3.3)

where ai0 > 0, ai1 ≥ 0, and bi1 ≥ 0 to ensure that the conditional variance is positive. All variance and mean parameters are regime-dependent. Finally, regime switching is assumed to be governed by a first-order Markov process with the following transition probability matrix: Pr(st = 1 st−1 = 1) = P Pr(st = 2 st−1 = 1) = 1 − P Pr(st = 2 st−1 = 2) = Q

(3.4)

Pr(st = 1 st−1 = 2) = 1 − Q where P (Q) is the transition probability for State st = 1 (2) conditional on State 1 (2). A similar definition applies to Regime (1 − P) and (1 − Q).6



As stated earlier, a special feature of this model is that in addition to capturing the stochastic volatility, it allows us to estimate the probability of regime shifting and shed some light on the duration of volatility. 3.2.2 Estimation Conditional on Regime i, the normal density function of the stock return, Rt , defined in (3.1) is given by: 1 1 (Rt − µit )2 . (3.5) exp − fit = f (Rt |st = i, t−1 ) = √ 2 2π hit h2 it

Following Gray (1996), regime probability pit can be expressed as a nonlinear recursive function of the transition probabilities and the conditional distribution of the return innovation such that: f1t−1 p1t−1 p1t = P f1t−1 p1t−1 + f2t−1 (1 − p1t−1 ) f2t−1 (1 − p1t−1 ) + (1 − Q) . (3.6) f1t−1 p1t−1 + f2t−1 (1 − p1t−1 ) Thus, the log-likelihood function of this model can be written as: L=

T

log[p1t f1t + (1 − p1t )f2t ]

(3.7)

t=1

where T is the number of observations. The above log-likelihood function can then be constructed recursively using the following expressions for εt and ht : εt = Rt − E[Rt |t−1 ] = Rt − [p1t µ1t + (1 − p1t )µ2t ]

(3.8)

ht = E[Rt |t−1 ]2 − {E[Rt |t−1 ]}2 = p1t (µ21t + h1t ) + (1 − p1t )(µ22t + h2t ) − [p1t µ1t + (1 − p1t )µ2t ]2 . (3.9) Equation (3.9) implies: ht−1 = p1t−1 (µ21t−1 + h1t−1 ) + (1 − p1t−1 )(µ22t−1 + h2t−1 ) − [p1t−1 µ1t−1 + (1 − p1t−1 )µ2t−1 ]2 .

(3.10)

Apparently, Equation (3.10) is not path dependent, as it does not depend on the entire past history of conditional variance. Therefore, it can be used recursively to construct hit via Equation (3.3). In addition, the estimation of the model gives the “smoothed probability”



prob(st RT , RT −1 , · · · ) , which provides information about the likelihood that the asset is in a particular volatility state at time t based on the full sample of observations. The model is estimated by using the BFGS (Broyden, Fletcher, Goldfarb, and Shanno) numerical algorithm as described in Gill and Murray (1972) and the estimation is implemented with GAUSS computation software and its constrained maximum likelihood (CML) module. Initial values for the optimization are based on estimates from a standard GARCH (1, 1) model. To obtain the negative minimum likelihood function values as well as to reduce the possibility of hitting a local minimum, we conduct an experiment by employing several starting values. Here, we report the estimates with the highest likelihood. Standard errors of parameters are computed using a White (1980) heteroskedastic consistent covariance estimator. As indicated by Gray (1996), Fong and Kim (2001, 2002), Fong and Koh (2002), and Fong (2003), some parameters might fall to the boundary during the estimation process.

3.3 The data and preliminary analysis 3.3.1 Sample data and study period The weekly price indices in this study are Shanghai A-shares (SHA), Shenzhen A-shares (SZA), Shanghai B-shares (SHB), and Shenzhen B-shares (SZB) taken from DataStream International and our sample covers January 1995 through June 2005. Weekly indices are used to avoid representation bias from some thinly traded stocks. These four stock indices are plotted in the Figure 3.1.

300 250 200 150 100 50 0 95

96

97

98

99 SHA SZA

00

01

02

03

04

05

SHB SZB

Figure 3.1 Stock indices of Chinese segmented stock market


56 Thomas C. Chiang, Zhuo Qiao and Wing-Keung Wong Table 3.1 Descriptive statistics for stock returns of the stock markets of China

Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera LB(5) LB(10) LB2 (5) LB2 (10)

SHA

SZA

0.101 0.071 30.485 −26.874 3.922 0.497 14.639 3115.460*** 12.779** 25.156*** 18.491*** 20.767**

0.121 −0.001 27.372 −30.877 4.328 −0.141 12.005 1853.540*** 10.532 22.282** 28.672*** 47.409***

SHB 0.011 −0.229 19.501 −17.819 4.902 0.494 5.391 152.783*** 29.778*** 37.429** 88.333*** 95.510***

SZB 0.170 −0.033 32.306 −32.590 5.689 0.919 10.575 1387.251*** 40.714*** 48.882*** 189.28*** 190.21***

This table contains a statistical summary for the stock returns for Shanghai A shares (SZA), Shenzhen A shares (SZA), Shanghai B shares (SHB), and Shenzhen B shares (SZB). *** and ** indicate significance at the 1 percent and 5 percent level, respectively. The Jarque-Bera statistic has a χ 2 distribution with two degrees of freedom under the null hypothesis of normally distributed errors. The critical value of χ 2 (2) at the 1 percent level is 9.210. LB (5) and LB (10) are the Ljung-Box statistics based on the levels of the time series up to the 5th and 10th order. LB2 (5) and LB2 (10) are the Ljung-Box statistics based on the squared levels. Both statistics on the levels and squared levels are asymptotically distributed as χ 2 (5) and χ 2 (10) respectively. The critical values of χ 2 (5) at the 1 percent and 5 percent levels are 15.086 and 11.071, respectively. The critical values of χ 2 (10) at the 1 percent and 5 percent levels are 23.209 and 18.307, respectively.

Table 3.1 contains information on the mean, standard deviation, skewness coefficient, kurtosis coefficient, the Jarque-Bera normality test (JB), and Ljung-Box test. As may be seen from the skewness coefficients, all the returns except SZA are skewed to the right. The kurtosis coefficients are in excess of 3.0, implying that the distributions of the series have fat tails. The JB statistic, following a Chi-squared distribution with two degrees of freedom, suggests that all of the stock returns fail to be normally distributed, which is quite typical for financial time series. To examine the serial correlations for the levels and level squares of stock returns, we perform the portmanteau tests denoted by LB (q) and LB2 (q), where q represents the number of lags included in the computation of the LB statistics. The evidence from the Ljung-Box Q statistics at lag 5 and lag 10 signifies strong serial correlations in the levels and in the squared levels, respectively. The significance of the autocorrelation of squared returns is consistent with the presence of time-varying volatility such as (G)ARCH effects. As pointed out by Lamoureux and Lastrapes (1990) and



confirmed by Hamilton and Susmel (1994), regime shifts in the volatility process can also induce a spuriously high degree of volatility clustering. With this evidence in hand, it is natural to inquire whether volatilities of stock returns in these segmented markets are time varying. Are the volatility patterns among the segmented stock markets different? What are the durations of volatility staying in the high versus the low regime for these markets? The following sections address these issues.

3.4 Empirical results of RS-GARCH model 3.4.1 Performance of RS-GARCH model VS. GARCH model For illustrative purposes and to set a basis for comparison, it is convenient to start with an estimation of a standard GARCH (1, 1) model. The results for the four Chinese stock index returns are reported in

Table 3.2 Estimates of the AR (1)-GARCH model Panel A

SHA

SZA

SHB

SZB

φ0 φ1 a0 a1 b1

−0.041(0.125) 0.047(0.048) 0.650(0.164)*** 0.141(0.024)*** 0.826(0.024)***

−0.135(0.135) 0.061(0.048) 0.426 (0.113) *** 0.116(0.017) *** 0.868(0.015) ***

−0.153(0.192) 0.099(0.049)** 1.535(0.367)*** 0.127(0.024)*** 0.812(0.028)***

−0.037(0.215) 0.134(0.051)*** 5.511(1.046)*** 0.204(0.037)*** 0.601(0.063)***

Panel B

Summary statistics and diagnostics

Log-likelihood LB(5) LB(10) LB2 (5) LB2 (10)

−1471.818 8.429 14.865 3.389 6.388

−1604.128 11.486** 17.270 8.652 11.582

−1646.329 8.036 12.358 4.493 5.601

−1511.416 5.423 10.243 3.270 5.289

The table shows estimates of the standard GARCH(1,1) model. Rt is the stock return, ht is the conditional variance, ϕ0 , ϕ1 ,a0 , a1 , and b1 are constant parameters, and εt is an error term. Rt = ϕ0 + ϕ1 Rt−1 + εt

where εt |t−1 ∼ N(0, ht )

2 +b h ht = a0 + a1 εt−1 1 t−1 , where a0 > 0, a1 ≥ 0 and b1 ≥ 0 to ensure that the conditional variance is positive.*** and ** indicate significance at the 1 percent and 5 percent level, respectively. Numbers in parentheses are standard errors. LB (5) and LB (10) are the Ljung-Box statistics based on the standardized residuals up to the 5th and 10th orders. LB2 (5) and LB2 (10) are the Ljung-Box statistics based on the squared standardized residuals. Both statistics on the level and squared level are asymptotically distributed as χ 2 (5) and χ 2 (10), respectively. The critical values of χ 2 (5) at the 1 percent and 5 percent levels are 15.086 and 11.071, respectively. The critical value of χ 2 (10) at the 1 percent and 5 percent level is 23.209 and 18.307, respectively.


58 Thomas C. Chiang, Zhuo Qiao and Wing-Keung Wong Table 3.3 Estimates of the Markov switching AR (1)-GARCH model Panel A

SHA

SZA

ϕ10 ϕ20 ϕ11 ϕ21 a10 a20 a11 a21 b11 b21 P Q d1 d2

−0.156(0.141) 1.277(0.930) 0.056(0.055) −0.093(0.184) 6.977(0.651)*** 26.073(11.815)** 0.031(0.064) 0.024(0.064) 0@ 0.749(0.344)** 0.973(0.012)*** 0.890(0.043)*** 37.037 9.091

−0.263 (0.148)* 1.732(1.005)* 0.038(0.058) −0.022(0.117) 7.203(0.664)*** 40.289(14.581)*** 0.101(0.077) 0.001(0.063) 0@ 0.593(0.296)** 0.976(0.011)*** 0.896(0.043)*** 41.667 9.615

Panel B

Summary statistics and diagnostics

Log-likelihood LB(5) LB(10) LB2 (5) LB2 (10)

−1388.832 8.540 13.402 3.558 4.698

−1425.681 6.862 10.452 2.999 3.902

SHB

SZB

−0.539(0.198)*** 0.551(0.477) −0.124(0.071)* 0.308(0.113)*** 5.467(1.292)*** 29.556(9.893)*** 0@ 0.046((0.113) 0.046(0.158) 0.564(0.281)** 0.822(0.007)*** 0.736(0.092)*** 5.618 3.788

0.463(0.235)** −0.152(0.648) −0.025(0.068) 0.219(0.156) 9.458(1.553)*** 27.660(6.453)*** 0.019(0.041) 0.526(0.341) 0@ 0.365(0.172)** 0.855(0.068)*** 0.740(0.091)*** 6.897 3.846

−1516.166 10.982 14.124 9.022 13.269

−1574.275 8.949 13.783 8.627 15.482

The table reports the estimates of a regime switching GARCH(1,1) model given below. Rt is the stock return,ht is the conditional variance, ϕi0 , ϕi1 , ai0 , ai1 , and bi1 are constant parameters, and εit is an error term. Ri,t = φi0 + φi1 Ri,t−1 + εit , where εit t−1 ∼ N(0, hit ) with i = 1, 2 states. 2 +b h hit = ai0 + ai1 εt−1 i1 t−1 , where ai0 > 0, ai1 ≥ 0 and bi1 ≥ 0 to ensure that the conditional variance is positive. ***, ** and * indicate significance at the 1 percent, 5 percent and 10 percent level, respectively. Numbers in parentheses are standard errors. @ indicates parameter fell to the boundary. P (Q ) is the transition probability for State st = 1 (2) conditional on State 1 (2). d1 and d2 are the duration of regimes 1 and 2, which equals 1/(1-P) and 1/(1-Q), respectively. LB (5) and LB (10) are the Ljung-Box statistics based on the standardized residuals up to the 5th and 10th order. LB2 (5) and LB2 (10) are the Ljung-Box statistics based on the squared standardized residuals. Both statistics on the levels and squared levels are asymptotically distributed as χ 2 (5) and χ 2 (10), respectively. The critical values of χ 2 (5) at the 1 percent and 5 percent level are 15.086 and 11.071, respectively. The critical values of χ 2 (10) at the 1 percent and 5 percent level are 23.209 and 18.307, respectively.

Table 3.2. The estimates of the AR (1) are positive and significant for Bshare markets. No comparable results are shown in the A-share markets. Consistent with most financial markets, the estimated coefficients of GARCH effects are highly significant for all of the markets. The volatility persistence measured by (a1 + b1 ) is very high. For instance, this measure for SHA equals 0.967 (0.141 + 0.826), which implies a mean duration of


Markov Regime-Switching Model of Stock Return Volatility 59 Table 3.4 The summary statistics for GARCH and RS-GARCH models No. of Parameters

Log-likelihood

Panel A: SHA GARCH (1,1) RS-GARCH (1,1)

5 12

−1471.818 −1388.832

/ −1487.584 −1476.818 165.972(0.000)*** −1426.670 −1400.832

Panel B: SZA GARCH (1,1) RS-GARCH (1,1)

5 12

−1511.416 −1425.681

/ 171.47(0.000)***

Panel C: SHB GARCH (1,1) RS-GARCH (1,1)

5 12

−1604.128 −1516.166

/ −1619.894 −1609.128 175.924(0.000)*** −1554.004 −1528.166

Panel D: SZB GARCH (1,1) RS-GARCH (1,1)

5 12

−1646.329 −1574.275

/ −1662.095 −1651.329 144.108(0.000)*** −1612.113 −1586.275

Model

LR Statistics

SBC

AIC

−1527.182 −1511.416 −1463.519 −1437.681

LR statistics refers to the likelihood ratio test statistic. The null hypothesis is the GARCH (1, 1) model. Numbers in parentheses next to the LR statistic are p-value. The likelihood ratio LR statistics is computed as follows: 2*(likelihood of H1 − likelihood H2 ), where H1 is the RS-GARCH (1, 1) model and H0 is the GARCH (1, 1) model. *** and ** denote statistical significance at the 1 percent and 5 percent level, respectively. SBC is the Schwarz Bayesian criterion for testing model adequacy and it is calculated as likelihood function value-(k/2) ln (T ). T is the number of samples. AIC is Akaike’s information criterion and AIC is calculated as the likelihood function value-k; k is the model parameter number.

1/(1 − 0.967) = 30.3 weeks for volatility shocks, indicating that shocks to the conditional variance are highly persistent. To highlight the feature of stochastic volatility embodied in Chinese segmented markets, we estimate the RS-GARCH model and compare its performance with a standard GARCH model. Table 3.3 reports the estimates of regime switching GARCH models for four stock markets. Apparently, volatilities in the two states are sharply diverse, indicating the existence of two distinct volatility regimes. The unconditional volatility parameters (a10 and a20 ) are statistically significant at the 5 percent and 1 percent levels, respectively. The evidence shows that the RS-GARCH models fit the data much better than standard GARCH models. The log-likelihood values for all RS-GARCH (1, 1) models are larger compared to their counterparts of the single regime GARCH (1, 1) models. For example, the log-likelihood value of the RS-GARCH (1, 1) model for SHA in Table 3.3 is −1388.832, which is much larger than the value −1471.818 to be obtained from a single regime GARCH (1, 1) model in Table 3.2.



To test the difference in performance, we first calculate the standard likelihood ratio (LR) statistic for each pair of models. The likelihood ˆ Log L(θ)], ˜ where (θ) ˆ is ratio LR statistic is given by: λLR = 2∗ [(Log L(θ)− ˜ is the the unconstrained estimator, the RS-GARCH (1, 1) model, and (θ) constrained model, GARCH (1, 1) model. Under the null hypothesis, the test statistic has a Chi-squared distribution with J degrees of freedom, where J is the number of restrictions. As may be seen in Column 4 in Table 3.4, the null hypothesis is decisively rejected and the results favor the regime switching GARCH (1, 1) model for all markets.7 To formally evaluate these two models, we further compute the statistics proposed by Schwartz (1978) and Akaike (1976). As shown in Table 3.4, the SBC and AIC values of RS-GARCH models are significantly higher than those of the standard GARCH models, supporting the data-fitting ability of the regime switching GARCH (1, 1) model over the standard GARCH(1,1) model in all markets. As part of the evaluation process, it is useful to compare the forecast abilities of RS-GARCH models and standard GARCH models. To this end, we evaluate the models based on their forecasting accuracy. The forecast errors of one-week-ahead forecasts are measured by MSE, MAE, LES, and

Table 3.5 One-week-ahead forecast errors of GARCH and RS-GARCH models MSE

MAE

LES

|LE|

Panel A: SHA GARCH (1,1) RS-GARCH (1,1)

3235.183 3130.0466

18.946 17.5914

9.230 9.3076

2.095 2.0782

Panel B: SZA GARCH (1,1) RS-GARCH (1,1)

3775.082 3608.849

22.096 20.627

9.961 8.843

2.193 2.134

Panel C: SHB GARCH (1,1) RS-GARCH (1,1)

2187.573 1983.374

25.893 25.078

9.186 8.840

2.204 2.189

Panel D: SZB GARCH (1,1) RS-GARCH (1,1)

6927.442 6872.797

36.115 36.168

11.539 10.540

2.508 2.371

Model

The four loss functions are defined as follows: MSE = T −1

T

(et2 − σt2 )2 ,

MAE = T −1

t=1

LES = T −1

T t=1

{ln(et2 ) − ln(σt2 )}2 ,

T

|et2 − σt2 |,

t=1

|LE| = T −1

T

| ln(et2 ) − ln(σt2 )|.

t=1



|LE|. The evidence in Table 3.5 consistently shows that the RS-GARCH models outperform the standard GARCH models in forecasting.

3.4.2 Additional empirical evidence from the regime switching GARCH model Having demonstrated the relative performance, the next question is: what other empirical regularities can be derived from the estimates of the RS-GARCH models? We shall summarize our findings as follows: First, the estimated volatilities in four markets are distinguished from each other in two different states. It can be shown that regime one is a low-variance regime, while regime two is a turbulent state, as evidenced by the ratios (3.73, 5.59, 5.41 and 2.92, respectively) of high variance to low variance (a20 /a10 ) for SHA, SZA, SHB, and SZB. Second, the transition probabilities, P and Q, are highly significant, suggesting the volatility regime is persistent. These probabilities measure the magnitude of persistence of each volatility state; a higher value suggests a longer length of stay. For instance, the duration of a lowvolatility state d1 for SHA is about 37 [1/ (1 − 0.973)] weeks. As indicated in Table 3.3, the values of P (the probability of staying at low-volatility) and Q (the probability of staying at high-volatility) are close to one for all of the markets.8 In contrast, (1-P) (the probability of shifting from a low-volatility state to a high-volatility state) and (1-Q) (the probability of shifting from a high-volatility state to a low-volatility state) is small. This indicates that volatility clustering, i.e., low (high) volatility usually followed by low (high) volatility, exists in all segmented markets. Nevertheless, the evidence of P being larger than Q for all markets signifies that the low-volatility regime dominates the market. Third, after allowing regime switching in the data generating process, the evidence shows that all coefficients in the ARCH are insignificant. This outcome is in line with the finding by Edwards and Susmel (2003). As they pointed out, the use of the SWARCH model tends to cause the ARCH effect to be suppressed or to disappear. Additionally, the degree of volatility persistence is reduced. This is shown in the measures of volatility persistence for SHA, SZA, SHB and SZB to be 0.749, 0.589, 0.564, and 0.365, respectively. These values are much smaller than those counterparts derived from the single regime GARCH models. Our empirical results are consistent with those reported in the studies by Hamilton and Susmel (1994) and Gray (1996): the high persistence of shocks implied by standard GARCH models is spuriously induced by structural breaks in the conditional volatility. This also justifies the warning made by Lamoureux and Lastrapes (1990) that “ignoring simple structural shifts in



unconditional volatility (i.e., model misspecification) can lead to the spurious appearance of extremely strong persistence in variance.” Although there are some features commonly shared by the estimates of four markets, two different points deserve our attention. First, despite the fact that not all of the unconditional mean return parameters (φ10 and φ20 ) are highly significantly different from zero in two regimes, it appears to have opposite signs among the four stock markets. For SHA, SZA, and SHB, we find that negative returns tend to be associated with a low variance regime, while positive returns are associated with turbulence. This phenomenon is different from Hamilton and Susmel’s (1994) finding, which shows that higher volatility regimes tend to be associated with recessions in the US stock market. Our results imply a rather interesting behavior of investors in Chinese stock markets: when stock returns are positive, investing activities vary dramatically, accelerating the stock volatility; when stock returns are negative, investors tend to be slowing down, rather than engaging in a short sell. This finding is consistent with the naïve, highly speculative, and information asymmetric behavior presented in the Chinese stock markets (Chen et al. 2001; Mei et al. 2004).9 In sum, investors in Chinese markets focus more on the speculative profits when the market moves upside, whereas investors in the US market concentrate on minimizing losses as the market moves downside. It appears that US markets are dominated by risk-averse, rational investors, whereas investors in Chinese markets tend to be speculators and assume excess risk. Second, the evidence clearly indicates that the volatility switching behavior of A-share markets differs from that of B-share markets. In particular, the values of P and Q for B-share markets are much smaller than those of A-share markets, implying that B-share markets are more volatile and more apt to shift between a high-volatility state and a low-volatility state. The state duration indicators, d1 and d2 , provide more straightforward evidence for illustration: for SHA and SZA, a low-volatility state lasts, on average, about 37–42 weeks, while a high-volatility state lasts, on average, 9–10 weeks; for B-share markets, however, both lowvolatility and high-volatility states have much shorter “survival” times: for SHB and SZB, a low-volatility state lasts, on average, about 6–7 weeks, while a high-volatility state lasts only about four weeks. One possible explanation for this is that the B-share holders are sophisticated foreign investors consisting of major international financial institutions. These investors hold more diversified international assets. It follows that their portfolio decisions and, hence, their adjustments are responsive



to a broader set of global information. Any shock that disturbs international asset return parity conditions or risk comportment will lead them to adjust their portfolio allocation, generating market volatility. On the contrary, A-share investors do not have sufficient market information, nor do they have alternative investing instruments. These restrictions prevent A-share investors from shuffling their portfolios and, hence, reducing the volatility switching. 3.4.3 Evidence from the smoothed probability The weekly stock return series and smoothed probability for each market are displayed in Figures 3.2 through 3.5. In each figure, the upper panel plots the stock return series, and the middle and bottom panels plot the smoothed probabilities for the market in State 1 (low-volatility) and State 2 (high-volatility), respectively. The plots follow Hamilton’s (1989) procedure of dating regime switches, which classifies an observation as staying at State i if the smoothed probability prob(st = i RT , RT −1 , · · · ) is greater than 0.5.

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64 Thomas C. Chiang, Zhuo Qiao and Wing-Keung Wong 30 20 10 0 –10 –20 –30 –40 95

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These figures offer us a visual illustration of the volatility patterns presented in different markets. Noticeably, the volatility patterns of the two A-share markets have quite contrasting dynamic variations as compared to those on the two B-share markets: the two A-share markets are relatively stable and dominated by a low-volatility state most of the time, while the two B-share markets are much more volatile and switch very frequently between a high-volatility and a low-volatility state. In addition, we observe that both SHA and SZA markets actually have a similar volatility pattern, especially for the period after October 1997. A similar volatility pattern also holds true for the two B-share series, although the stocks are traded in different currencies and in different places. Figure 3.2 through Figure 3.5 also provide us a very convenient instrument for tracking the regime switching in the segmented markets. By visualizing these figures, the volatility switching of A-share and B-share markets is subject to major “domestic” events that have occurred in recent history. For instance, the adoption of “price limits” (16 December, 1996) by SHSE and SZSE as a regulatory tool helped the markets shift to a low-volatility state effectively. In contrast, the death of Chinese leader


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Den Xiaoping (19 February, 1997), a policy change to issue new stocks to the investors on the secondary market (13 February, 2000),10 and suspension of the sale of state shares via stock markets (24 June, 2002) pushed the markets to a high-volatility state. With respect to the B-share market volatility switching, it appears that the spell of high-volatility states on two B-share markets corresponds closely to the major international financial crises, while the two A-share markets seem immune to these events: as international crises take place, these two A-share markets stay at a low-volatility state or switch to a low-volatility state rapidly. It is of interest to check some of the recent international events and their impact on volatility switching. As the Asian crisis started to transpire in early July 1997, the two B-share markets switched to a high-volatility state immediately, while the two A-share markets switched from high-volatility to low-volatility. During the Asian financial crisis, the two A-share markets stayed at a low-volatility state up to May 1999, while the two B-share markets remained sensitive to the volatility changes in the global markets. For example, during the global stock market collapse caused by the drastic change in Hong Kong’s Hang


66 Thomas C. Chiang, Zhuo Qiao and Wing-Keung Wong 40 30 20 10 0 –10 –20 –30 –40 95

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Seng Index in late October 1997, the two A-share markets remained calm, while the two B-share markets switched to high-volatility quickly and remained highly volatile for about 2 months. When the Russian financial crisis broke out in mid-July 1998, the two B-share markets also switched to a high-volatility regime and hung on there until early September 1998. Again, in the Brazil crisis (January 15, 1999), the two A-share markets remained at a low-volatility state, while SHB (SZB) shifted to a highvolatility state on January 21, 1999. The same scenario holds true for the incidence of global stock market collapses triggered when the IT bubble burst on April 14, 2000; both SHB and SZB moved to a high-volatility state on May 4 and May 11, 2000, respectively, and remained in a highvolatile state for about one month. However, the two A-share markets remained in a low-volatility state. All of these incidences provide strong support for the market segmentation argument that international volatility spillovers do affect the segmented stock markets profoundly and the sterilization policy in China has been an effective instrument for shielding the A-share markets from external turbulence. Nevertheless, it can be argued that A-share investors are insensitive to external shocks owing to



a lack of investment sophistication or simply because they fail to switch to alternative investment instruments.

3.5 Volatility linkages among segmented stock markets Although the above analyses offer some significant insight into the nature of volatility associated with different markets, the possibility of volatility linkages among the four segmented markets has been abstract. In this section, we fill in this gap. The way we handle it is to modify the variance equation by adding a conditional variance derived from a cross-market. More subtly, we shall investigate regime-dependency of the volatility linkages among the four segmented markets by introducing an indicator ID to the following specifications: Rt = φ0 + φ1 Rt−1 + εt where εt t−1 ∼ N(0, ht ) 2 +b h ht = a0 + a1 εt−1 1 t−1 + δ · hr,t · ID

(3.11)

where Rt is an index return of a particular stock market, hr,t is the conditional variance derived from the estimation of the RS-GARCH model based on a cross stock market. ID is an indicator variable, which takes the following forms: ID = 1, if prob(st = 2 RT , RT −1 , · · · ) > 0.5 (3.12a) ID = 0, if prob(st = 2 RT , RT −1 , · · · ) ≤ 0.5 and ID = 1, if prob(st = 1 RT , RT −1 , · · · ) > 0.5 ID = 0, if prob(st = 1 RT , RT −1 , · · · ) ≤ 0.5

(3.12b)

Equation (3.12a) [3.12b] states that when a cross stock market is in a high[low-] volatility regime, i.e., the smoothed probabilities are greater than 0.5, ID equals one; otherwise, it equals zero. It follows that conducting a significance test on parameter δ provides us a direct test for analyzing the volatility linkages among the four segmented stock markets. If δ is significantly different from zero, it suggests the existence of cross-market spillover; the sign, however, will signify the direction of impact. The estimates of volatility linkages under a high-volatility regime and a lowvolatility regime are, respectively, reported in Panel A and Panel B of Table 3.6.


68 Thomas C. Chiang, Zhuo Qiao and Wing-Keung Wong Table 3.6 Analyses of volatility linkages among four segmented stock markets Panel A

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/ 1.647 (0.203)*** −0.005(0.008) 0.010(0.006)

1.823(0.422)*** / −0.005(0.006) 0.003(0.004)

0.027(0.008)*** 0.019(0.005)*** / 1.178(0.214)***

0.231(0.049)*** 0.161(0.035)*** 1.975(0.215)*** /

Panel B SHAIDLV SZAIDLV SHBIDLV SZBIDLV

SHA / −0.292 (0.020)*** −0.241(0.038)*** −0.142(0.028)***

SZA −0.510(0.033)*** / −0.363(0.009)*** −0.184(0.038)***

SHB −0.206(0.026)*** −0.059(0.006)*** / −0.234(0.021)***

SZB −0.350 (0.036)*** −0.221(0.017)*** −0.609 (0.040)*** /

This table contains correlation coefficients of volatility linkages for Shanghai A shares (SHA), Shenzhen A shares (SZA), Shanghai B shares (SHB), and Shenzhen B shares (SZB). Panel A is based on estimation from equations (11) and (12a). Panel B is based on estimation from equations (3.11) and (3.12a). HV indicates high-volatility and LV indicates low-volatility. *** and ** indicate significance at the 1 percent and 5 percent level, respectively. Numbers in parentheses are standard errors.

The findings are interesting. As we inspect the impact of A-shares on other markets, the estimated values of the coefficient δ are positive in high-volatility regimes, and negative in low-volatility regimes. More important, all of the coefficients from A-share markets are statistically significant. This suggests that the A-share markets’ volatility not only has a spillover effect on all of the markets but exhibits different impacts under different regimes. Specifically, when the highly volatile regime is prevailing, it provokes more volatility over other segmented markets; in contrast, when the tranquil regime is underway, it tends to stabilize the other markets. We find similar evidence between the two B-share markets. That is, the volatility coefficients are positive between the two B-share markets in the high-volatility regime, and negative in the low-volatility regime. However, there is no impact from B-share markets to A-share markets in the high-volatility regime. One possible explanation is that in the high-volatility regime, the volatile movements in B-share markets are often more sensitive to the disturbances from international markets. The A-share markets, however, do not react in the same way owing to a lack of information or overconfidence. Generally speaking, we find that the volatility spillover is positive in the high-volatility regime and negative in the low-volatility regime. In addition, there is evidence that volatility linkages among the four segmented markets are regime-dependent.



3.6 Conclusion This study adopts a regime switching GARCH model to examine the volatile nature among the four major segmented Chinese stock indices. We also conduct statistical tests to examine the volatility spillover effects among these four segmented markets at different volatility regimes. Our empirical findings are consistent with the following notions. First, there is strong evidence of regime shift in the volatility of the four segmented markets, and the RS-GARCH model appears to outperform the single regime GARCH model in modeling the volatility of stock markets in China. Second, although there are some common features of volatility switch in A-share and B-share markets, B-share markets appear to be more volatile and to shift more frequently between a high-volatility state and a low-volatility state. Third, for SHA, SZA, and SHB, we find that negative returns tend to be associated with a low variance regime, while positive returns are associated with a turbulent market. This phenomenon is different from Hamilton and Susmel’s (1994) finding, which shows that higher volatility regimes tend to be associated with recessions in the US stock market. Our finding suggests that investors’ behavior in light of profit maximization between Chinese and US markets is quite diverse. US investors are more sensitive to the downside of the market due to risk aversion, while Chinese investors are more excited by the upside of the market because they are more apt to pursue a speculative opportunity. Fourth, the volatility switch of A-share markets and B-share markets is subject to different major events. The volatile movements in B-share markets are sensitive in reacting to international shocks. A-share markets seem to be immune to the volatility spillover from international financial markets. Finally, evidence strongly indicates that A-share volatility has a significant impact on all of the alternative markets, regardless of whether it is in a high- or low-volatility regime. However, the volatility spillover of B-share markets occurs only between the B-share markets and has no impact on the A-share markets in a high-volatility regime. We find that volatility linkages among the four segmented markets are regime-dependent.

Notes 1. As these four shares are the main components in the Chinese markets, our investigation shall focus on these four markets. 2. By reviewing the documents promulgated by the China Securities Regulatory Commission (CSRC) and the State Administration of Foreign Exchange (SAFE), it appears that, on June 1, 2001, Chinese authorities lifted the last



3.

4. 5.

6. 7.

8. 9.

10.

restrictions on their citizens’ B-share investments, although the policy went into effect on February 19 (Webb and Clifford, 2001). However, foreign currency in cash is still forbidden in B-share trading. The B-share market is still segmented due to its limitation of available foreign currencies. Following this line of reasoning, Chakravarty et al. (1998) and Su and Fleisher (1999) argue that domestic investors are better informed than foreign investors about the value of local assets because of the familiarity of the language, culture, and institutional setting. However, no supportive evidence is found by Chui and Kwok (1998) and Mok and Hui (1998). The properties of the regime switching GARCH model and its strengths will be discussed in more detail in the next section. Although the power ARCH model has been designed to take care of the long memory property of the stock return volatility, the regime shift issue has not been resolved. In his recent chapter, Engle (2002) provides alternative stochastic volatility models. For example, the LR statistic for SHA is 165.972; its p-value is significant at the 1 percent level. Notice that the LR tests employed in the text should not be viewed as a formal diagnostic checking, since the two models are not strictly nested. The existence of unidentified parameters under the null of a singleregime model may cause the RS-GARCH model to violate the assumption for justifying the use of standard LR tests. Moreover, standard likelihood ratio (LR) statistic tests may be biased, since it no longer follows the standard χ 2 distribution. Hansen (1992) derives the limiting distribution for the statistic. However, its implementation requires complicated computing work because of the large number of parameters needed for the grid. The fact that these probabilities are relatively large also suggests a meaningful decomposition of the time series in terms of volatility regimes. It is widely recognized that China’s stock market is a “policy oriented market,” meaning that it is government policy, rather than economic fundamentals, that drives the stock market. Chen et al. (2001) and Mei et al. (2004) found that China’s stock returns cannot be explained by fundamental factors. To improve the method of issuing new stocks, the China Securities Regulatory Commission (CSRC) adopted a new policy that allows new shares to be purchased by the investors on the secondary market. Before this date, new securities were only sold to some “special” investors, who can obtain much higher profit than investors on the secondary market (security transaction market). This policy innovation has a significant impact on fund flows, stimulating transactions on two exchanges.

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Part II Persistence and Nonlinear Cointegration



4 Nonlinear Persistence and Copersistence Christian Gourieroux and Joann Jasiak

4.1 Introduction Theoretical research on long-term relationships between economic time series has a history that spans several decades during which various linear and nonlinear comovements were unveiled, such as the Phillips curve, and the purchasing power parity. In contrast, the econometric analysis of long-term relationships is much more recent, and has been conducted mainly in the linear framework. This is the case of the cointegration theory for nonstationary time series (see Granger, 1986; Engle and Granger, 1987; Johansen, 1998), and the codependence theory for stationary series (Gourieroux and Peaucelle, 1992; Engle and Kozicki, 1993; Kugler and Neusser, 1993). Under both approaches, the dynamics of the time series of interest (i.e. VAR model) as well as their long-term relationships are assumed to be linear. Over the last two decades, some extensions of the early concepts to nonlinear dynamics have been proposed. We distinguish two types of such extensions. 1. The first type of extensions concerns autocorrelograms and was inspired by the empirical evidence on significant differences between the autocorrelograms of a time series and of some of its nonlinear transformations. A typical example is the autocorrelogram of squared asset returns that reveals long memory, while the levels of asset returns are uncorrelated. This observation led to new definitions of nonlinear autocorrelograms for stationary series (Ding, Engle and Granger, 1993; Gourieroux and Jasiak, 2002), designed to capture nonlinear serial dependence. Note that nonlinear transformations of nonstationary processes, can have surprising outcomes. For example, a Gaussian 77


78 Christian Gourieroux and Joann Jasiak

random walk is nonstationary, while its sign is not. Thus, there exist nonlinear transformations that eliminate nonstationary features. 2. The second type of extensions concerns the error correction model (ECM) for a linear cointegrating system. Some authors proposed to replace the linear long-run cointegrating relation by a nonlinear model, while others considered an equilibrium error with nonlinear dynamics. For example, the long-run cointegrating relation can be specified as a nonlinear regression model: y t = a x t , θ + ut , where (ut ) and x0 are independent, (ut ) is a stationary error term, and x0 an I(1) vector of regressors (see e.g., Choi and Saikkonen, 2005). Let us point out two drawbacks of this approach. First, the dynamics of the regressors and of the equilibrium errors are still linear, although the regressors have a nonlinear effect on (yt ). Second, the cointegrating relation is partly specified ex-ante and relies on a strong assumption. More precisely, even though the regression function a(.) is nonparametric, the nonlinear cointegrating relation b(yt , xt ), say, is assumed to be additively decomposable as yt − a xt . The same remark is valid for other extensions, such as the basic threshold cointegration model, where the cointegrating relation is linear, while the equilibrium error follows a threshold autoregressive model (see e.g., Balke and Fomby, 1997). It is clear that prior to imposing any formal specification, one needs to be able to define and measure nonlinear dependence and co-dependence. In this chapter we address this issue and propose persistence measures for stationary processes with short and long memory, and for nonstationary processes. The chapter is organized as follows. First, we examine nonlinear persistence and copersistence in stationary Markov processes. We observe that in a nonlinear dynamic framework, a stationary Markov process may feature long memory,1 as depicted by autocorrelations that decrease at a slow hyperbolic rate (see e.g., Davy doff, 1973). Section 2, defines the notion of a transformed autocorrelogram that improves upon the nonlinear autocorrelogram in Gourieroux and Jasiak (2002), as it is more convenient for persistence analysis in nonlinear processes. The computation of the transformed autocorrelogram, which is quite straightforward and based on nonlinear canonical decomposition, is


Nonlinear Persistence and Copersistence 79

discussed in Section 2. Section 3 describes how the space of nonlinear transformations of a process can be decomposed into supplementary spaces of functions with different persistence decay patterns. Section 4 provides the examples. These include Gaussian processes, time discretized diffusion processes and deterministic autoregression. Section 5 describes the practical implementation of persistence analysis based on a sample canonical decomposition. For illustration, Section 5 also includes a Monte-Carlo study of finite sample properties of the estimators. In a multivariate framework, some nonlinear trasnformations of time series can feature less persistence than the original time series themselves. This observation leads to the concept of copersistence, which is defined and discussed in Section 6. Section 7 extends the analysis to nonstationary time series. Section 8 concludes.

4.2 Transformed autocorrelogram Serial dependence in univariate [resp. multivariate] time series is commonly measured by the autocorrelation [resp. cross-correlation] functions. By definition, these functions account for the linear serial dependence only. This section extends the notion of autocorrelation to nonlinear serial dependence and describes how the transformed autocorrelogram is derived from the nonlinear canonical decomposition. 4.2.1 Definition Let us consider a strictly stationary process (Xt ) of dimension n and a scalar function g defined on Rn . Definition 2.1. The transformed autocorrelation at lag h for transformation g is denoted by: VarE g Xt+h |Xt . (2.1) ρh (g) = Corr g Xt+h , E g Xt+h |Xt = V arg Xt There exist as many transformed autocorrelograms {ρh (g), h ∈ N} as possible transformations g. By construction, the transformed autocorrelations are nonnegative: Proposition 2.1. ρh (g) = Maxg2 Corr g(Xt+h ), g2 (Xt ) , where the maximization is carried over all scalar functions g2 .



Proof. The analytical expression of the transformed autocorrelogram follows from the definition of correlation: Cov E g Xt+h |Xt , g2 Xt Corr g Xt+h , g2 Xt = Var g Xt+h Var g2 Xt Var E g Xt+h |Xt = Corr E g Xt+h |Xt , g2 Xt . Var g Xt+h The correlation attains its maximum for g2 Xt = E g Xt+h |Xt . Then Var E g Xt+h |Xt Corr E g Xt+h |Xt , g2 Xt = 1, and ρg (g) = . Var g Xt+h Q.E.D. The definition of transformed autocorrelation differs from the standard nonlinear autocorrelation existing in the literature,2 i.e., rh (g) = Corr g Xt+h , g Xt = Corr g Xt+h , LE g Xt+h |g Xt , where LE g Xt+h |g Xt denotes the linear regression of g Xt+h on g Xt . In the expression of the transformed autocorrelation, the linear regression is replaced by the conditional expectation, in order to accommodate nonlinear features. From Proposition 2.1 it follows that ρh (g) ≥ |rh (g)|.

4.2.2 Nonlinear canonical analysis Let us consider a strictly stationary process (Xt ) with a continuous distribution and marginal density f . Under weak conditions,3 the joint density of Xt , Xt+h , denoted by fh , can be written as (Barrett and Lampard, 1955, Lancaster, 1968):   ∞

  fh xt , xt+h = f (xt )f xt+h 1 + λj,h aj,h xt+h bj,h xt ,  

(2.2)

j=1

where λj,h are the canonical correlations, which are ranked λ1,h ≥ λ2,h ≥ . . . ≥ 0. The canonical directions at lag h are scalar functions denoted by aj,h () and bj,h (), which satisfy the orthogonality and normalization



conditions given below: E aj,h Xt ak,h Xt = 0, ∀k = j, E bj,h Xt bk,h Xt = 0, ∀k = j, Eaj,h Xt = Ebj,h Xt = 0, ∀j Var aj,h Xt = Var bj,h Xt = 1, ∀j. Equation (2.2) provides the nonlinear canonical decomposition of the joint density. The proposition below shows how the transformed autocorrelogram can be derived from the nonlinear canonical decomposition. Proposition 2.2. We have: ∞ 2 2 j=1 λj,h < g, aj,h > 2 ρh (g) = ∞ , 2 j=1 < g, aj,h >

where = E g Xt aj.h Xt = Cov g Xt , aj.h Xt .

Proof. For any nonlinear function g, we get: ∞

E g Xt+h |Xt = Eg Xt+h + λj,h E g Xt+h aj,h Xt+h bj,h Xt . j=1

Since bj,h , j = 1, . . . , is an orthonormal basis of functions, it follows that: ρh (g) =

 ∞ λ2 < g, a >2 1/2 j,h VarE g Xt+h |Xt j=1 j,h  =  ∞ . 2 Var g Xt+h j=1 < g, aj,h >

Q.E.D. Thus, for any lag h, the transformed autocorrelations can be computed from the canonical correlations λj,h and the covariances between transformation g and canonical directions aj,h . Proposition 2.2 extends the existing formulas derived for Gaussian time series in Granger, and Newbold (1976) (see Section 4.1) to any Markov process.4

4.3 Persistence characteristics and components This section introduces the concepts of persistence decay pattern and space that are used to characterize persistence in strictly stationary processes. Next, these characteristics are distinguished as elements of



persistence decomposition which is defined later in this section and illustrated by various examples. 4.3.1 Persistence decay pattern and persistence space The scope of serial dependence revealed by the traditional autocorrelation functions depends on the rate at which the autocorrelations taper off to zero when lag h increases. For example, the autocorrelations (rh ) of a stationary AR(1) process decay at a geometric rate αh = r h , with 0 ≤ r ≤ 1, while the autocorrelations of a long memory stationary process decay at a hyperbolic rate αh = h2d−1 , d < 0.5. In general, a persistence decay pattern α = αh , h ≥ 0 is a positive sequence converging to zero. Henceforth, the persistence decay pattern will be used to measure how serial dependence of a transformed series evolves with the lag. Definition 3.1. The persistence set for decay pattern α is the set of transformations defined by Eα = g : ρh (g) = 0 αh , (3.1) where symbol 0 indicates that there exists a positive constant c such that ρ (g) ≤ cα , ∀h. h h Proposition 3.1. The persistence set Eα is a vector space. Proof. Let us consider a linear combination of two elements g and g* in Eα and compute its correlation with an arbitrary scalar transformation g2 . We have: Cov ag + a∗ g ∗ Xt+h , g2 Xt ∗ ∗ Corr ag + a g Xt+h , g2 Xt = Var (ag + a∗ g ∗ ) Xt Var g2 Xt aCov g Xt+h , g2 Xt = Var (ag + a∗ g ∗ ) Xt Var g2 Xt a∗ Cov g ∗ Xt+h , g2 Xt + Var (ag + a∗ g ∗ ) Xt Var g2 Xt Var g Xt = a Corr g Xt+h , g2 Xt ∗ ∗ Var (ag + a g ) Xt Var g ∗ Xt ∗ ∗ +a Corr g Xt+h , g2 Xt . Var (ag + a∗ g ∗ ) Xt



By applying Proposition 2.1, we find that:

ρh ag + a∗ g ∗ ≤ a

Var g Xt

ρh (g) Var (ag + a∗ g ∗ ) Xt Var g ∗ Xt ∗ ∗ +a ρh g = 0 αh . Var (ag + a∗ g ∗ ) Xt

Q.E.D. The persistence space is easily interpretable for a zero persistence decay pattern. Corollary. E0 = g : ρh (g) = 0, ∀h = g : E g Xt+h |Xt = Eg Xt+h , ∀h . When (Xt ) is a Markov process, transformation g is in E0 if and only if the transformed process [g(Xt )] is a martingale difference sequence with respect to the filtration generated by process (Xt ).

4.3.2 Persistence decomposition This section defines the notion of persistence decomposition. Definition 3.2. Process (Xt ) admits persistence decomposition if there exists a countable set of sequences αn,h , n ∈ N, such that: (i) αn,h = o αn−1,h , ∀n , and (ii) For any transformation g in the supplement of Eαn in Eαn−1 , we have: ρh (g) ∼ αn−1,h , for h large. Sequences αn,h , ∀n, define the set of persistence decay patterns, and Eαn defines the associated persistence space. As the set of persistence decay patterns cannot be completely ordered, persistence decomposition does not necessarily exist for all processes. However, some processes admit persistence decomposition, and are described in the next section.



4.4 Examples of persistence decomposition 4.4.1 Process with stable canonical decompositions Let us consider a stationary process, with canonical directions, independent of horizon h: ∞

fh xt , xt+h = f xt f xt+h {1 + λj,h aj xt+h bj xt },

(4.1)

j=1

where λ1,h ≥ λ2,h . . . ≥ 0 and functions aj , bj , j = 1, 2, . . . satisfy the orthogonality and normalization conditions given in subsection 2.2. Proposition 4.1. Any with a stable canonical decomposition process (4.1), where λj,h = o λj−1,h , ∀j, admits a persistence decomposition. The set of persistence decay patterns consists of sequences λj,h , j = 1, 2, . . . , while the associated persistence spaces are: Eλj = {g : = 0, ∀k, k ≤ j − 1}. Proof. This result follows directly from Proposition 2.1. Q.E.D. For simple processes, the persistence decay patterns coincide with sequences of canonical correlations of various orders (see Gourieroux and Jasiak, 2002). Example 1: Gaussian processes Let us consider a Gaussian process with zero mean, unit variance and autocorrelation function ρh . The canonical decomposition is given by (Barrett and Lampard, 1955; Cramer, 1963): ∞

j fh xt , xt+h = ϕ xt ϕ xt+h {1 + ρh Hj xt+h Hj xt }, j=1

where φ denotes the p.d.f. of the standard normal distribution, Hj is the Hermite polynomial of order j, and ρ1 ≥ ρ2 ≥ · · · ≥ ρh ≥ · · · ≥ 0. j The persistence decay patterns are: αj,h = ρh , j = 1, 2, . . . , and the persistence spaces Eαj are generated by the Hermite polynomials of degrees greater or equal to j. In particular, let us consider an AR(1) Gaussian process with ρh = ρ h , 1 ≥ ρ ≥ 0 and a fractional Gaussian process


Nonlinear Persistence and Copersistence 85 Table 4.1 Persistence decay patterns for Gaussian processes Process

Pattern

Parametrization

Gaussian AR(1)

geometric: αj,h = rjh

rj = |ρ|j 2dj−1

1−j

hyperbolic: αj,h = Aj h

Gaussian I(d)

Aj = A, dj = dj = dj + 2

j h for the (1 − L)d Xt = εt , d < 0.5, with ρh ∼ Ah2d−1 . Since ρ h = ρ j autoregressive process and

j 1−j 2 dj+ 2 −1 h (2d−1)j ρ ∼ Ah = Ah ,

for a fractional process, we can derive the persistence decay patterns. In the long memory process, the fractional orders of integration dj decrease arithmetically at rate d − 0.5. Example 2: Mixture of Gaussian processes This example illustrates non-Gaussian long memory processes obtained as mixtures of AR(1) Gaussian processes with stochastic autoregressive parameters. More precisely, let us consider an AR(1) Gaussian process, with a positive stochastic autoregressive parameter with distribution π . The joint distribution of Xt , Xt+h is:   ∞ 

 fh xt , xt+h = ϕ xt ϕ xt+h 1 + ρ jh Hj xt+h Hj xt dπ (ρ)   = ϕ xt ϕ xt+h

j=1

 

1+



∞

j=1

  Eπ ρ jh Hj xt+h Hj xt . 

j

In general Eπ ρ jh = Eπ ρ h . This implies that the joint bivariate distri bution of Xt , Xt+h is not Gaussian, although each of the components has a marginal Gaussian distribution and the canonical directions are the same as for a Gaussian process. The persistence decay patterns are easily found when the stochastic autoregressive parameter is beta distributed B (δ, 1 − δ) , 0 < δ < 1 (Granger and Joyeux, 1980): π (ρ) =

ρ δ−1 (1 − ρ)−δ 1 (ρ) .

(δ) (1 − ρ) [0,1]


86 Christian Gourieroux and Joann Jasiak Table 4.2 Persistence decay patterns of a beta mixture of Gaussian processes Process

Pattern

beta mixture

hyperbolic: αj,h = Aj h

Parametrization 2dj −1

Aj =

1 , dj = δ/2

(δ)jδ−1

δ+k 1 In this case Eπ ρ k = ( (δ) ) 1+k . We find the persistence decay patterns: ( ) δ + jh 1 1 1 1 ∼ αj,h = Eπ ρ jh = .

(δ) 1 + jh

(δ) jδ−1 hδ−1

It is interesting to note the difference between the nonlinear dynamics of this beta mixture illustrated in Table 4.2 and the Gaussian I(d) process. Although the canonical directions of these processes are identical and they both feature long memory, the fractional integration order dj = δ/2 of the beta mixture process is independent of j. 4.4.2 Deterministic autoregression Let us consider a deterministic stationary autoregression: Xt+1 = c Xt ,

(4.2)

where function c is non-constant, and the initial condition X0 is random. Property 4.2. The stationary process (Xt ) is a deterministic autoregression if and only if ρh (g) = 1, ∀h, for any non-constant function g. Proof. Necessary condition: Without loss of generality, we can choose h = 1. We get: ρ1 (g) = max Corr g Xt+1 , g2 Xt g2

= max Corr g c Xt , g2 Xt g2

≥ Corr g c Xt , g c Xt = 1. Sufficient condition: If we consider the identity function g =Id, there exists g2∗ such that: Corr Xt+1 , g2∗ Xt =1. It follows that ∃a, b : Xt+1 = ag2∗ Xt + b, a.s.. Q.E.D.



The above property gives a characterization of the deterministic autoregressions in terms of persistence decomposition. This characterization follows from definition (2.1) of the transformed autocorrelogram. Indeed, let us consider the well-known quadratic mapping: Xt+1 = 4Xt 1 − Xt , an initial condition X0 uniformly distributed on [0,1], (Tong, 1990, Section 3.3.2) and the identity transformation g =Id. We get: ρh (g) = 1, ∀h, whereas rh (g) = Corr Xt+h , Xt = 0, ∀h. The quadratic mapping represents a weak white noise process with zero conventional autocorrelations rh (g) at all lags, which additionally features strong nonlinear dependence revealed by ρh (g). Also note that a stationary deterministic autoregression features a “unit root” property (since limh→∞ ρh (g) = 1). Therefore the relationship between unit root and nonstationarity requires a careful characterization in nonlinear processes. 4.4.3 Time discretized unidimensional diffusion process Let us now consider a stationary unidimensional diffusion process: dXt = µ Xt dt + σ Xt dWt ,

(4.3)

where (Wt ) is a Brownian motion and µ and σ denote the drift and volatility, respectively. Under compactness conditions,5 this process admits the following canonical decomposition:   ∞

h   fh xt , xt+h = f xt f xt+h 1 + λj aj xt aj xt+h , (4.4)   j=1

where the canonical elements are obtained from the spectral analysis 2

d + 1 σ 2 d . More precisely, they of the infinitesimal generator: A = µ dx 2 dx2 satisfy:

µ (x)

daj (x) dx

d 2 aj (x) 1 = log λj aj (x) . + σ 2 (x) 2 dx2

(4.5)

We find geometric persistence decay patterns: αj,h = λhj , ∀j.



4.4.4 Discretized reflected Brownian motion The Brownian motion reflected on the interval [0, l] is another example of a stationary Markov continuous time process, which is not a diffusion process, but nevertheless admits persistence decomposition. Its canonical decomposition is (Darolles and Gourieroux, 2001):  ∞

 h jπ 2 fh xt , xt+h = f xt f xt+h 1 + exp −  2 l cos

j=1

jπ jπ cos x x l t+h l t

.

(4.6)

4.5 Statistical Inference Let us denote by X1 ,…, XT a sample of observations on process (Xt ). The elements of persistence decomposition can be obtained from the empirical nonlinear canonical decomposition. 4.5.1 Empirical nonlinear canonical analysis Let us consider the covariance operator at lag h: h = ϕ Xt ψ Xt+h fh xt , xt+h dxt dxt+h . (5.1) This operator can be approximated by replacing the unknown joint p.d.f. fh by a kernel estimator. The approximated operator is: h+T = ϕ Xt ψ Xt+h fˆh+T xt , xt+h dxt dxt+h , (5.2) X Xt−x 1 where fˆh+T x, y = T1 T K t−x−y , n is the dimension of t=1 η2n K η η Xt , K is a kernel, and η is the bandwidth. Nonlinear canonical analysis of this approximated operator provides the estimates of canonical correlations and canonical directions:

∞

fˆh,T xt , xt+h = fˆh,T xt fˆh,T xt+h {1 + µˆ j,h aˆ j,h xt+h bˆ j,h xt }. j=1

(5.3) The consistency and asymptotic distributional properties of the estimated canonical correlations and of the functional approximations of canonical directions have been derived in Darolles, Florens and



Gourieroux (2004), for stationary geometrically mixing processes. In particular these results can be used to check if the canonical directions become invariant at large lags h. When this condition is satisfied, it is possible to proceed to persistence decomposition as shown in subsection 4.1. 4.5.2 Monte Carlo analysis of fractional Gaussian process The theoretical properties of kernel-based nonlinear canonical components of long memory processes have not been yet derived. To provide some insights on the performance of this method, we examine simulated realizations of a Gaussian fractionally integrated process with parameter d. From Example 1 in Section 4, we know that its canonical directions are Hermite polynomials, and that the canonical correlations display hyperbolic decay. We investigate the performance of the following correlation estimators: (1) (i) λˆ j,h = Corr Hj xt , Hj xt+h , is the autocorrelation computed from the seriestransformed by a Hermite polynomial of degree j; (2) ˆ (ii) λ = ρˆh Hj xt+h , is the associated transformed autocorrelation; j,h

(3) (iii) λˆ j,h is the kernel based canonical correlation of order j.

These estimators require different amounts of information on canonical directions. In particular λˆ (1) requires the knowledge of the current and lagged canonical directions, λˆ (2) requires only the knowledge of the current directions, and λˆ (3) requires no information about the canonical directions at all. In a nonparametric setup, estimation by λˆ (3) is the only implementable method. In the second step, the estimated correlations are used to obtain regression-based estimators of the fractional order d. The Monte-Carlo experiment is as follows. Process (Xt ) is simulated as a fractionally integrated Gaussian process (1 − L)d Xt = εt , εt ∼ N (0, 1), with d = 0.45, of length T = 4000. Figures 4.1, 4.2, 4.3 display the three types of autocorrelograms, for j = 1,…,4, and lags h = 1,…,500. The estimated autocorrelograms are used to derive the estimators of the fractional parameter d as follows. We consider each autocorrelogram (k) (k) λˆ , k = 1, . . . , 3, j = 1, . . . , 4 and regress log λˆ on 1 and log h for large h. j,h

j,h

The regression coefficient on log h provides an estimator of 2dj − 1, from which d is computed, according to formulas in Table 4.1. The results are reported in Table 4.3. For comparison, we repeat the whole experiment



0.6

j=1 j=2 j=3 j=4

acf

0.4

0.2

0.0

0

100

200

300

400

500

lags

Figure 4.1 Autocorrelogram of Hj (X)

0.6

j=1 j=2 j=3 j=4

acf

0.4

0.2

0.0

0

100

200

300

400

500

lags

Figure 4.2 Transformed Autocorrelogram of Hj (X)



0.5

j=1 j=2 j=3 j=4

0.4

acf

0.3

0.2

0.1

0

100

200

300

400

500

lags

Figure 4.3 Nonlinear Autocorrelogram of Hj (X) Table 4.3 Estimation of d (true value d = 0.45) j

acf

T-acf

Nonlinear acf

1

0.465 (2.72) 0.45 (6.62) 0.44 (3.00) 0.45 (2.65)

0.475 (3.66) 0.48 (8.53) 0.485 (3.94) 0.48 (3.88)

0.485 (3.57) 0.485 (6.85) 0.49 (3.76) 0.49 (3.71)

2 3 4

for a fractionally integrated Gaussian process with d = 0.3, and report the estimated dˆ in Table 4.4. The regression-based estimator of the fractional order is known for its lack of accuracy. Thus, as expected, the estimates based on the standard autocorrelation functions of Hermite polynomials differ significantly from the true values of d. Especially in Table 4.4, for polynomials of orders higher than j = 2, we find estimated d from beyond the stationarity region. This can be explained by the effect of Hermite polynomials on extreme values of Xt . Indeed, these polynomials enhance the extreme


92 Christian Gourieroux and Joann Jasiak Table 4.4 Estimation of d (true value d = 0.3) j

acf

T-acf

Nonlinear acf

1

0.34 (3.07) 0.54 (1.41) 0.50 (0.72) 0.50 (0.01)

0.42 (1.44) 0.50 (1.79) 0.52 (2.42) 0.49 (0.71)

0.41 (2.08) 0.45 (1.90) 0.47 (2.05) 0.49 (1.35)

2 3 4

j=1

j=2

4 2

15

0

10

–2

5

–4

0

0

1000

2000

3000

4000

0

1000

j=3

2000

3000

4000

3000

4000

j=4

40 20 0 –20

250 150 50 0

–60

0

1000

2000

3000

4000

0

1000

2000

Figure 4.4 Extremes of Hermite Polynomials, j = 1, 2, 3, 4

values. The large number of extremes reduces the precision of autocorrelation estimators, while their serial correlation induces a finite sample positive bias in the estimated parameter d (see Deo and Hurvich (2001) for a similar analysis). The effect of transformation by Hermite polynomials is illustrated in Figure 7.4, where we display the extreme values of Hermite polynomials for a simulated long memory process with d = 0.3. The extremes are defined as observations differing from the mean by more than three standard deviations.



Table 4.4 shows that this spurious effect is attenuated by using the nonlinear autocorrelation function, where the (unknown) canonical directions are kernel smoothed, and hence extreme values of the transformed series are less frequent.

4.6 Copersistence After defining the notion of persistence in nonlinear, dynamic and stationary processes, we can search for nonlinear functions of a multivariate series that feature less persistence than its individual components. First, we define the notion of copersistence. Next, we discuss the problem of identification of copersistence directions. For ease of exposition, we consider two series only. 4.6.1 Definition Let us consider a bivariate stationary series Xt = (X1t , X2t ), and define the maximal persistence decay pattern (MPDP) of univariate series Xj , j = 1, 2 say, as: ρh∗ Xj = max ρh g Xjt = max Corr g Xj,t+h , g2 Xj,t , j = 1, 2. g

g,g2

(6.1) The MPDP is computed with respect to the whole universe generated by both series (X1t , X2t ). Definition 6.1. Processes (X1t ) and (X2t ) are copersistent if and only if there exists a scalar transformation Yt = a(X1t , X2t ), which depends both on X1 and X2 (non-degeneracy condition), is a one-to-one of function

one of the arguments, and such that: ρh∗ (Y) = o ρh∗ (X1 ) = o ρh∗ (X2 ) . Such a transformation is called a copersistence direction. In the nonlinear framework, it is important to impose the nondegeneracy condition. Indeed, there exist some nonlinear transformations of (X1t ) that feature less persistence than (X1t ) itself, which do not generate any relationship between X1 and X2 . If transformation a is a one-to-one function of X1 (say), we can write: a X1t , X2t = ηt , or X1t = a−1 1 ηt , X2t ,

(6.2)



where α1−1 denotes the inverse of a with respect to the first argument. Therefore, it is possible to express X1t as a nonlinear function of X2t and ηt with different persistence decay patterns. The definition of copersistence is invariant with respect to one-to-one nonlinear transformations of either (X1t ), or (X2t ). 4.6.2 The multiplicity of copersistence directions In the stationary linear dynamic framework, it is easy to characterize codependent series and find the codependence directions (see Gourieroux and Peaucelle, 1992). In particular, the set of codependence directions is a vector space. The aim of this section is to examine if standard results on codependence can be extended to the nonlinear framework. Consider a bivariate Markov model with finite dimensional dependence (Gourieroux and Jasiak, 2001). The joint distribution of Xt , Xt+h = X1t , X2t , X1,t+h , X2,t+h is: √ √ 12 x1t − 1/2 , fh xt , xt+h = 1 + λh 12 x1,t+h + x2,t+h − 1 where λ ∈ [0, 1/12] . In particular, the marginal distribution of Xt is such that X1t and X2t are independent, with a marginal uniform distribution on [0,1]. The canonical decomposition involves a single √ √ term, with canonical directions a1 (x) = 12 (x1 + x2 − 1) , b1 (x) = 12 (x1 − 1/2) . The parameter λ is upper bounded to ensure the positivity of the joint p.d.f. For any nonlinear transformation a(x), we have: Var E a Xt+h |Xt ρh [a (x)] = Var a xt √ λh E 12 X1t + X2t − 1 a Xt . = 1/2 Var a xt Therefore, there are two different persistence decay patterns for transformed autocorrelograms: √ (i) If E 12 X1t + X2t − 1 a Xt = 0, then a(Xt ) features a geometric decay λh ; √pattern (ii) If E 12 X1t + X2t − 1 a Xt = 0, the persistence decay pattern is equal to zero. We can easily verify that the transformed autocorrelograms of series (X1t ) and (X2t ) (and also the MPDP) have the same geometric decay λh .



Let us, for example, consider nonlinear transformations of the type: a (X) = b (X1 ) + αc (X2 ) , where b is an invertible function, c is a given function such that Cov [X2 , c (X2 )] = 0, and α is a scalar. We get: √ √ E 12 X1t + X2t − 1 a Xt = Cov 12 X1t + X2t − 1 , b Xt + αc X2t √ = 12 Cov X1t , b X1t + αCov X2t , c X2t . It follows that transformations: Cov X1t , b X1t c (X2 ) , a (X) = b (X1 ) − Cov X2t , c X2t

(6.3)

are the copersistence directions. We get an infinite number of relations of the type: X1t = b−1 d X2t + ηt , which are interpretable in terms of copersistence. The above analysis can be directly extended to more complicated processes. Proposition 6.1. Let us assume that (Xt ) is a stationary process with canonical decomposition   ∞ 

 f xt+h , xt = f xt+h f xt 1 + λhj aj xt+h bj xt ,   j=1

where λ1 > λ2 . (i) The MPDP’s of (X1t ) and (X2t ) are identical to the MPDP of (Xt ) if E [a1 (X)X1 ] = 0, E [a1 (X)X2 ] = 0. (ii) If the MPDP’s of (X1t ) and (X2t ) are equal to the MPDP of (Xt ), then series (X1t ), (X2t ) are always copersistent. The copersistence directions are transformations which are uncorrelated with the first canonical direction E [a1 (X)a(X)] = 0, and are one-to-one functions of one of the series. In brief, in a linear framework, time series are generally not codependent, and when they are codependent, the space of codependence directions is of a finite dimension. In a nonlinear framework, two series are generally copersistent and the set of copersistence directions is of dimension infinity.



Proposition 6.1 provides a methodology for estimating the copersistence directions from an estimated canonical decomposition [see Section 5].

4.7 Nonlinear integration and nonlinear cointegration In stationary processes, the MPDP measures the maximal decay rate of persistence which is valid for both correlations and predictors [see equation (2.3)]. The aim of this section is to extend this notion to homogenous processes, which may feature nonstationarities. We first discuss the Gaussian random walk to show that persistence assessment in terms of correlations and predictions may differ significantly in the presence of nonstationarities. This leads to the notion of persistence by trajectory, which is defined in the second subsection, followed by nonlinear cointegration. 4.7.1 Gaussian random walk Let us consider a Gaussian random walk: xt =

t

ετ ,

(7.1)

τ =1

where the components of the noise are i.i.d., standard Normal. This is a homogenous Markov process. It has been proven in Ermini and Granger (1993) that the exponential transformations of this Gaussian random walk are such that: ! hµ2 E exp µXt+h |Xt = exp (7.2) exp µXt , 2 and ρt,h = max Corr exp µXt+h , g2 Xt g2

= Corr exp µXt+h , exp µXt " % # # exp µ2 t 2 − 1 # . =$ % exp µ2 t + h 2 − 1

(7.3)

The exponential Gaussian random walk has an explosive behavior, implied by the autoregressive representation (7.2) with autoregressive



2 larger than one. However, the limiting behavior coefficient exp hµ 2 of the autocorrelation function for large t: ! !h µ2 h µ2 ρ¯h = lim ρt,h = exp − , = exp − 2 2 t→∞ resembles the autocorrelation of a stationary AR(1) process. This implies that, in general, a definition of persistence based on the asymptotic properties of the predictors can be preferable. 4.7.2 Persistence by trajectory Let us consider a process (Xt ) and a possibly extended universe (Zt ). We assume that the process is homogenous with respect to the universe, that is, the conditional distribution of Xt given Zt−1 do not depend on date t. Then, the conditional expectations E g Xt+h |Zt ∈ are also time independent. Let us now introduce a positive sequence α = αh , h ≥ 0 converging to zero at infinity. Definition 7.1. The by trajectory (b.t.) persistence space of persistence decay pattern α is defined by: Eαb.t. = g : E g Xt+h |Zt − c(g) = 0 αh a.s., for some scalar c(g) . Therefore we have E g Xt+h |Zt − c(g) ≤ α Zt αh a.s., say. Eαb.t. is a vector space, which is monotone with respect to persistence decay pattern. The b.t. persistence decay patterns can be defined along the lines of Subsection 3.3. However, we also need to separate the stationary transformations of process (Xt ) from the nonstationary ones. For this purpose, we can consider the vector space: b,t

Eb.t. (0) = ∪α Eα ,

(7.4)

where the union is taken over all persistence decay patterns. A transfor mation g belongs in space Eb.t. (0) if and only if predictor E g Xt+h |Zt becomes independent of current information Zt when horizon h tends to infinity. In Section 3, we noted that a regular, stationary process is such that Eb.t. (0) is equal to the set of all scalar functions. In general, this space is strictly smaller, and the behavior of some transformations of a process can resemble the behavior of a stationary process, while the behavior of other transformations may look nonstationary. This leads to the following definition of nonlinear integration.



Definition 7.2. If g belongs in Eb.t. (0), the transformed process is nonlinearly integrated of order 0 (NLI(0)). If g does not belong in Eb.t. (0), the transformed process is NLI (nonlinearly integrated). The above definition of nonlinear integration is used in the sequel to define nonlinear cointegration. 4.7.3 Nonlinear cointegration

Let us consider a bivariate process Xt = X1t , X2t homogenous with respect to information (Zt ) = (Xt ). Definition 7.3. Component series X1 and X2 are nonlinearly cointegrated if: (i) (X1t ) and (X2t ) are NLI with respect to the whole universe (Zt ) = (X1t , X2t ); (ii) There exists a transformation Yt = a(X1t , X2t ), which depends both on X1 and X2 , is a one-to-one function of one of the arguments, and such that (Yt ) is NLI(0). As in the stationary case examined in Section 6, there can exist a set of cointegration directions of dimension infinity. An example of nonlinear cointegration is given below. Example 7.1. Let us consider three independent Gaussian white noises ε1,t , ε2,t , ε3,t , and define: Z1,t = ρ1 Z1,t−1 + ε1,t (Gaussian AR(1) process), Z2,t = ρ2 Z2,t−1 + ε2,t (Gaussian AR(1) process), Z3,t =

t

ε (Gaussian random walk), τ =1 3,t

X1,t = sgn ε1,t Z1,t Z3,t , X2,t = sgn ε2,t Z2,t Z3,t , Component series (X1,t) and (X2,t ) are NLI due to the presence of the random walk Z3,t . Moreover any nonlinear transformation of & X1,t X2,t , sgn X1,t , sgn X2,t that satisfies the partial invertibility condition is NLI(0).



In practice, the multiplicity of nonlinear cointegration directions can cause an identification problem. It comes as no surprise that additional identification constraints have appeared implicitly in the literature on nonlinear cointegration. For example, Park and Phillips (1999, 2002) and Karlsen, Myklebust and Tjostheim (2007) considered the estimation of nonlinear relations of the type X1,t = f X2,t + u1,t , where (X2,t ) features nonstationarities, and introduced various assumptions on the error term u1,t . Park and Phillips (2002), Karlsen, Myklebust, and Tjostheim (1999) assumed the independence between processes X2 and u1 . This assumption is an identification condition of regression function f . Indeed, if we consider two admissible decompositions X1,t = f X2,t + u1,t = f˜ X2,t + u˜ 1,t , we deduce u1,t = f˜ X2,t − f X2,t + u˜ 1,t , and (u1,t ) is independent of (X2,t ) if and only if f = f˜ . 4.7.4 Markov process with finite dimensional dependence Let us consider a homogenous Markov process with transition function: J

bj xt+1 aj xt p xt+1 |xt =

(7.5)

j=1

= b xt+1 a xt , say. Elements a and b in decomposition (7.5) are defined up to an invertible linear transformation. It is equivalent to assume decomposition (7.5) or a finite dimensional predictor space (Gourieroux and Jasiak, 2001). The predictors, such as E g Xt+h |Xt , ∀g, h, belong in the finite dimensional space generated by transformation aj Xt , j = 1, . . . , J. The transition function h-step ahead is given by: p(h) xt+h |xt = b xt+h Ch−1 a xt ,

(7.6) '

where the elements of matrix C are ci,j = ai (x) bj (x) dx. The predictors are: E g Xt+h |Xt = g xt+h p(h) xt+h |Xt dxt+h =

g (x)b (x) dx Ch−1 a Xt .

(7.7)

To simplify the discussion, let us assume that matrix C can be diagonalJ ized and has real eigenvalues λj , j = 1, . . . , J. Let C = j=1 λj uj vj denote its



spectral decomposition. We get: J

h−1 λj E g Xt+h |Xt =

g (x) b (x) uj dx vj a Xt .

(7.8)

j=1

Let us now search for the NLI(0) directions. Two cases have to be distinguished, depending on the eigenspace associated with a unit eigenvalue, when there is one.6 Case 1: If there is no eigenvalue equal to 1, or if the eigenspace associated with the unit eigenvalue does not contain the constant function. The space Eb.t. (0) is: Eb.t. (0) = {g : g(x) b (x)uj dx = 0, for any j with λj ≥ 1}. Case 2: The eigenspace associated with the unit eigenvalue contains the constant function. Up to a change of basis we can always assume that λ1 = 1 is associated with v1 a(xt ) = 1. We get: Eb.t. (0) = {g : g(x) b (x)uj dx = 0, for any j ≥ 2, with λj ≥ 1}. In both cases, space Eb.t. (0) is of infinite dimension, and there is a multiplicity of cointegration directions for a bivariate process Xt = X1t , X2t . Note that there exist NLI(0) transformations which are linear combinations of the bj functions. Therefore, after an appropriate change of factors a and b, we can write (in case 1): J1 J

p xt+1 |xt = bj xt+1 aj xt + bj xt+1 aj xt , j=1

(7.9)

j=J1 +1

where bj , j = J1 + 1, . . . , J [resp. j = 1, . . . , J1 ] are NLI(0) transformations [resp. NLI transformations]. Then, it is possible to separate the stationary and “nonstationary” components of the transition density function.

4.8 Conclusion In this chapter, we investigated various aspects of persistence in nonlinear time series. We introduced the concept of persistence space and persistence decomposition for nonlinear stationary processes. Several



examples of persistence decomposition were discussed, including the long memory processes where we highlighted differences between a fractionally integrated process, a beta mixture of AR(1) processes, continuous time processes and chaos. In a similar way, the integration directions can be characterized for nonstationary nonlinear processes. The notion of persistence [resp. integration] was used in this chapter to discuss nonlinear persistence [resp. nonlinear cointegration] directions in a multivariate framework. We emphasized the multiplicity of copersistence [resp. cointegration] directions, which render difficult the identification and interpretation. In particular, we pointed out the identifying constraints introduced implicitly in the literature on nonlinear cointegration to solve this difficulty in a rather ad-hoc manner. In stationary processes without long memory, the estimation of persistence decomposition and the analysis of copersistence follow directly from the sample canonical correlation analysis. Although the Monte Carlo study shows that the approach seems to be consistent for stationary processes with long memory, a comprehensive theoretical study of processes with nonlinear long memory and nonlinear nonstationary processes is still to come. It is left for future research.

Notes Christian Gourieroux – CREST and CEPREMAP, e-mail: [email protected] Joann Jasiak – York University, e-mail: [email protected] The authors gratefully acknowledge financial support of the Natural Sciences and Engineering Research Council of Canada, and of the Chair AXA/Risk Foundation: “Large Risks in Insurance”. 1. In the linear dynamic framework, a stationary Markov process is a linear autoregressive process: ρXt−1 +εt , where εt is a strong white noise and |ρ| < 1. The autocorrelations of this process decrease to zero at a geometric rate and feature only short memory. 2. See e.g. Ding, Engle and Granger (1993), Ding and Granger (1996), Granger and Terasvirta (1997), He and Terasvirta (1997), Gourieroux and Jasiak (2002). % '' 2 3. For instance, if f (xt )f xt+h dxt dxt+h < +∞, ∀h. fh (xt , xt+h )

4. In fact, ρh (g) is the formula, Abadir and Talmain (2005) were not aware of when they wrote “No other exact formulas have been found to solve the problem in nonnormal settings”. The approach based on the canonical decomposition seems preferable to ad-hoc expansions based on power functions, for example. 5. See e.g. Hansen, Scheinkman and Touzi (1998). 6. Markov processes with finite dimensional dependence are direct extensions of Markov chains with a finite state space. Let us denote by j = 1, . . . , J the admissible states and by P = (pi,j ) the transition matrix. The transition function


102 Christian Gourieroux and Joann Jasiak can be written: p xt+1 |xt = a xt+1 Pa xt , where aj xt = 1, if xt = j, 0 otherwise. In this particular case b = P a, C = P, and p(h) xt+1 |xt = a xt+1 P h a xt . For a Markov chain, the persistence analysis is based on the eigenvalues of P.

References Abadir, K. and Talmain, G. (2005) “Autocovariance Functions of Series and their Transforms,” Journal of Econometrics, 124: 227–252. Balke, N. and Fomby, T. (1997) “Threshold Cointegration,” International Economic Review, 38: 627–645. Barrett, J. and Lampard, D. (1955) “An Expansion for Some Second Order Probability Distributions and its Application to Noise Problems,” I.R.E. Trans, PGIT, IT-1: 10–15. Bierens, H.J. (2000): “Nonparametric Nonlinear Co-Trending Analysis, With an Application to Inflation and Interest in the U.S.,” Journal of Business and Economic Statistics, 18: 323–337. Chang, Y., Park, J. and P. C. B. Phillips (2001) “Nonlinear Econometric Models with Cointegrated and Deterministically Trending Regressors,” The Econometrics Journal, 4: 1–36. Choi, I. and Saikkonen, P. (2004) “Test for Nonlinear Cointegration,” Working Paper, University of Helsinki, Helsinki. Corradi, V. (1995) “Nonlinear Transformations of Integrated Time Series: A Reconsideration,” Journal of Time Series Analysis, 16(6): 537–549. Corradi, V. and White, H. (1994) “Testing for Stationarity, Ergodicity and for Comovements Between Nonlinear Discrete Time Markov Processes,” Working Paper, University of Pennsylvania, Pennsylvania. Cramer, H. (1963) Mathematical Methods of Statistics, Princeton University Press, Princeton. Darolles, S., Florens, J.P. and C. Gourieroux (2004) “Time Reversibility and Kernel Based Nonlinear Canonical Analysis,” Journal of Econometrics, 119: 323–353. Darolles, S. and Gourieroux, C. (2001) “Truncated Dynamics and Estimation of Diffusion Equations,” Journal of Econometrics, 102: 1–22. Davydoff, J. (1973) “Mixing Conditions for Markov Chains,” Theory of Probability and its Applications, 18(2): 312–328. Deo, R. and Hurvich, C. (2001) “On the Log-Periodogram Regression Estimator of the Memory Parameter in Long Memory Stochastic Volatility Models,” Econometric Theory, 17: 686–710. Ding, Z., Engle, R, and C. Granger (1993) “A Long Memory Property of Stock Market Returns and a New Model,” Journal of Empirical Finance, 1: 83–106. Ding, Z. and Granger, C. (1996) “Modeling Volatility Persistence of Speculative Returns: A New Approach,” Journal of Econometrics, 73: 185–216. Engle, R. and Granger, C. (1987) “Cointegration and Error Correction Representation, Estimation and Testing,” Econometrica, 55: 251–276.


Nonlinear Persistence and Copersistence 103 Engle, R. and Kozicki, S. (1993) “Testing for Common Features,” Journal of Business and Economic Statistics, 11: 369–380. Ermini, L. and Granger, C. (1993) “Some Generalizations on the Algebra of I(1) Processes,” Journal of Econometrics, 58: 369–384. Gourieroux, C. and Jasiak, J. (2001) “State-Space Models with Finite Dimensional Dependence,” Journal of Time Series Analysis, 22: 665–678. Gourieroux, C. and Jasiak, J. (2002) “Nonlinear Autocorrelograms: An Application to Inter Trade Durations,” Journal of Time Series Analysis, 23: 1–28. Gourieroux, C. and Peaucelle, I. (1992) “Series Codependantes: Application à l’hypothèse de Parité du Pouvoir d’Achat,” Revue d Analyse Economique, 68: 283–304. Granger, C. (1986) “Developments in the Study of Cointegrated Economic Variables,” Oxford Bulletin of Economics and Statistics, 48: 213–228. Granger, C. (1995) “Modelling Nonlinear Relationships Between Extended Memory Variables,” Econometrica, 63: 265–279. Granger, C. and Hallman, J. (1991) “Nonlinear Transformations of Integrated Time Series,” Journal of Time Series Analysis, 12: 207–234. Granger, C. and Joyeux, R. (1980) “An Introduction to Long Memory Time Series Models and Fractional Differencing,” Journal of Time Series Analysis, 1: 15–29. Granger, C. and Newbold, P. (1976) “Forecasting Transformed Series,” Journal of the Royal Statistical Society, B, 38: 189–203. Granger, C. and Terasvirta, T. (1993) Modelling Nonlinear Economic Relationships, Oxford University Press, Oxford. Granger, C. and Terasvirta, T. (1999) “A Simple Nonlinear Time Series Models with Misleading Linear Properties,” Economic Letters, 62: 161–165. Hansen, L., Scheinkman, J and N.Touzi (1998) “Spectral Methods for Identifying Scalar Diffusions,” Journal of Econometrics, 86: 1–32. He, C. and Terasvirta, T. (1999) “Statistical Properties of the Asymmetric Power ARCH Process,” in R. Engle and White H. (Ed.), Cointegration, Causality and Forecasting, In Honor of W.J. Granger, Oxford University Press, Oxford: 462–474. Johansen, S. (1988) “Statistical Analysis of Cointegration Vectors,” Journal of Economic Dynamics and Control, 12: 231–254. Karlsen, H., Myklebust C. and D. Tjostheim (2007) “Nonparametric Estimates in a Nonlinear Cointegration Type Model,” Annals of Statistics, 35: 252–299. Kugler, P. and Neusser, K. (1993) “International Real Interest Rate Equalization,” Journal of Applied Econometrics, 8: 163–174. Lancaster, H. (1968) “The Structure of Bivariate Distributions,” Annals of Mathematical Statistics, 29: 719–736. Park, J. and Phillips, P.C.B. (1999) “Asymptotics for Nonlinear Transformations of Integrated Time Series,” Econometric Theory, 15: 269–298. Park, J. and Phillips, P.C.B. (2002) “Nonlinear Regression with Integrated Time Series,” Econometrica, 69: 1452–1498. Tong, H. (1990) Nonlinear Time Series: A Dynamical System Approach, Oxford University Press, Oxford.


5 Fractionally Integrated Models for Volatility: A Review Dean Fantazzini

5.1 Introduction The main motivation to use fractionally integrated I(d) models is that the propagation of shocks in these processes occurs at a slow hyperbolic rate of decay, as opposed to the exponential decay associated with the I(0) stationary and invertible ARMA class of processes, or the infinite persistence resulting from an I(1) process. In this regard, many empirical studies have showed the extreme degree of persistence of shocks to the conditional variance process. Therefore, fractionally integrated models allow for a proper modelling of the long-run dependencies in the modelling of the conditional variance. The rest of the chapter is organized as follows. Section 5.2 introduces some important definitions in this field together with the basic tools to be used with fractional models, while Section 5.3 presents the class of ARFIMA models. Section 5.4 discusses the most used fractional model for volatility modelling, that is the Fractional Integrated GARCH model, while Section 5.5 shows recent interesting extensions of this model. Section 5.6 presents an empirical application with stock market indexes and Section 5.7 concludes.

5.2 Definitions and basic tools Let us analyze the fractional differencing operator by using a TaylorMacLaurin expansion around L = 0:

∞ d(d − 1) 2 d (1 − L)d = 1 − dL + (−1)j Lj L + ··· =

2!

j=0

j

104


Fractionally Integrated Models for Volatility 105

where d! d = j (d − j)!j! If we consider the Gamma function given by (j) =

∞ xj−1 exp(−x)dx

j > 0, with (j + 1) = j! = j(j)

0

(j − d) d! d = = j (d − j)!j! (−d)(j + 1) then, we can express the fractional differencing lag operator as follows: d(d − 1) 2 d(d − 1)(d − 2) 3 L − L + ··· 2! 3! ∞ ∞ (j − d) = φj Lj Lj = (−d)(j + 1)

(1 − L)d = 1 − dL +

j=0

=

∞

 

j=0



j=0

k−1−d  Lj = F(−d, 1, 1; L) k

0 0 a.s. ∀t, if all the coefficients in the infinite ARCH representation are non-negative, i.e., λk > 0 for k = 0, 1, 2, . . . . For 0 < d ≤ 1, we have that F(−d, 1, 1, ; L = 1) = 0, so that λ(1) = 1. Consequently, the second moment of the unconditional distribution of εt is infinite, and the FIGARCH process is not weakly stationary. However, we know that IGARCH models are strictly stationary and ergodic: since the higher order lag coefficients in the infinite ARCH representation of any FIGARCH model may be dominated in absolute value sense by the corresponding IGARCH coefficients, it follows that the FIGARCH(p, d, q) class of processes is strictly stationary and ergodic for 0 ≤ d ≤ 1. However, see Conrad (2010) for more details about stationarity conditions. Considerable care should be exercised in interpreting persistence in nonlinear models. In this context in which the conditional variance is parameterised as a linear function of past εt2 , the persistence is simply characterized in terms of the impulse response coefficients for the optimal forecast of the future conditional variance as a function of the time t innovation, νt , γk ≡

2 ] ∂Et [εt+k

∂νt

−

2 ] ∂Et [εt+k−1

∂νt

In more general conditional variance models the γ i ’s will depend on the time t information set. For the FIGARCH models the impulse response coefficients are independent of t. The impulse response coefficients may be found from the coefficients in the following lag polynomial obtained from equation (5.3): (1 − L)εt2 = (1 − L)1−d φ(L)−1 ω + (1 − L)1−d φ(L)−1 [1 − β(L)]νt = ζ + γ (L)νt The long-run impact of past shocks for the volatility process may now be assessed in terms of the limit of the cumulative impulse response



weights, see, for example, Rossi (2010): γ (1) = lim

k→∞

k

γi = lim λk = F(d − 1, 1, 1; 1)φ(1)−1 [1 − β(1)]

i=0

k→∞

For the covariance stationary GARCH(p, q) model and the FIGARCH (p, d, q) model with 0 < d < 1, shocks to the conditional variance will ultimately die out in a forecasting sense. Whereas shocks to GARCH process die out as a fast exponential rate, for the FIGARCH model λk will eventually be dominated by a hyperbolic rate of decay. Thus, even though the cumulative impulse response function converges to zero for 0 ≤ d < 1, the fractional differencing parameter provides important information regarding the pattern and speed with which shocks to the volatility process are propagated. In contrast, for d = 1, we have that F(d − 1, 1, 1; 1) = 1, and the cumulative impulse response weights will converge to the non zero constant γ (1) = φ(1)−1 [1 − β(1)]. Thus, from a forecasting perspective, shocks to σt2 persist indefinitely. For d > 1, F(d −1, 1, 1; 1) = ∞, resulting in an unrealistic explosive conditional variance process and γ (1) being undefined. See Rossi (2010) for more details and some examples. 5.4.1 An extension Chung (1999) highlights that the parallel between the FIGARCH specification for the variance and the ARFIMA framework of the conditional mean equation is not perfect, leading to difficult interpretations of the estimated parameters. Particularly, the fractional differencing operator applies to the constant term in the mean equation for ARFIMA models, whereas it does not in the variance equation in the case of FIGARCH models. As a consequence, Chung (1999) proposes a modified FIGARCH process: φ(L)(1 − L)d (εt2 − σ 2 ) = [1 − β(L)](εt2 − σ 2 ) where σ 2 is the unconditional variance of ε t . Then, we can apply variance targeting to this model and replace σ 2 with its sample counterpart. If we keep the same definition of λ as in (5.4), we can formulate the conditional variance as follows: σt2 = σ 2 + {1 − [1 − β(L)]−1 φ(L)(1 − L)d }(εt2 − σ 2 ) ⇒ σt2 = σ 2 + λ(L)(εt2 − σ 2 )

(5.5)


110 Dean Fantazzini

Chung (1999) shows that σ 2 > 0 and 0 ≤ ϕ1 ≤ β1 ≤ d ≤ 1 is sufficient to ensure the positivity of (5.5) when p = q = 1. In practice, λ(L) is an infinite summation which has to be truncated. Baillie et al. (1996) propose to truncate λ at 1000 lags and replace the

2 unobserved ε2t ’s by the empirical counterpart of E(ε 2t ), i.e., 1/T T t=1 εt . Instead, Chung (1999) proposes to truncate λ at the size of the information set (T − 1) and to initialise the unobserved (ε2t – σ 2 ) at 0, since this quantity is usually very small in absolute value and has a zero mean.

5.5 Other recent fractional models 5.5.1 The HYGARCH model Davidson (2004) notes the interesting and counterintuitive fact that the memory parameter of the FIGARCH model is −d, and is increasing as d approaches zero, while in the ARFIMA model the memory increases when d increases. According to Davidson (2004), the unexpected behavior of the FIGARCH model may be due less to any inherent paradoxes than to the fact that, embodying restrictions appropriate to a model in levels, it has been transplanted into a model of volatility. We already know that for the FIGARCH model λ(1) = 1 by construction, for any value of d, and therefore, it belongs to the same “knife-edge-nonstationary” class represented by the IGARCH model. Davidson (2004) proposes a generalized version of the FIGARCH and calls it the HYperbolic-GARCH (HYGARCH), allowing for a (possible) faster non-geometric rate of decay of λk , for which weak stationarity would still be possible. His model is given by σt2 = ω[1 − β(1)]−1 + {1 − [1 − β(L)]−1 φ(L)[1 + θ[(1 − L)d − 1]]}εt2 (5.6) The λk coefficients are thus weighted by θ, and the HYGARCH nests the FIGARCH when θ = 1. The process is weakly stationary when θ < 1, in which case the GARCH component observes the usual covariance stationarity restrictions (see Davidson (2004) and Conrad (2010) for more details). 5.5.2 The FIEGARCH model The EGARCH model put forward by Nelson (1991) can be extended to allow for fractional orders of integration. Let us consider a standard



EGARCH(p, q) model,   p q 1 − βi Li  ln(σt2 ) = ω + αi Li [φzt + ψ(|zt | − E|zt |)] ⇒ i=1

i=1

 ln(σt2 ) = 1 −

p

−1 βi 

 ω + 1 −

i=1

p

−1  βi L i 

i=1



q

 αi Li  g(zt ),

i=1

where zt = εt /σt and re-write it as follows by using lag polynomials: ln(σt2 ) = ω∗ + [1 − β(L)]−1 [α(L)]g(zt ) If we factorise the autoregressive polynomial [1 − β(L)] = ϕ(L) (1 − L)d where all the roots of ϕ(L) = 0 lie outside the unit circle, the FIEGARCH(p, d, q) model by (Bollerslev and Mikkelsen (1996)) can be written as ln(σt2 ) = ω∗ + φ(L)−1 [1 − L]−d [α(L)]g(zt )

(5.7)

This FIEGARCH(p, d, q) formulation nests the conventional EGARCH model for d = 0 and the Integrated EGARCH (IEGARCH) model for d = 1. By analogy to the ARFIMA class of models for the conditional mean, {ln(σ 2t )} is covariance-stationary and invertible for d in the interval between −0.5 and 0.5. However, shocks to the optimal forecasts for future values of {ln(σ 2t )} will dissipate for all values of d < 1. It follows also from Theorem 2.1 in Nelson (1991) that {ln(σt2 )} is strictly stationary and ergodic for d < 0.5. Furthermore, it is worth noting that in contrast to the FIGARCH formulation, the parameters for the FIEGARCH model do not have to satisfy any nonnegativity constraints in order for the model to be well-defined. 5.5.3 The FIAPARCH model The idea of fractional integration has been also extended to the APARCH model by Tse (1998). Particularly, the FIAPARCH(p, d, q) combines the FIGARCH(p, d, q) model given in (5.4) and the Asymmetric Power ARCH (APARCH) model by Ding et al. (1993). Taylor (1986) and Schwert (1990) introduced the standard deviation GARCH model, where the standard deviation is modelled rather than the variance. This model, along with several other models, was generalized by Ding et al. (1993) with the Power ARCH specification. In the


112 Dean Fantazzini

APARCH(p, q) model, the power parameter of the standard deviation can be estimated rather than imposed, and the optional parameters are added to capture asymmetry of up to order q: σtδ = ω +

q

αi (|εt−i | − γi εt−i )δ +

i=1

p

δ βi σt−1

(5.8)

i=1

where δ > 0 and –1< γ i 0.5tφ →d N(0, 1). 6.4.3 The modified GPH test In Smith (2005) the properties of d, estimated (incorrectly) from a fairly general Mean-plus-Noise (MN) model, are considered. The general MN model has the form: yt = µt + εt

t = 1, 2, . . . , T

(6.6)

and µt = (1 − p)µt−1 + pηt

0 0), zt−d is the transition variable. εt = νt ht and νtiid → N(0, 1).

−1 zt−d − c F zt−d , γ , c = 1 + exp −γ ht−d  2   zt−d − c  F zt−d , γ , c = 1 − exp −γ   ht−d

2 + · · · + φ ε2 + θ h 2 ht = φ0 + φ1 εt−1 q t−q 1 t−1 + · · · + θq ht−q

(7.3)

(7.4) (7.5)

According to Teräsvirta (1994), the transition function may take several possible forms but the most useful functions are either logistic or exponential which define respectively the Logistic STEC-GARCH model (i.e., system of equations 7.2 and 7.3: LSTEC-GARCH) and the Exponential STEC-GARCH model (i.e., system of equations 7.2 and 7.4 ESTC-GARCH). Equation (7.5) describes the conditional volatility ht that is assumed to follow a q-order GARCH model. It is worth noting that both LSTEC-GARCH and ESTEC-GARCH models imply the existence of two different regimes with potentially different dynamic properties while the transition between these regimes is smooth (Westerhoff and Retiz, 2005). The aforementioned models can be applied to examine the nonlinear cointegration linkages between French and German stock market indices as well as their nonlinear adjustments toward the long-run relation. The estimation of STEC-GARCH models enables us to test for contagion


Nonlinear Shift Contagion Modeling 149

between these European markets. Indeed, the significant impact of the French (respectively German) stock market returns in the French (respectively German) stock price adjustments implies that contagion effects characterize both markets, while the evidence of their cointegration relationships in both regimes is indicative of the shift contagion.

7.4 Data and empirical results 7.4.1 The data The aim of our chapter is to test the contagion hypothesis between the French and German stock markets in a new nonlinear framework using intraday data of two main market indicators: the CAC40 and DAX30. Previous studies that focus on contagion between stock markets have used at best daily prices. Several authors show that stock prices tend to adjust immediately to new information such as macroeconomic announcements (Buwens, Ben Omrane and Giot, 2005) and unexpected events (Brooks, Patel and Su, 2003). To further investigate the speed and the nonlinear pattern of stock price adjustment in this empirical study, our study relies on the use of intraday data which is particularly suitable for reproducing instantaneous and dynamic linkages of stock market prices. Data used in this chapter is obtained from Euronext and covers the period from December 1, 2004 through April 22, 2009. In practice, only continuous order driven session is considered to avoid biases of empirical results due to the differences in trading protocols, i.e., we do not consider the pre-opening and pre-closing periods. Moreover, as mentioned above, in Euronext Paris (Frankfurt stock exchange respectively), stocks are traded continuously from 9:00a.m. (9:00a.m.) to 5:25p.m. (5:30p.m.). To make our analysis possible, we use the common denominator period, which is the window running from 9:00a.m. to 5:25p.m. In our database, CAC40 and DAX30 quotes are available in every five minutes intervals. So, each trading day is considered as 101 periods of five minutes. 7.4.2 Preliminary results Given that we are concerned by testing for the shift contagion between French and German stock markets during the current international financial crisis, we divide our studied period in two subsamples: from December 1, 2004 to December 29, 2006 and from January 1, 2007 to April 22, 2009. Thus, all the tests and estimations are done twice for these two subperiods. The evolution of the CAC40 and DAX30 is depicted against observation number for both subsamples in Figures 7.1 and 7.2. It suggests


150 Mohamed El Hedi Arouri, Fredj Jawadi, Wael Louhichi and Duc Khuong Nguyen

6,800 6,400 6,000 5,600 5,200 4,800 4,400 4,000 3,600 10000

20000

30000

DAX

40000

50000

CAC

Figure 7.1 Stock price dynamics (Subsample 1)

9,000 8,000 7,000 6,000 5,000 4,000 3,000 2,000 10000

20000

30000 DAX

40000

50000

CAC

Figure 7.2 Stock price dynamics (Subsample 2)

further evidence of market comovements between two considered stock markets over two possibly different regimes (increasing and decreasing). In addition, stock prices seem not to be I(0) for both subsamples. We then check the stationarity of our variables using several unit root tests. The null hypothesis of unit root cannot be rejected over the two subsamples, indicating that both indices are integrated of order one or I(1).3 We also


Nonlinear Shift Contagion Modeling 151 Table 7.1 Correlation matrix RF

RG

Subsample 1 RF RG

1.00 0.77

0.77 1.00

Subsample 2 RF RG

1.00 0.82

0.82 1.00

Table 7.2 Descriptive statistics of intraday returns Subsample 1 Mean RF RG

2.87E-06 2.78E-06

Max.

JarqueMin. Std. Dev. Skewness Kurtosis Bera

0.006 −0.012 0.008 −0.013

0.0006 0.0007

−0.363 −0.491

15.21 16.41

No. of ARCH(q) obs.

319980.4 5485.3(10) 51320 386456.8 3013.8(15) 51320

Subsample 2 RF −1.31E-05 RG −4.74E-06

0.033 −0.019 0.031 −0.023

0.0015 0.0015

0.161 0.184

19.51 655373.0 6762.7(15) 57710 24.53 1114553. 8600.9(16) 57710

compute the correlation matrix between French and German stock market indices in order to provide a preliminary assessment of their existing linkages. From Table 7.1, we note the high correlation between the two European stock markets. The fact that market correlations have increased over the last years indicates further comovements between the CAC40 and DAX30 during the current international financial crisis, partly due to higher policy coordination between the two countries’ monetary and financial authorities, and suggests the potential of contagion effects between French and German stock markets. Table 7.2 reports descriptive statistics of French and German stock market intraday returns for both subsamples. Several interesting remarks can be made. First, stock return averages are positive in the first subsample but negative for the second one, this is essentially because of the impact of the current international financial crisis. Second, intraday returns appear to be highly volatile over the recent period. Third, results of the statistical tests strongly reject the symmetry and normality of stock returns.



ARCH tests are also performed to examine the autoregressive conditional heteroscedasticity of the return variance for the CAC40 and DAX30 indices. Results of the 10-order ARCH test confirm the existence of ARCH effects in raw return data (see,Table 7.2). However, the volatility persistence and clustering are likely to be more significant and apparent over the second subsample, due to long swings in stock price variations. In what follows, we propose to explicitly investigate the dependence structure between French and German markets as well as their timevarying contagion effects in the short- and long-run in a nonlinear framework by taking into account their conditional volatility. For this end, a STEC-GARCH model (Switching Transition model combined with a Generalized Autoregressive Conditional Heteroscedasticity model) that offers the possibility of capturing the potential of regime changes in mean and variance processes is estimated. 7.4.3 STEC-GARCH estimation procedure and results The STEC-GARCH estimation procedure that we presented in Section 7.3 can be described in several steps. First, using information criteria, autocorrelation functions and Ljung-Box tests, we determine the optimal lag length (p) for the empirical model. Second, the null hypothesis of linear cointegration is tested against the alternative hypothesis of nonlinear cointegration of smooth transition type, by examining the linearity of zt using Multiplier Lagrange tests, denoted by LM. Third, we apply the Teräsvirta (1994) and Escribano and Jorda (1999) choice tests in order to specify the transition function that governs the transition between regimes. Fourth, we use the Quasi-Maximum Likelihood (QML) method to estimate the already-identified STECM model (i.e., p and transition function are known) with GARCH modeling for conditional variance as our statistical tests reject the null hypothesis of no ARCH effect (see Table 7.2). Finally, the robustness of the estimation results of the STECGARCH model can be checked by carrying out several misspecification tests of autocorrelation, conditional heteroscedasticity and no remaining nonlinearity. The existence of contagion in the aforementioned nonlinear framework is investigated on the basis of two different levels. On the one hand, contagious effect is detected via the presence of the German stock market returns in the estimated nonlinear model for France for different regimes (i.e., recession or expansion). On the other hand, it is examined through the stochastic properties of the disequilibrium between the CAC40 and the DAX30 given by zt . The more the latter approaches a white noise and the transition function is close to unity, the more the contagion is



activated and its effects are important. However, when the contagion is low, the stock price deviations zt approaches a random walk and the estimated transition function converges toward zero. When the transition function is bounded between the extreme values one and zero, the contagion is more or less likely to be activated. To illustrate more explicitly these different contagion regimes, we define the transition variable by the lagged stock price deviation zt−d . With regard to empirical results, the use of information criteria, autocorrelation functions and Ljung-Box tests leads us to retain an optimal lag length value of 2 (p = 2) for both subsamples. After verifying that both the CAC40 and DAX30 are I(1) integrated, the application of several unit root tests to check the stationarity of zt in Equation (7.1) show that the linear cointegration hypothesis cannot be rejected for both subsamples, suggesting some evidence of significant long-run dependence between German and French stock markets. Although the linear framework is useful for describing the inter-market linkages, it might not be able to fully capture the regime shifts and possibly time-varying contagion. Therefore, we now extend it to the nonlinear context. To start, the hypothesis of nonlinear stock price adjustment is examined using LM tests introduced by Lukkonen, Saikkonen and Teräsvirta (1988) and previously employed by several authors such as Teräsvirta (1994), and Van Dijk, Teräsvirta and Franses (2002). Note that these LM tests rely on the introduction of several values of the delay parameter (1 ≤ d ≤ 5) which define the transition variable in the nonlinear model as in Equation (7.2). According to the test results, the null hypothesis of linear adjustment is rejected at the conventional levels over the two subsamples, which support our intuition that stock prices can be tied by some nonlinear links. In particular, the rejection linearity is more apparent for zt−2 over the first subsample and for zt−4 over the second one. Van Dijk, Teräsvirta and Franses (2002) report that the rejection of linearity after confirming some evidence of the linear cointegration may be an indicator of a nonlinear mean-reverting mechanism in the data. Accordingly, a STECM seems to be suitable for our stock price data, but an accurate transition function needs to be specified to describe at best the dynamics of stock price adjustment. Our choice tests favour a logistic function which is almost expected since one would rationally argue the existence of at least two possibly different stock price states over the recent period. The remaining task consists of specifying the appropriate GARCH model to reproduce the conditional volatility of stock market indices. Our results from commonly used information criteria and



general-to-specific estimation procedure of GARCH models retain a GARCH (1,1) and a GARCH (2,2) over the two subsamples respectively. Altogether, our empirical model take the form of a LSTEC-GARCH (2,2,1,1) model and a LSTEC-GARCH (2,2,2,2) over the first and the second sample respectively. As discussed above, the estimation of these models is based on the QML method which generates consistent estimators even though the conditional mean is misspecified (i.e., assumption of normality in return innovations). The optimization strategy follows the numerical algorithm of Broyden, Fletcher, Goldfarb, and Shanno (BFGS). In practice, we suppose the equality between the Autoregressive parameters in both regimes to simplify the algorithm convergence and we rewrite the STEC-GARCH model as follows: p

2 yt = α0 + λ1 zt−1 + λ1 zt−1 × F zt−d , γ , c + αi yt−i + βj xt−j + εt i=1 2 + · · · + φ ε2 + θ h 2 ht = φ0 + φ1 εt−1 q t−q 1 t−1 + · · · + θq ht−q

j=1

(7.6)

We estimate the above model and report the obtained results in Table 7.3. We observe significant and negative dependence of the current stock return on its previous levels for the two markets considered. Notify that this predictability is not in line with the weak-form market efficiency and that all the stock markets studied might be currently in a phase of correction. Our findings show several important results. First, the volatility is appropriately reproduced using GARCH models since the estimators satisfy the stationary condition and are statistically significant. Second, the selected STEC-GARCH representations seem, however, to be more efficient to reproduce stock price fluctuations over the first sample since the misspecification tests indicate that nonlinearity was completely captured, which is not the case for the second sample. Besides, the adjustment terms in the first and second regimes indicate significant evidence of nonlinear mean reversion only for the first sample. This implies that stock prices over the more recent period are not mean-reverting yet perhaps because of the strong devaluation induced by the current international financial crisis. Third, the estimation of the logistic function shows significant estimators for the transition speed while the threshold parameter provides strong evidence in favor of a nonlinear relationship and contagion effects between French and German stock market returns. As a result, there exist significant comovements between these markets


Nonlinear Shift Contagion Modeling 155 Table 7.3 LSTEC-GARCH estimation results Subsample 1

Subsample 2

0.1 10−5 [3.9]

0.0005 [27.4]

α1

−0.17 [−10.7]

3.48 [20.7]

α2

−0.05 [−8.97]

1.58 [6.8]

β1

0.22 [13.7]

−1.67 [−10.3]

β2

0.04 [4.85]

−2.06 [−7.9]

β3

0.01 [1.67]

–

λ1

0.0002 [0.72]

−0.034 [−36.1]

−0.001 [−2.25]

0.011 [18.5]

γ

0.37 [2.98]

0.60 [24.1]

c

−0.02 [−3.97]

−0.004 [−19.9]

1.35 10−8 [3.57]

8.6 10−6 [16.3]

φ1

0.15 [21.7]

0.11 [15.3]

φ2

–

0.10 [13.02]

θ1

0.82 [118.3]

0.06 [11.7]

θ2

–

0.68 [105.3]

0.68 0.79 0.28

0.11 0.05 0.003

Linear part α0

Nonlinear part λ2

GARCH parameters φ0

Misspecification tests AR(12) ARCH(12) NRNL

Note: AR(12), ARCH (12) and NRNL designate the p-value of LjungBox Test for 12th order autocorrelation, the p-value of ARCH test for 12th order and the p-value for no-remaining nonlinearity. Values between [.] are based on robust estimates of the covariance matrices of the parameter estimates.



due to the significance of the autoregressive parameters. More importantly, when stock prices exceed some threshold, contagion becomes stronger implying the transition from one regime to another. The estimated transition function conditionally on stock price deviations is plotted in Figure 7.3 to illustrate these different regimes of contagion as well as the continuum of intermediate states reproducing the comovements between stock markets. Accordingly, we identify clearly two extreme regimes: an outer regime in which contagion is

Transition function

1.00

0.75

0.50

0.25

0.00 –0.04

–0.03

–0.02

–0.01

0.00

0.01

0.02

12500

15000

0.03

Transition variable

Figure 7.3 Transition function

1.00

0.75

0.50

0.25

0.00 2500

5000

7500

10000

Figure 7.4 Time-varying transition function



low and an upper regime of strong contagion effects. The fact that the transition function reaches the unity level and persists for several times indicates that the contagion effects are almost activated. The transition between these regimes seems to be abrupt albeit we adopt a smooth transition modeling. We also investigate the speed of shifts between the different contagion regimes through the estimation of a time-varying transition function as depicted in Figure 7.4. Two important remarks can be made. On the one hand, the relative importance of variability indicates that the adjustment was activated all the time. On the other hand, the asymmetry in stock price adjustment is also captured through the time spent in the upper regime when the transition function converges toward the unity.

7.5 Conclusion The aim of this chapter was to investigate the changing nature of interdependencies between French and German stock markets during the recent period (2004–2009) marked by the advent of the global financial crisis and as a result by the strong volatility of financial markets. More precisely, contagion and shift-contagion effects between these markets are investigated using a STEC-GARCH model and intraday stock market returns. Our analysis was conducted over two subperiods: before the subprime crisis (2004–2006) and during the subprime crisis (2007–2008). Our main results show evidence of significant shift-contagion effects between French and German stock markets as their comovements were subjected to changes in nature characterized by two different regimes (an outer and an upper). We also established the existence of a nonlinear time-varying error-correcting mechanism that moves the crossmarket linkages toward their long-run equilibrium. It is interesting to note that the latter is indeed more pronounced over our second subperiod.

Notes 1. We refer to Classens, Dornbush and Park (2000); Forbes and Rigobon (2002); Dungey, Fry and Martin (2003); Caramazza, Ricci and Salgado (2004) and Bekaert, Harvey and Ng (2005) for a comprehensive review and discussion of different contagion definitions. 2. Lisbon, Porto and the LIFFE have joined the group later. 3. Results of unit root tests are not reported here to save space, but they are available upon request addressed to the corresponding author.



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Nonlinear Shift Contagion Modeling 159 Edwards, S. (1998) “Interest Rate Volatility, Capital Controls, and Contagion,” NBER Working Paper 6756, National Bureau of Economic Research, Cambridge. Eichengreen, B., Rose, A. and C. Wyplosz. (1996) “Contagious Currency Crises,” Scandinavian Economic Review, 98(4): 463–484. Escribano, A, and Jordã, O. (1999) “Improving Testing and Specification of Smooth Transition Regression Models,” in: P. Rothman (Ed.), Nonlinear Times Series Analysis of Economic and Financial Data, Kluwer Academic Publishers: Boston. Forbes, K. and Rigobon, R. (2002) “No Contagion, Only Interdependence: Measuring Stock Market Co-movements,” Journal of Finance, 57(5): 2223–2261. Garcia, P.A. (2003) “Assessing European Stock Markets (Co)Integration,” Economics Letters, 78(2): 197–203. Glick, R. and Rose, A.K. (1999) “Contagion and Trade: Why are Currency Crises Regional?,” Journal of International Money and Finance, 18(4): 603–617. Granger, C.W.J. and Teräsvirta, T. (1993) Modeling Nonlinear Economic Relationships. Oxford University Press: New York. Gravelle, T., Kichian, M. and J. Morley. (2006) “Detecting Shift-contagion in Currency and Bond Markets,” Journal of International Economics, 68(2): 409–423. Hamao, Y., Masulis, R.W. and V. Ng. (1990) “Correlations in Price Changes and Volatility across International Stock Markets,” Review of Financial Studies, 3(2): 281–307. Jawadi, F. (2009) “Essay in Dividend Modelling and Forecasting: Does Nonlinearity Help?” Applied Financial Economics, forthcoming. Lundbergh, S. and Teräsvirta, T. (1999) “Modelling Economic High Frequency Time Series with STAR-STGARCH models,” SSE/EFI Working Paper Series in Economics and Finance, No. 291. Lundbergh, S. and Terasvirta, T. (2002) “Evaluating GARCH Models,” Journal of Econometrics, 110(2): 417–435. Luukkonen, R., Saikkonen, P. and T. Teräsvirta. (1988) “Testing Linearity against Smooth Transition Autoregressive Models,” Biometrika, 75(3): 491–499. Michael, P., Nobay, A.R. and D.A. Peel. (1997) “Transaction costs and Nonlinear Adjustment in Real Exchange Rates: An Empirical Investigation,” Journal of Political Economy, 105(4): 862–879. Rigobon, R. (2003) “On the Measurement of the International Propagation of Shocks: Is the Transmission Stable?” Journal of International Economics, 61(2): 261–283. Teräsvirta, T. (1994) “Specification, Estimation and Evaluation of Smooth Transition Autoregressive Models,” Journal of the American Statistical Association, 89(425): 208–218. Teräsvirta, T. and Anderson, H.M. (1992) “Characterizing Nonlinearities in Business Cycles using Smooth Transition Autoregressive Models,” Journal of Applied Econometrics, 7(S): 119–136. Tong, H. (1983) Threshold Models in Nonlinear Time Series Analysis, Springer Verlag: New York. Tong, H. (1990) Nonlinear Time Series: A Dynamical System Approach. Oxford University Press: Oxford. Tong, H. and Lim, K.S. (1980) “Threshold Autoregression, Limit Cycles and Cyclical Data,” Journal of Royal Statistical Society, 42(B): 245–292. Tsay, R. (1989) “Testing and Modelling Threshold Autoregressive Processes,” Journal of the American Statistical Association, 84(405): 231–240.


160 Mohamed El Hedi Arouri, Fredj Jawadi, Wael Louhichi and Duc Khuong Nguyen Van Dijk, D., Teräsvirta, T. and F. Franses. (2002) “Smooth Transition Autoregressive Models: A Survey of Recent Developments,” Econometric Reviews, 21(1): 1–47. Westerhoff, F. and Reitz, S. (2005) “Commodity Price Dynamics and the Nonlinear Market Impact of Technical Traders: Empirical Evidence for the US Corn Market,” Physica A: Statistical Mechanics and its Applications, 349(3-4): 641–648.


8 Sparse-Patterned Wavelet Neural Networks and Their Applications to Stock Market Forecasting Jack Penm and R.D. Terrell

8.1 Introduction to Wavelet analysis A new approach adopted in this chapter is to apply sparse-patterned wavelet neural networks to simulate emerging stock market price movements. The approach is based on wavelet analysis, which is a relatively new and quite powerful mathematical tool for non-linear financial econometrics. Like conventional Fourier time series analysis, it involves the projection of a time-series onto an orthogonal set of components: in the case of Fourier analysis sine and cosine functions; and in the case of wavelet analysis wavelets. A critical difference is that wavelet analysis exhibits the characteristics of the local behavior of the function, whereas Fourier analysis presents the characteristics of the global behavior of the function. Compared to Fourier analysis, wavelet analysis offers several advantages. Fourier analysis decomposes a given function into sinusoidal waves of different frequencies and amplitudes. This is an effective approach when the given function is stationary. However, when the characteristics at each frequency change over time or there are singularities, Fourier analysis will give us the average of the changing frequencies over the whole function, whereas wavelet analysis can tell us how a given function changes from one time period to the next. It does this by matching a wavelet function, of varying scales and positions, to that function. Wavelet analysis is also more flexible, in that we can choose a specific wavelet to match the type of function we are analyzing. Contrary to Fourier analysis, the basis involved is fixed to be sine or cosine waves.

161


162 Jack Penm and R. D. Terrell

The continuous wavelet transform (CWT) is used to transform a function, f(x) ∈ L2 (Rd ), that is defined over continuous time. Hence, we have d x−t w(a, t) = f(x)a− 2 ψ dx, (8.1) a Rd where the parameters a ∈ R+ and t ∈ R d are dilation and translation parameters respectively. Those parameters are used for creating the wavelet family where both vary continuously. The idea of the transform is, for a given d times translation td times and a given dilation λ and of the mother wavelet ψ, to make the wavelet functions best preserve all the information from f(·). Then we can recover f(·) from its CWT using the following inverse transform: ∞ d 1 x−t f(x) = a−(d+1) w(a, t)a− 2 ψ dt da. Cψ 0 a Rd In reality, all time-series observations are collected discretely. Therefore, there exists a need to use the discrete wavelet transform to undertake time-series analysis in a discrete case. We can then recover f(·) from its discrete wavelet transform using the following inverse transform: f(x) =

i

− d2

wi a i

ψ

x − ti . ai

Multi-resolution analysis (MRA) provides an ability of discrete wavelet transformation to decompose a time-series into a high-frequency wavelet coefficient (W) component and a low-frequency residual (U) component. The high-frequency W indicates the dynamic characteristics, and the low-frequency U shows the trend characteristics. The low-frequency U component will then be decomposed into the next level high-frequency W and low-frequency U. The decomposition process will repeat until the final level U becomes smooth and stationary. Figure 8.1 shows a wavelet decomposition tree (WDT) to describe this process. MRA outlines the wavelet decomposition tree approach to decompose the original time-series to many level W components and a final level U component. Those level W components and the final level U component can then follow the inverse WDT approach to reconstruct the original time-series. The original time-series variable X will then be decomposed to many W


Sparse-Patterned Wavelet Neural Networks 163

X : original time-series

W1

U1

U2

W2

W3

U3 Figure 8.1 A wavelet decomposition tree

variables and one U variable by using MRA and WDT analysis. Those new W and U variables will facilitate modeling and forecasting, and improve predictability. MRA outlines the wavelet decomposition tree approach to decompose the original time-series to many level W components and a final level U component. The redundant Haar wavelet transform is adopted in the wavelet decomposition tree, with the following relations at the k-th point of the j-th level. Uj (k) = 0.5 Uj−1 (k) + Uj−1 (k − 2j−1 ) ; Wj (k) = Uj−1 (k) − Uj (k) (8.2) where U is the residual and W are the wavelet coefficients. We observe that the Haar wavelet function is asymmetric and resembles a step function. Haar wavelets have an ability to overcome weaknesses of symmetric wavelets and effectively capture the scale characteristics of a time-series which will be modeled for forecasting purposes. The time-series will be decomposed to high- and low-frequency components. Those high- and low-frequency components can then follow the inverse WDT approach to reconstruct the original time-series. This chapter is organized as follows. Section 8.2 outlines the construction of sparse-patterned wavelet neural networks, which demonstrate the ‘presence and absence’ restrictions on the coefficients of subset timeseries systems, including full-order systems. To demonstrate the effectiveness of the proposed non-linear financial econometrics approaches,



Section 8.3 presents an application to illustrate the practical use of the proposed sparse-patterned wavelet neural networks, which models the relationship between financial time-series, and then undertakes forecasting. In this example, the relationship between the Taiwanese stock market index and the Taiwanese future market index is used. A brief summary is provided in Section 8.4.

8.2 Sparse-patterned Wavelet neural networks Wavelet neural networks combine the theories of wavelets analysis and neural networks. A three-layer wavelet neural network generally consists of a feed-forward neural network, with one hidden layer. A neural network is a non-linear statistical data modeling tool composed of highly interconnected nodes that can model complex relationships between inputs and outputs. A neural network processes information and generates some form of response based upon the relationship identified within a panel data system, using both time-series and cross-section data. Each elementary node of a neural network can receive input signals from other nodes, which activates the algorithmic procedure in each node to transform the input signal into an output signal to other nodes. These nodes are arranged in a series of layers that connect nodes in different layers, but not with nodes within the same layer. Thus, these nodes are usually divided into the input layer, the output layer and one or more hidden layers. The input layer receives the inputs while the final layer is the output layer, as it provides the target output signal. The hidden layers are any layers that lie between the input and output layers. Neural networks also exist as computer-based systems operating many non-linear computational units or nodes interconnected by links with adjustable weights. Multi-layer networks with one or more hidden layers allow neural networks to classify functions that are not linearly separable. Historically, neural networks were not used extensively until they could solve non-linear problems. Several pruning algorithms for performance improvement have been proposed to eliminate non-significant connections. Conventionally linear time-series approaches have been adopted in modelling financial time-series in econometrics. However, the modeling power of linear approaches is weak in relation to the complexities of financial markets. This chapter focuses on non-linear models to improve performance in modeling and simulation. It is proposed that non-linear models, in particular wavelet neural networks, have the



ability to improve the performance of modeling and simulate the movements of financial variables, including equity market indicators. These networks have the flexibility to account for potentially complex nonlinear relationships which cannot be fully captured by linear models. Subset time-series models are often necessary when variables exhibit some form of periodic behavior, such as strong seasonality. If the underlying true process has a subset structure, the suboptimal full-order model specification can give rise to inefficient estimates and inferior projections. It is also possible that zero coefficients could exist in time-series models, particularly when periodic responses are likely. Conventional neural networks have difficulty when used for practical modeling of subset time-series systems, as they cannot demonstrate the ‘presence and absence’ restrictions on the coefficients of a system. In order to increase the modeling and analysis power of neural networks to be applied to subset time-series systems including full-order systems, Penm and Terrell (2003) and Chen, Penm and Terrell (2006) incorporate two types of connection (synapse), namely inhibitor arc and switchable connection, to the neural network structure. Constraints on the connection strength (synaptic weights) are imposed on the extended network structure. The inhibitor arc is connected to neural network theory from Petri nets, and the associated connection strength for all these inhibitor arcs is constrained to zero at all times. The switchable connection is obtained from switching theory, and the strength is switchable between zero and non-zero at any time. Further, O’Neill, Penm and Penm (2007), Penm and Terrell (2003) and Chen, Penm and Terrell (2006) have extended the relevance of multi-layered neural networks and so more effectively model a greater array of decomposed wavelet functions to construct wavelet neural networks. In order to increase estimation power of a wavelet neural network, the input layer variables comprise high-frequency wavelet coefficient variables and low-frequency residual variables. This approach thus recognizes that many connections between nodes in layers are unnecessary and can be deleted to construct sparse-patterned wavelet neural networks. Those inhibitor arcs – reflecting inhibitive synapses – in patterned wavelet neural networks focus zero strength at all times to node connections. The patterned network also allows for connections between nodes in layers that have variable strengths at different points of time by introducing additionally excitatory arcs – reflecting excitatory synapses. Also, the innovative and sophisticated learning algorithm we have developed (Penm and Terrell, 2003; Chen, Penm and Terrell, 2006) is



simple to use and can avoid cumbersome matrix inversion, and therefore results in better numerical accuracy. Patterned neural networks with inhibitor arcs and switchable connections are intuitively the most direct approach to increasing the estimation power of neural networks. These extensions provide neural networks with an ability to estimate sequentially changing panel data systems with a subset structure. This dynamic sparse-pattern illustrates the ‘presence and absence’ restrictions on the coefficients of the panel system, and can update the specification each time a new observation becomes available. This specification is superior to the conventional static one in which all synapses are considered to be excitatory, as no ‘absence’ restrictions are imposed on the coefficients. The sparse-patterned network approach reveals that both the modeling and simulation performance of patterned neural networking can be improved by the chosen optimal specification, using model selection criteria. Figure 8.2 indicates the structure of a polynomial network with a single hidden-layer for the predictor of a time-series system, where all y(t), u(t) and w(t) denote input neuron vectors. If the neural input-vector y(t) includes the first-order and second-order terms y1 (t), y2 (t), y1 (t)y2 (t), y21 (t) and y22 (t), a three-layered polynomial neural network can be constructed. The hidden-node transfer function in this network consists of a quadratic regression polynomial of two variables used by the group method of data handling (GMDH) algorithm.

zˆ (t)

output layer

Σ –g1y(t)

– g2u(t)

– g3w(t)

fh(y1(t),y2(t))

hidden layer

fh(w1(t),w2(t)) fh(u1(t),u2(t))

y1(t)

y2(t)

u1(t)

u2(t)

w1(t)

w2(t)

input layer

Figure 8.2 A neural network with a single hidden-layer for a time-series system with input vectors y(t), u(t) and w(t)



The general connection between the mean-corrected input and output vectors of a hidden-node is represented by fh (y1 (t), y2 (t)) = h11 y1 (t) + h12 y2 (t) + h13 y1 (t)y2 (t) + h14 y21 (t) + h15 y22 (t), where yi (t), i = 1, 2, are the input variables, and h1i , i = 1, 2, . . . , 5 are zero or non-zero coefficients. For an output-vector z(t) with three neural input-vectors, y(t), u(t) and w(t), the prediction vector zˆ (t) will be: zˆ (t) = −g1 y(t) − g2 u(t) − g3 w(t), where y(t) = y1 (t) y2 (t) y1 (t)y2 (t) y21 (t) y22 (t) , u(t) = u1 (t) u2 (t) u1 (t)u2 (t) u12 (t) u22 (t) , w(t) = w1 (t) w2 (t) w1 (t)w2 (t) w12 (t) w22 (t) , and

gi = hi,1 hi,2 · · · hi,5 , i = 1, 2 and 3.

(8.3)

If any hi,j = 0, j = 1, 2, 3, 4, 5, and i = 1, 2 and 3 is missing, (8.3) becomes a time-series system without full-order. If an input vector, say u(t), in (8.3) is missing, (8.3) becomes a subset time-series system. In this case, the hidden unit operating fh (u1 (t), u2 (t)) becomes inoperative and the corresponding incoming and outgoing arcs become inhibitive. The resulting neural network will be a sparse-patterned neural network. Sparse-patterned neural networks with inhibitor arcs and switchable connections are intuitively the most direct approach to increasing the modelling power of neural networks. These extensions provide neural networks with an ability to model sequentially changing time series systems with a subset structure. Section 8.3 presents an application of sparse-patterned wavelet neural networks with input layer variables, comprising those high-frequency wavelet coefficient variables and low-frequency residual variables, decomposed from wavelet decomposition. The application concerns prediction relationships between the stock and futures markets.

8.3 Empirical application to stock market forecasting This section gives an illustration of the practical use of the proposed sparse-patterned wavelet neural network. The three-layer patterned wavelet neural network described in Section 8.2 can be used to model



the relationship between financial time-series, and then undertake forecasting. In this example, the relationship between the Taiwanese stock market index and the Taiwanese future market index is used. A future contract is one of the most important hedging instruments for the underlying asset. Stock index futures have many attractive hedging benefits for a trader who wishes to trade the underlying stock portfolio corresponding to the index. In Taiwan the stock market at the Taiwanese Stock Exchange is an important East Asia emerging market, and the main stock market indicator is the Taiwanese equity index (TAIEX). The index is calculated on the basis of market capitalization of the constituent stocks traded on the Taiwanese Stock Exchange. The Taiwanese Futures Exchange offers a futures contract on the TAIEX. This contract is available on a quarterly expiry date and is known as the TAIEX Futures contract (TX). There is already a considerable literature examining the relationship between stock and future market prices. The literature has either examined the theoretical relationship between the markets through models such as the cost-of-carry, or examined the causality between the markets through lead-lag relationships, cointegrating tests or bi-variate spillover models. The general findings confirm a strong causality between the markets. This relationship is not unexpected given the pricing relationship between the markets and the fact that the basis reduces to zero at the maturity of the future contract. However, there has been debate about the direction of causality, with the evidence generally indicating that the futures market leads the stock market. In particular, Penm and Terrell (2003) have examined the lead-lag relationship between stock index futures and cash index prices in Australia, and document that the futures price is a leading indicator for the spot, when spot prices move together under market-wide movements. We sampled the data on the TAIEX and TX daily between 27 January and 29 June 2008 (T = 98), the TAIEX data being observed as the daily market closing index values, and the TX data being observed as the last traded price on each day in the June 2008 contract. We use u(t) = log (TAIEXt−1 ) and w(t) = log (TXt−1 ) to predict y(t) = log (TAIEXt ). To demonstrate the usefulness of the proposed approach in a small sampling environment, forecasting for period (T + 1) is carried out by building a patterned wavelet neural network on the first T period, using three-layer neural networks. In the course of constructing the patterned network, the Haar wavelet function is adopted. Augmented DickeyFuller tests are selected to examine the most suitable scale to undertake wavelet decomposition. Further, the Wavelet Toolbox of Matlab is used



to undertake wavelet transformation. For both u(t) and w(t), MRA and WDT analysis associated with a wavelet decomposition tree indicate that each time-series involved has been decomposed to seven highfrequency W components and a low- frequency R component. Therefore, a (16–16–1) wavelet neural network with 16 neuron variables in the input layer is initially constructed. The ‘presence and absence’ patterns, indicating switchable connections and inhibitor arcs, are then examined by using the model selection algorithm proposed in Penm and Terrell (2003). One-step ahead forecasts based on each selected optimal patterned network specification are undertaken for T = 93 through T = 98. It is commonly agreed that accurate forecasting in share markets is a difficult task. The approach adopted here focuses on models which can valuably simulate share price movements. The trading profits gained from the outcome of these models can still be offset by stock market frictions, such as stamp duty costs, capital gain taxes and broker’s fees. For brevity, the forecast performance outcomes are summarized in Table 8.1. The forecast performances based on the full-order neural network with two neural variables, both u(t) and w(t), are also shown for (RMSE) denotes comparison purposes. The root mean squares error

98 1 (Pi −Ai )2 2 (yˆ i − yi )2 , and Theil’s inequality coefficient, , where Ai 6 i=93 (Pi , Ai ) stands for a pair of predicted and observed changes. The RMSE error of the patterned wavelet neural network forecast is 28.05 percent of the prediction error of the full-order neural network forecast. The Theil’s inequity coefficient is 0.0071 for the sparse-patterned wavelet neural network forecast, and 0.0118 for the full-order neural network forecast. Of course, many other non-parametric econometric techniques and dynamic forgetting factor methods could play a significant role in stock

Table 8.1 Forecasting performance comparison – The sparsepatterned wavelet neural network forecast versus the full-order neural network forecast

Neural network

RMSE over test period

RMSE improvement

Theil’s inequality coefficient

Patterned wavelet Full-order

0.005233 0.006701

28.05% –

0.0071 0.0118



market price simulations. The interest of this chapter is mainly to investigate how the model developed may be utilized to improve the accuracy of financial simulations of share prices by using the proposed sparse-patterned wavelet neural network approach. Possible approaches to improve the accuracy of forecast magnitude are being investigated by our researcher team.

8.4 Summary In this chapter, a numerically robust specification approach has been developed to select the best specification of a sparse-patterned wavelet neural network. The proposed construction method is simple to use and can be applied to an M-layered wavelet neural network with hidden layer nodes in layer m ∈ [1,M − 2]. Section 8.3 presents a case study to demonstrate the effectiveness of the proposed approach and widen the possible use of sparse-patterned wavelet neural networks in financial simulations and/or econometric forecasting.

References Chen, A., Penm, J. and R.D. Terrell. (2006) “An Evolutionary Recursive Algorithm in Selecting Statistical Subset Neural Network/VDL Filtering,” Journal of Applied Mathematics and Decision Sciences, Article ID 46592, 1–12. O’Neill, T., Penm, J. and J.S. Penm. (2007) “A Subset Polynomial Neural Networks Approach for Breast Cancer Diagnosis,” International Journal of Electronic Healthcare, 3(3): 293–302. Penm, J. and Terrell, R.D. (2003) Collaborative Research in Quantitative Finance and Economics, Evergreen Publishing: ACT, Australia.


9 Nonlinear Cointegration and Nonlinear Error-Correction Models: Theory and Empirical Applications for Oil and Stock Markets Mohamed El Hedi Arouri, Fredj Jawadi and Duc Khuong Nguyen

9.1 Introduction Recently national markets around the world have become more interdependent due to the increasing process of economic globalization as well as to the rapid development of internet and telecommunication technologies. One of the major facts of the global finance is that asset price movements in one market can now spill over easily and quickly to another market. For this reason, one must consider the joint behavior of financial variables to better understand the dynamic nature of market interrelations. Traditionally, the relationship between any two variables can be investigated within the causality framework of Granger (1969) which bases itself upon on the predictability of time series. That is, if forecasts of a variable, say Y, using both past values of Y and those of another variable, say X, are better than forecasts obtained by using only past realizations of Y, then X is said to cause Y in the sense of Granger causality. Of course the direction of causality can be inversed. Notice also that the vector autoregressive (VAR) model popularized in econometric literature by Sims (1980), coupled with block exogeneity tests (or also referred to as Granger causality tests) is central to situations where the analysis requires a simultaneous equations framework rather than a single equation. It is however worth noting that the application of the standard Granger causality framework appears to be limited under several circumstances. First, any causal linkages will be invalid if the variables included in the VAR system are not stationary as required, or in other words, they exhibit cointegrating properties. In this regard, the regression of one variable on the others leads to spurious results. Some authors have suggested 171


172 Mohamed El Hedi Arouri, Fredj Jawadi and Duc Khuong Nguyen

the use of differencing technique to transform nonstationary variables into stationary before analyzing the causality, but this filter might result in the loss of information about their true relationship. Second, the Granger causality tests impose a linear relationship between considered variables while they may well take nonlinear forms of linkages. Finally, the economic theory generally suggests that many economic and financial variables, while nonstationary individually, would be bound by some relationships in the long run. The essential idea is that the convergence of these variables towards some equilibrium values would return under the impact of the economic and market forces despite their short-term deviations from the long-run relationship. In finance the relations between spot and futures prices for a given commodity or asset, between interest rates on bonds of different maturities, and between security prices and dividend yields are examples of possible cointegrating relationships. Thus, the concept of cointegration, introduced by Granger (1981) and further developed by Engle and Granger (1987), is central to the understanding of the long-run relationship between cointegrating variables at equilibrium. It is particularly important in that it shows how to deal with nonstationarity, and provides a useful tool to explore the full properties of possibly cointegrated variables within linear models. Subsequently, a number of studies focusing on cointegration techniques have been moving towards nonlinear extensions of the basic theory. They can be divided into two streams of research. The first stream extends error-correction models so as to allow for nonlinear short-run dynamics as well as potentially nonlinear adjustment mechanisms to deviations from the long-run relationship, while the second attempts to make the cointegrating relations nonlinear themselves. Studies that apply nonlinear cointegration models to macroeconomics and finance include, for example, Balke and Fomby (1997), and Liu (2001) for interest rates; Franses and Van Dijk (2000) for stock prices; Jawadi and Prat (2009) for stock indices and Escribano (2004) for the money demands in the United Kingdom. The purpose of this chapter is twofold. We first provide a comprehensive review of the theoretical aspects of nonlinear cointegration and nonlinear error-correction models (NECMs). We then show how they can be implemented to explore the dynamic linkages between oil and equity prices using international data. To achieve these objectives, in Section 9.2 we introduce the theoretical foundations of the nonlinear cointegration and major versions of NECMs up to date. Section 9.3 illustrates the empirical application of these models to the case of oil-equity price relations after discussing their


Nonlinear Cointegration and Nonlinear Error Correction Models 173

stylized facts. Section 9.4 summarizes the findings and concludes the chapter.

9.2 Theoretical background of nonlinear cointegration and NECMs This section aims to briefly present the recent developments of nonlinear cointegration theory and NECMs. We focus particularly on nonlinear modeling as a useful framework to investigate the relationship between oil and stock markets. Working with the nonlinearity is fully justified by the existence of market frictions characterizing both oil and stock markets (e.g., information barriers, noise trading, transaction and information costs, market segmentation, shocks, etc.). Since these frictions may induce asymmetries, discontinuities and persistence in the oil-equity price relationship, a nonlinear time-varying correcting mechanism is more relevant than a linear framework. Indeed, the nonlinear cointegration approach enables the investigation of the relationship between oil and stock prices in both the short- and the long-term, while reproducing the asymmetry and persistence of the said relationship. Moreover, the NECMs allow us to reproduce all types of oil-equity interactions as well as their dynamic adjustment towards an equilibrium situation that may be neglected when using linear models. 9.2.1 Nonlinear cointegration The traditional framework of linear cointegration, introduced by Granger (1981), and further developed by Engle and Granger (1987), and Johansen (1988), has been recently extended to the case of nonlinear cointegration models by a number of studies in econometric literature. A partial list of references includes Kunst (1992), Granger and Teräsvirta (1993), Balke and Fomby (1997), Escribano and Granger (1998), Escribanon and Pfann (1998), Escribano and Mira (2002), and Dufrénot and Mignon (2002). On the one hand, new notions such as “attractors”, “transients”, “equilibration” are introduced to linear cointegration models in order to capture nonlinear dynamics. On the other hand, the stationarity of the linear combination between two first-order integrated variables, which is the necessary condition for linear cointegration framework, was extended to the mixing hypothesis. Accordingly, it is possible to test for both linear and nonlinear cointegration relationships. In addition, several bound conditions are imposed on the nonlinear functions modeling the adjustment mechanism since the dependency between variables may take different forms of nonlinearities



(Escribano and Mira, 2002). Finally, recent developments in nonlinear cointegration techniques also cover a battery of new tests related to nonlinear co-trending as presented in Dufrénot and Mignon (2002). In what follows, the fundamental concepts of nonlinear cointegration will be briefly introduced before we turn to present the relevant framework that can be applied to reproduce the dynamic linkages between oil and stock markets. •

Brief preliminary definitions: attractor, transient and equilibration

Attractor: the attractor concept defines the long-run equilibrium, or equivalently called long-run attractor for nonlinear dynamical systems. It substitutes the equilibrium state required for usual linear cointegration framework (see Guckenheimer and Holmes, 1983 for detailed discussions). Transient: the “transient” refers to the distance from the “attractor”. Therefore, the well-known mean-reversion phenomenon in linear cointegration models is now represented by the transient-correction mechanism towards the basin of attractors in nonlinear cointegration framework. Equilibration: the generalization of linear cointegration leads to the apparition of the concept of equilibration. According to Markellos (1997), n stochastic processes are said to be equilibrated on a common attractor if they are topologically conjugacy and are characterized by a transient-correction mechanism. The topological conjugacy implies that the attractors Xt and Yt have similar topological properties such as the same entropies, topological dimension, and Lyapunov exponents (see Dufrénot and Mignon, 2002 for more details) Note that the topological properties conduct to an equivalent definition of the I(1) in nonlinear framework. •

Co-mixing processes and mixing tests1

We define the co-mixing processes according to Dufrénot and Mignon ∞ ∞ (2002). Let Xt t=1 and Yt t=1 be two no-mixing processes and f a nonlinear function, these processes are said to be mixing if ∞ (i) there exists a sequence f (Xt , Yt , γ ) t=1 that is mixing for γ = γ ∗ and non-mixing for γ = γ ∗ . ∞ (ii) or there exists a sequence f (Xt , Yt , γ ) t=1 that is non-mixing but with a dependence structure weaker than the dependence character ∞ ∞ izing the processes Xt t=1 and Yt t=1 .



In the econometric literature, several mixing tests have been developed, but the most powerful ones are the KPSS nonparametric test and the R/S parametric test of Lo (1991). The former tests the null hypothesis of “mixing” against the alternative of “non-mixing”, whereas the latter tests the null hypothesis of null or short-range dependence (mixing) against the alternative of “non-mixing” or long-range dependence. As far as the KPSS is concerned, we empirically retained the values recommended for the by Schwert (1989) truncation parameter: l4 = 1 1 4 4 T T int 4 100 and l12 = int 12 100 , where T is the number of observations and int[.] denotes the interior part. Concerning the choice of q for the R/S test, we used the value of Andrews (1991) as defined by the following formula: qt = [KT ], 2 1 3 3 2ρˆ where KT = 3T , [KT ] = int(KT ) and ρˆ is the first-order 2 2 1−ρˆ

autocorrelation coefficient. •

Nonlinear cointegration conceptions

∞ ∞ Let Xt t=1 and Yt t=1 be two no-mixing processes or I(1) and a nonlinear function f (Xt , Yt, γ )such that the sequence f {Xt , Yt, γ }∞ t=1 is mixing for γ = γ ∗ and non-mixing for γ = γ ∗ . Then, Xt and Yt are said to be nonlinearly cointegrated with cointegrating functionf . In the literature, the definition of nonlinear cointegration was presented differently depending upon whether the nonlinearity characterizes the long-run attractor or/and the nonlinear adjustment mechanism. We identify three types of definition or conceptions of nonlinear cointegration. Definition 1 Formally, let Xt and Yt be two non-mixing series in level and mixing when they are differenced d times, I(d). The first definition of nonlinear cointegration stipulates that the attractor is linear as in the usual cointegration framework, but it enables the convergence towards the equilibrium relationship through nonlinear adjustment. Precisely, we have: zt = Y t − β 0 − β 1 Xt

(9.1)

where (β0 , β1 ) is the cointegrating vector and zt is the disequilibrium error. The nonlinearity is then introduced in error-correction models to develop the NECMs. The main motivation stems from the fact that the



asymmetry, persistence and discontinuity characterizing the adjustment of several economic and financial variables are ignored in linear ECMs. Formally, considering two non-mixing series Xt and Yt generated by linear processes, the nonlinear vector error-correction model (NVECM) can be written as: Xt = λ1 zt−1 +

p1

α1i Xt−i +

i=1

Yt = λ1 zt−1 +

p2

p2

α2j Yt−j + λ2 F(zt−1 ) + εXt

j=1

α1j Yt−j +

j=1

p1

α2i Xt−i + λ2 F(zt−1 ) + εYt

(9.2)

i=1

(9.2) where (λ1 , λ1 ) and (λ2 , λ2 ) are respectively the linear and nonlinear adjustment terms for Xt and Yt and F(.) is a nonlinear function. Definition 2 The second conception of nonlinear cointegration introduces directly the nonlinearity in the long-run cointegration relationship and specifies the nonlinear attractor as follows: Xt = Yt

(9.3)

where and represent two nonlinear functions. The disequilibrium error corresponds to zt = Yt − Xt , whereas the representation of the ECMs is only slightly different from the model proposed by Engle and Granger (1987), given that the attractor of nonlinear models can be defined by a fixed point as well as by a stationary cycle. According to Dufrénot and Mignon (2002), if we assume that the y variables Xt and Yt follow two nonlinear processes where εtx and εt refer to two stationary processes; i = 1, . . . , p1 ; and j = 1, . . . , p2 : Xt = x Xt−i + εtx y Yt = y Yt−j + εt

(9.4)

then, Xt and Yt are said to be integrated, and their long-run stationary relation is defined as: Xt = Xt − X Xt−i Yt = Yt − Y Yt−j

(9.5)



In this case, the relative VECM which allows for a mean-reversion process toward the long-run attractor corresponds to:

Xt = λ1 zt−1 +

p1

α1i Xt−i +

i=1 p2

Yt = λ2 zt−1 +

p2

α2j Yt−j + ηXt

j=1 Y α1j t−j +

j=1

p1

(9.6) X α2i t−i + ηYt

i=1

Note that ηXt and ηYt are both stationary processes. Definition 3 This definition combines the two previous conceptions of nonlinear cointegration. Indeed, it enables us to introduce nonlinearity both in the long-run attractor and in the adjustment processes of the ECMs. We focus in what follows on the nonlinear cointegration modeling and the first type of NECMs presented in Equation (9.2) since these models appear to be the most useful in practice and suitable to apprehend the linkages between oil and equity markets. Our discussions centre particularly on the conditions for nonlinear cointegration, statistical inferences and estimation procedure. However, for the sake of simplicity and conciseness, we propose to limit the presentation to the case of single equation NECMs that we shall apply to investigate the responses of stock market to oil price shocks. •

Conditions for nonlinear cointegration: The theorem of Escribano and Mira (2002)

According to Escribano and Mira (2002), a NECM can be defined as follows: yt = α0 + λzt−1 +

p j=1

y α1,j t−j +

q

X α2,i t−i + f zt−1 , γ + µt

i=1

Xt = νt zt = y t − β X t

(9.7)

where yt is the endogenous variable and Xt is a vector of K explanatory and α are vectors of parameters; λ refers to variables that are I(1); α1,i 2,j the adjustment terms; zt is the residual term of the linear cointegration relationship and f (.) designates a nonlinear function.



The following conditions must be satisfied in order to validate a nonlinear cointegration relationship: i. µt and νt are mixing processes with finite second-order moments and crossmoments; ii. f(.) is a nonlinear function that is continuously differentiable and ∂f zt−1 , γ that satisfies some regularity conditions: −2 < < 0 ∂zt−1 (stability condition);

p

P iii. the roots of 1 − α1,j L = 0, all lie outside the unit circle;

i=1 iv. µt is a martingale difference process with zero mean and constant variance. Under these assumptions. As pointed out by Escribano and Mira (2002), zt is NED (Near Epoch Dependent) and yt and Xt are nonlinearly cointegrated with the coin tegrating vector 1, −β . The existence of nonlinear cointegration is investigated through testing the null hypothesis of linear adjustment, H0 : f zt−1 , γ = 0, against its nonlinear cointegration alternative, H1 : f zt−1 , γ = 0. The rejection of H0 leads to favor nonlinear cointegration. However, this nonlinear cointegration testing procedure using either a Wald test, a Lagrange Multiplier test or a log-likelihood ratio test is somewhat difficult to apply, particularly when f zt−1 , γ is not identified under H0 .Moreover, as suggested by Dufrénot and Mignon (2002), the condition f zt−1 , θ = 0 under H1 is not sufficient because this function must also characterize an error-correction mechanism. That is why the stability conditions on f (.) are quite important. In summary, the above conditions have to be satisfied to ensure the consistence of the nonlinear modeling. Whenever it is difficult to check for nonlinear cointegration or simulation techniques are required to identify the nonlinear forms of f zt−1 , γ , Escribano (2004) suggests the use of the rational polynomial function, the cubic function or the smoothing functions which obviously satisfy the stability conditions. In this chapter, we propose to study two particular kinds of NECMs: the NECM-Rational Polynomial (NECM-RP) and the Switching Transition Error Correction Models (STECMs). 9.2.2 Threshold cointegration, STECMs and NECM-RPs The threshold cointegration concept was initially introduced by Balke and Fomby (1997). It extends the linear cointegration case by allowing the adjustment process to occur only when the deviations from long-run



relationship exceed some critical threshold. Further, threshold cointegration models permit us to capture mean-reverting mechanism as well as nonlinearity and asymmetries in the adjustment process where negative and positive deviations do not have the same behavior. Anderson (1997) extended the model of Balke and Fomby (1997) so that it takes into account gradual transition from one regime to another rather than an abrupt shift. By doing so, we obtain the Smooth Transition ECMs (STECMs) whose statistical inferences were recently developed by Franses and Van Dijk (2000), and Van Dijk, Teräsvirta and Franses (2002). This type of modeling is highly relevant in financial studies for taking into account asymmetrical and discontinuous adjustments in the presence of market frictions such as transaction costs or stickiness of stock prices (see Jawadi and Prat, 2009 for a partial list of financial applications of the STECMs and modeling approach). However, to the best of our knowledge, such a model has never been applied to investigate the linkages between oil and equity markets. As we mentioned above, the STECMs specify different regimes with possibly different adjustment processes across regimes. They also enable us to capture temporal paths of nonlinear adjustment governed by smooth transition of regimes and to account for a gradual adjustment mechanism. Formally, they can be represented by: yt = α0 + λ1 zt−1 +

q i=0

α1,i yt−i +

+ λ2 zt−1 × F zt−d , c, γ + εt

p

α2,j xt−j

j=1

(9.8)

where zt−d , c and γ denote respectively the transition variable, the threshold parameter and the transition speed. The parameter d designates the delay parameter governing the dynamics of the transition variable. According to Van Dijk, Teräsvirta and Franses (2002), the transition function describing the transition between STECM regimes may be either logistic as in Equation (9) or exponential as in Equation (10). Therefore, depending on the choice of the transition function, we get either LSTECMs or ESTECMs respectively. −1 F zt−d , c, γ = 1 + exp −γ zt−d − c (9.9) 2 (9.10) F zt−d , c, γ = 1 − exp −γ zt−d − c The NECM-RP is a more general nonlinear model which is able to adequately capture many potential sources of nonlinearities such as abrupt



changes in adjustment speeds, effects of negative and positive shocks on stock price adjustment, and multiple long-run attractors. It can also reproduce asymmetric dynamics between the overvaluation and undervaluation regimes. This model was employed by Chaouachi, Dufrénot and Mignon (2004) to investigate the dynamic misalignments of the dollar-sterling exchange rate over the 1057–2002 period. Formally, the NECM-RP has the following form: yt = α0 + λ1 zt−1 +

q

α1,i yt−i +

i=0

q

α2,j xt−j

j=1

3 zt−1 + a + b + λ2 × + εt 2 zt−1 + c + d

(9.11)

where a, b, c and d ∈ 4 . Given their flexibility and nonlinear features, both switching and rational RP functions check the statistical properties and nonlinear cointegration stipulated by Escribano and Mira’s theorem (Escribano and Mira, 2002). In addition, both STECMs and NECM-RP seem to be more appropriate than linear ECMs in modeling the stylized facts of the oil-equity market relationships that we will briefly present in Section 9.3. 9.2.3 Econometric modeling procedure for single equation NECMs In practice, the empirical strategy for the estimation of a single equation NECM evolves the following steps: •

Step 1: We test for the unit root on individual series Xt and Yt to check for their integration order. If we accept the stationarity hypothesis, then we reject the possibility of cointegration relationship between the variables studied. But, if it is not possible to reject the I(1), we go to the second step. • Step 2: We estimate the parameters of the Equation (1): (β0 , β1 )using the OLS estimators and store the residuals of the long-run attractor that we denote: zˆ t = Yt − βˆ0 − βˆ1 Xt . Then, we check for linear cointegration by using mixing tests such as Lo (1991) test and KPSS test. The purpose of the test is to check whether zˆ t has a unit root. If the linear cointegration cannot be rejected (i.e., absence of unit root), we estimate a linear ECM and verify that the model residuals are mixing to insure its correct specification. Inversely, if the test rejects the linear cointegration hypothesis, go to the next step.


Nonlinear Cointegration and Nonlinear Error Correction Models 181 •

•

•

•

•

Step 3: In this step, we test the null hypothesis of linear cointegration against nonlinear cointegration by the use of linearity tests based on Wald, LM or likelihood ratio statistics. The acceptation of linear cointegration hypothesis enables us to go to step 4, whereas its rejection implies to go to step 6. Step 4: If the linear cointegration is not rejected in step 2 while being rejected in step 3, this may be due to the presence of nonlinearity only in the short-run dynamics. Then, we go to step 5. Step 5: We test for dynamic nonlinearity using nonlinear causality tests and estimate short-run models by nonlinear ARIMA for example (see Péguin-Feissole and Teräsvirta, 2001 for more details). Step 6: We substitute zˆ t−1 for zt−1 in f zt−1 , γ and estimate a NECM by nonlinear least squares (NLS) method to find an estimation γˆ of γ that appropriately fits the nonlinear function f zt−1 , γ . The estimation of the NECMs allowing for nonlinear mean-reversion process can be now carried out. Then, we go to step 7. Step 7: Finally, to insure the robustness of the empirical results, it is necessary to check the mixing hypothesis for the residuals of the estimated NECMs through using several diagnostics and misspecification tests such as tests for parameter constancy, tests for reaming nonlinearity, conditional heteroscedasticity tests, normality tests, and autocorrelation tests.

9.3 Empirical application to oil-equity price dynamics After presenting the recent developments in nonlinear cointegration and nonlinear ECMs, this section is devoted to discussions about major findings of the energy finance literature on oil-equity market linkages and shows how much nonlinear cointegration tools are useful in understanding such dynamics in both the short- and long-run. 9.3.1 Stylized facts related to oil-equity market relationships Over the last three decades, oil prices have changed with sequences of very large increases and decreases. Obviously, it is natural to question whether oil price movements affect macroeconomic variables, and especially the economic growth rate. The majority of research studies that focus on this issue found significant effects of oil price changes on economics activity for several developed and emerging countries (Cunado and Perez de Garcia, 2005; Balaz and Londarev, 2006; Gronwald, 2008; Kilian, 2008; and Cologni and Manera, 2008, 2009). Furthermore, some chapters established that the link between oil and economic activity is



not entirely linear, and that negative oil price shocks (price increase) tend to have larger impact on growth than positive shocks do (Hamilton, 2003; Zhang, 2008; and Lardic and Mignon, 2006, 2008). It is however important to note that in sharp contrast to the volume of works investigating the relation between oil price shocks and economics activity, there has been relatively little study into the oil-equity price relationships. In addition, most of these works have mainly focused on most advanced countries. Within the universe of emerging markets, not only does there exist very few studies, but also they are essentially concerned with the short-term interaction between energy price shocks and stock markets. The rationale of using oil price fluctuations as a risk factor affecting stock prices is that value of stock in theory equals sum of expected future cash-flows discounted at investors’ required rate. These cash-flows are affected by macroeconomic events that might be, in turn, influenced by oil shocks. As a result, oil price changes may influence stock prices. On the oil-stock market relation, the pioneer chapter by Jones and Kaul (1996) examines the reaction of international stock markets (Canada, UK, Japan and US) to oil price shocks by using the standard cash flow dividend valuation model. They found that, for the US and Canada, this reaction can be accounted for entirely by the impact of the oil shocks on cash-flows. Their results for Japan and the UK were, however, inconclusive. Using an unrestricted vector autoregressive (VAR), Huang, Masulis and Stoll (1996) show a significant link between stock returns of some American oil companies and oil price changes. Nevertheless, they find no evidence of causal linkages between oil prices and market indices such as the S&P500. In contrast, when applying an unrestricted VAR with GARCH effects to American monthly data, Sadorsky (1999) shows a significant relationship between oil price changes and aggregate stock returns in the US. In particular, the author points out that oil price shocks have asymmetric effects since negative oil shocks exert a greater impact on stock returns than positive oil price shocks. Relying on nonlinear causality tests, Ciner (2001) provides evidence that oil shocks affect in a nonlinear manner stock index returns in the US, which is consistent with the well-documented influence of oil prices on economics activity. Recently, some chapters have shifted their attention to major European, Asian and Latin American emerging markets. The empirical results of these studies indicate a significant short-run link between oil price changes and emerging stock markets. For instance, using a VAR model, Papapetrou (2001) reports that oil prices are important in explaining equity price movements in Greece. Basher and Sadorsky (2006) find strong evidence that oil prices impact stock price returns in emerging



markets using an international multifactor arbitrage pricing theory model. To summarize, one should note that previous works suffer from a methodological limitation in the sense that they explicitly assume a linear relationship between oil and stock markets, whereas they can take nonlinear forms of relations as well. This chapter contributes to the above literature by using the nonlinear cointegration techniques that allow capturing both linear and nonlinear dimensions of the oil-equity price linkages. It is expected that oil prices are nonlinearly associated with stock market returns. 9.3.2 Data and preliminary results The relationship between oil and stock markets is investigated over the period May 1987 to January 2009 using the monthly world stock market index and world oil market index. All data are obtained from Morgan Stanley Capital International (MSCI). They are expressed in US dollar in order to avoid the effects of currency risk and they are transformed using the natural logarithm formula in order to reduce their variance. First of all, we test the integration order of these indices using ADF, PP, and Andrews and Zivot (1992) tests. While the second test is robust to autocorrelation and heteroscedasticity, the latter is more robust to the eventual presence of structural breaks. Results of these tests indicate that both oil and stock market indices seem not to be stationary in levels, but they are stationary in the first difference, which suggests that they are I(1).2 Second, the dynamic evolution of oil and stock returns (i.e., natural log difference of prices at times t and t − 1) reproduced on Figure 9.1 yields some evidence of linkages between two variables. Besides, both return series seem to have downward trends and negative values over the last year period, which is potentially due to the financial crisis. Finally, as we can observe, the price fluctuation is more remarkable for oil market and both markets are likely to be more linked over the recent period. To check the relationship between oil and equity markets, we computed the bilateral correlation. The results highlight a negative, but a low level of statistical association between oil and equity market of about −5%. This may be due to the fact that linear modelling and Pearson correlation coefficients underestimate the linkages of the two markets because factors such as structural shifts resulting potentially from recent economic and political events are not accounted for. Descriptive statistics of the data used and their stochastic properties are reported in Table 1. The normality hypothesis is strongly rejected for both oil and equity market indices. The significance of excess



.5 .4 .3 .2 .1 .0 –.1 –.2 –.3 –.4 88

90

92

94

96

98

RM

00

02

04

06

08

RB

Figure 9.1 Dynamic distribution of oil and stock returns Note: RM and RB respectively designate stock and oil returns. Table 9.1 Descriptive statistics and normality test

Series Mean Maximum Minimum Std. Dev. Skewness Kurtosis RM RB

0.002 0.003

0.10 0.45

−0.21 −0.31

0.04 0.09

−1.12 −0.03

6.0 5.52

JarqueBera (P-value) 152.8(0.0) 69.15(0.0)

kurtosis, coupled with negative skewness leads to conclude that oil and equity return dynamics seem to follow an asymmetric and leptokurtic distribution which might be subjected to nonlinear adjustments over time. Linear Cointegration Tests After showing that both oil and stock market indices are I(1), our actual task consists of checking for linear cointegration hypothesis. For this purpose, we estimate the long-run relationship between oil and equity market price indices using Equation (9.1) and test the stationary property of zt . The obtained results are provided in Table 9.2. The lecture of these results leads us to reject the linear cointegration hypothesis since the absolute value of ADF statistic is higher than the critical value provided by Engle and Yoo (1987). The rejection of this hypothesis implies the rejection of linear linkages between oil and equity


Nonlinear Cointegration and Nonlinear Error Correction Models 185 Table 9.2 Linear cointegration tests 2

Series

Constant

LB

R

ADF (p, model)

Z&A

LMSCI

3.70 (32.7)*

0.45 (13.3)*

0.40

−1.86 (0, a)

−4.50

Note: The values between brackets are the t-statistics. (*) and (a) designate respectively the significance at 5% level, and a model without constant and linear trend. p denotes the lag length for ADF tests. Z&A denotes the statistic of Zivot and Andrews (1992) test.

markets. Accordingly, one could say that the oil market is entirely independent from the global stock markets, and as a result not affected at all by recent economic downturn. From an economic point of view, this story is far from believable. Besides, according to Andrews and Zivot (1992) test, we suggest significant linkages between oil and equity markets. We would then think that the oil-equity price relationship can be hold by some nonlinear forms. Note that we may then test the linear cointegration against the nonlinear cointegration using either LM or Wald tests or both according to our econometric modelling procedure in Sub-section 2.3. It is however essential to note that these tests are not sometimes enough to decide between linear and nonlinear cointegration when the sample data are generated by stochastic processes with the potential of nonlinearity, asymmetry, and structural breaks. For this reason, we propose to test the cointegration hypothesis in a nonlinear framework using “mixing” tests to check the above results. Mixing tests The purpose of this testing procedure is to test the null hypothesis of mixing against the alternative hypothesis of non-mixing. Rejecting the alternative hypothesis implies that oil and equity market indices are a priori nonlinearly mean-reverting towards the long-run equilibrium defined by the attractor. In fact, we performed two “mixing” tests, namely KPSS and modified R/S tests, and report the empirical results in Table 9.3. Results from both tests indicate the acceptance of the mixing hypothesis, which typically suggests a nonlinear cointegration relationship between oil and equity markets. The robustness of these results is checked by performing two linearity tests, Keenan (1985) test and LM3 test (Van Dijk, Teräsvirta and Franses, 2002), of which the first one tests the null hypothesis of linearity against the nonlinearity without specifying the model under the alternative, and the second explicitly specifies a STECM


186 Mohamed El Hedi Arouri, Fredj Jawadi and Duc Khuong Nguyen Table 9.3 Results tests KPSS

of

mixing

R/S

l4

l12

Andrews

0.70

0.32

0.1*

Note: (*) designates the value of V = [modified (R/S)/Root(T )] where T denotes the observation number.

Table 9.4 Linearity tests Keenan test (p-value)

LM3 test (p-value, d)

0.02

(0.01, 1)

Note: d designates the optimal value of the delay parameter for which the linearity is mostly rejected.

under the alternative. Table 9.4 shows the obtained results, and as we can see, the linearity is strongly rejected. In summary, our testing procedure reveals the presence of nonlinearity in the linkage between oil and stock markets, we now propose to reproduce this nonlinear mean-reversion process in their relationship using two types of NECMs: the LSTECM and the NECM-RP. Apart the mixing test conclusion, the choice of these models is also based on the results of LM and Keenan tests. NECM estimations The NECMs are estimated through the NLS method using a nonlinear optimization program following the procedure proposed by Escribano and Mira (1998), and Van Dijk, Teräsvirta and Franses (2002). The number of lags (p) for the NECM is specified using the usual information criteria, the autocorrelation functions and the Ljung-Box tests. We retain p = 1 according to the results while two lags are selected for oil returns when specifying the linear model, suggesting thus some evidence of significant stock price time dependence. We report the estimation results for the NECM-RP and the three-regime LSTECM in Table 9.5. It is important to remark that the three-regime LSTECM fits better the oilequity relationship than the NECM-RP does according to choice tests of transition function (Teräsvirta, 1994).


Nonlinear Cointegration and Nonlinear Error Correction Models 187 Table 9.5 NECM estimation results Coefficients

LSTECM (1,1)

NECM-RP

α0 λ1 λ2 α1,1 α2,1 α2,2 c1 c2 γ ADFGLS R/S σNECM σLECM ARCH (p-value) JB (p-value)

0.005** (1.82) 0.177* (2.36) −0.188* (−2.5) 0.052 (0.81) −0.016 (−0.51) 0.049* (2.51) −0.274* (−13.21) −0.207* (−8.11) 105.78* (7.63) −11.67 1.10 0.77 0.65 0.00

0.008 (0.23) −0.023 (−0.49) −0.0108** 0.111** (1.75) −0.04* (−1.22) 0.056* (1.98)

−11.46 1.30 0.90 0.04 0.00

Note: The values in brackets are the t-statistics of nonlinear estimators. (*) and (**) denote the significance at 5% and 10% levels respectively. ARCH, JB, ADFGLS , R/S and σNECM σLECM denote empirical statistics of conditional heteroscedasticity test, Jarque-Bera test, ADF-GLS test, Lo test, and the ratio of residual variances of nonlinear and linear models.

Several suggestions and implications regarding the linkages between oil and equity markets can be made on the basis of the obtained results. First, for both NECMs, lagged oil market returns significantly affect the dynamics of world market returns, indicating some evidence of dependence of stock market on the oil market. Moreover, this oil-dependence degree is higher than the dependence of stock market on its past values. Second, the analysis of the NECM-RP estimation shows some evidence of nonlinear and asymmetric adjustment between oil and equity markets. This is confirmed by Figure 9.2 that plots the histogram of the rational polynomial function of the estimated misalignment values zˆ t−1 . Indeed, the rational polynomial function strongly rejects the normality hypothesis and it captured the asymmetry inherent to the relationship between equity and oil markets. More interestingly, Figure 9.2 shows a bimodal density with two modes of unequal heights. This confirms the negativity of the skewness and the degree of asymmetry. The presence of these unequal modes suggests important oil-equity deviations from their equilibrium depending on the financial cycle. In addition, the asymmetry observed in the distribution of the rational polynomial function is also informative of the asymmetric


188 Mohamed El Hedi Arouri, Fredj Jawadi and Duc Khuong Nguyen 1.6

1.75

1.4

1.50

1.2

1.25

1.0 1.00 0.8 0.75 0.6 0.50

0.4

0.25

0.2

0.00

0.0 0.5

1.0

1.5

2.0

2.5

3.0

Figure 9.2 Rational polynomial function

time-varying linkages between oil and equity markets. The increase of cross-market linkages over the recent period, also suggested by Figure 9.1, may be explained by the effect of the current international financial crisis. However, it is equally important to remark that the residuals of the NECM-RP are mixing, non-normal and heteroscedastic. Furthermore, the second adjustment term is only significant at 10% level. This may be indicative of the fact that the rational polynomial function is not able to reproduce the whole nonlinearity inherent to the oil-equity market relationship. With regard to the three-regime LSTECM which is motivated by the potential of regimes and regime changes over the study period 1987– 2009, some findings are particularly interesting in relation to the nonlinear adjustment and regime transition mechanism. Indeed, the threshold parameters as well as the transition speed coefficient are strongly and statistically significant, which clearly justifies our methodological choice of the LSTECM. The transition between regimes is somewhat abrupt due to important shocks affecting these markets (e.g., 1987 stock crash, 1991 first gulf war, 2003 second Gulf war, and 2007 financial crisis) since the estimation value of γ is higher. We also find strong evidence of nonlinear adjustment according to which the adjustment process may be absent in the first regime (i.e., λ1 is significant and positive). Within this regime, stock market deviations may be uncorrected and can be characterized as random walk. As a result, the stock market would be segmented from the oil market. However, in cases of a large distortion, the stock price dynamics should converge towards the equilibrium because the attractor should



be activated to insure the convergence between oil and equity markets. For this reason, the adjustment would be more active in the other regimes (i.e., λ2 is significant and negative, and the sum of λ1 +λ2 is also negative, confirming the nonlinear mean-reversion process). This means that both oil and equity markets may move independently, but would nonlinearly converge to their long-run relation after a shock affecting one market (e.g., shocks related to the current international financial crisis). Notice that all diagnostic tests confirm the robustness of these estimations. To illustrate more explicitly this dynamic adjustment and meanreversion process between oil and equity markets, we plot in Figure 9.3 the logistic transition function conditional on the disequilibrium error, while we represent in Figure 9.4 the time-variation of this transition function. Figure 9.3 suggests the existence of three different regimes: two upper regimes and a central regime. This is quite similar to the exponential representation. Globally, we observe that the adjustment process is significantly active all the time since equity price return series is located in the upper regimes for a lot of time. Indeed, the duration of the stock prices in the central regime, given by c1 < zt−d < c2 for which the adjustment is not active, is very short. Moreover, in addition to the fact that the logistic transition function reaches the extreme value 1 (i.e., indicator of strong adjustment), it persists much more in the second upper regime (third regime), which suggests that the adjustment is stronger over the last year notably because of the current international crisis.

1.0

Transition function

0.9 0.8 0.7 0.6 0.5 0.4 –1.0

–0.5

0.0

0.5

1.0

Transition variable

Figure 9.3 Three-regime logistic transition function


190 Mohamed El Hedi Arouri, Fredj Jawadi and Duc Khuong Nguyen 1.0 0.9 0.8 0.7 0.6 0.5 0.4 1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

Figure 9.4 Time-variation of the three-regime logistic transition function

Figure 9.4 shows that the three-regime LSTECM adequately captures the asymmetric effects characterizing the oil-equity relationship. The reason is that the adjustment toward the long-run equilibrium is activated per regime especially when a shock affects one market. Events such as the 1987 stock crash, the 1991 first gulf war, the 2003 second Gulf war and the 2007–2009 financial crisis are of particular interests.

9.4 Conclusion In this chapter, we have two main objectives. First, we present the recent developments of nonlinear cointegration and nonlinear errorcorrection models. The new concepts recently introduced in financial econometrics such as attractors, transient, equilibration, mixing tests and threshold cointegration are also discussed. Second, we show how these nonlinear cointegration models can be applied to the study of oil-equity market relationships based on the general specifications and estimation procedure. Since our testing procedure reveals the presence of nonlinear dynamics and asymmetric effects in the relationships between oil and world stock markets, we decided to estimate two nonlinear ECMs, the threeregime LSTECM (Logistic Smooth Transition Error-Correction Model) and the NECM-RP (Rational Polynomial Nonlinear Error-Correction Model) which allow for capturing the potential of nonlinearities, asymmetries and regime changes. Our findings show the superiority of nonlinear models over useful linear cointegration tools in reproducing the short- and



long-term linkages between oil and stock markets over the last three decades. More interestingly, the dynamic adjustment process of these markets toward their long-run relationship is subject to regime shifts, and activated per regime when short-term price deviations of either oil or stock market from the equilibrium exceed some threshold.

Notes 1. “Mixing” and “non-mixing” are respectively replaced by I(1) and I(0) in the usual framework of linear cointegration. 2. Unit root results are not presented to save space but are available upon request from the corresponding author.

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Index adaptive market hypothesis, 124 Akaike, 28, 34, 59–60, 70 algorithm, 25, 27–28, 55, 140, 154, 164–166, 169–170 Andrews, D., 191 Ang, A., 47 APT, 3, 10 arbitrage, 3, 7–9, 20, 49, 126–127, 147, 183 ARCH, 22, 28–29, 37, 47, 61, 70–1, 103, 108, 111–112, 116, 118, 120, 123, 151–152, 155, 187 ARFIMA, 104–107, 109–111 A-Share, 49–51, 55, 58, 62–66, 68–69, 71 asymptotic, 25–29, 56–58, 88, 97, 103, 106, 114–115, 132–133 attractor, 173–177, 180, 185, 188, 190 auction, 145–146 autocorrelogram, 77–81, 87, 89–91, 94, 103 autoregression, 4, 14–16, 48, 79, 86–87, 159

Bai, J., 140 Balke, N., 102, 191 Bekaert, G., 47, 158 Bierens, H., 102 bivariate, 22, 27, 50, 85, 93–94, 98, 100, 103, 145, 147 Black, F., 140 Bollerslev, T., 71, 122 Borel, 6 Box-Jenkins, 27–28 Brownian motion, 3, 87–88 B-Share, 49–51, 55, 62, 64–66, 68–71, 73

calibration, 12 Calvo, S., 158 Campbell, J., 19, 140

canonical, 70, 78–81, 84–89, 93–96, 101–102 CAPM, 3, 10, 12, 30 Chapman-Kolmogorov, 6–7 Chen, A., 170 Chi-square, 56, 60 chaos, 101 Choleski, 23, 25 Cochrane, J., 19 cointegration, 19, 75, 77–78, 96, 98–103, 148–149, 152–153, 158, 171–181, 183–185, 187, 189–193 comovement, 23, 72, 77, 102, 144, 150–151, 154, 156–157 contagion, 143–149, 151–159 covariance, 21–25, 27–28, 34–35, 37, 42, 47, 51, 55, 73, 81, 88, 102, 109–112, 128, 155, 191 Davidson, J., 123 Diebold, F., 71 differencing, 103–107, 109, 128, 172 diffusion, 4, 8, 20, 79, 87–88, 102–103 discount, 3–5, 7–15, 17–19, 50, 182 Duffie, D., 19 Efficient market hypothesis, 124, 141–142 Eichengreen, B., 159 eigenspace, 100 Engle, R., 71, 102–103, 123, 192 ergodic, 23, 35–36, 41–42, 102, 108, 111 error correction model, 78, 144, 147, 171–173, 175–179, 181, 183, 185, 187, 189–193 Escribano, A., 192 Expectation-Maximization, 25 Fama, E., 140–141 filter, 25, 28, 38, 40, 42, 170, 172

194

GREGORIOU-3: “INDEX” — 2010/11/11 — 17:49 — PAGE 194 — #1

Index 195 Fourier, 161 fractional, 84–86, 89, 91, 101, 103–107, 109–111, 113, 115, 122–125, 128–129, 131–136, 138, 140–141 gamma, 105–106 GARCH, 51–53, 55, 57–61, 63–67, 69–72, 104, 106–113, 115, 122–123, 144, 146–148, 152–155, 157–159, 182 Gaussian, 23–24, 31, 34–35, 42, 77, 79, 81, 84–86, 89, 91, 96, 98, 115–122, 130 Geweke, J., 141 GMM, 18 Granger, C., 71, 102–103, 123, 141–142, 159, 192 Granger causality, 171–172 Grossman, S., 141 Hamilton, J., 48, 72, 192 Hannan-Quinn, 28, 34 Hansen, B., 48, 72, 123 Hansen, L., 19, 103 heteroskedastic, 23, 28, 31, 34–35, 48, 55, 72–73, 114–115, 122–123, 191 hidden-layer, 166 homogeneous, 23 hyperbolic, 78, 82, 85–86, 89, 104, 106–107, 109–110, 122 hypergeometric, 105–106 infinitesimal, 8, 87 Ingersoll, J., 19 integrable, 5–7 Jarque-Bera, 31, 56, 151, 184, 187 Johansen, S., 103, 192 Kilian, L., 192 KPSS, 133, 136, 141, 175, 180, 185–186 kurtosis, 13, 31–32, 38, 56, 151, 184

Lagrange Multiplier (LM), 26 lead-lag, 168 Likelihood Ratio (LR), 27, 60, 70 linkage, 23, 48, 50–52, 67–69, 72, 143–145, 148–149, 151, 153, 157, 171–172, 174, 177, 179, 181–188, 191 Ljiung-Box, 28 Lo, A, 140–141, 193 log-likelihood, 25, 34, 54, 57–59, 112, 178 long memory, 70–71, 77–78, 82, 85–86, 89, 92, 101–103, 106, 115, 122–124, 126, 128, 131–133, 135, 138, 140–142

Markov, 1, 3–25, 27–31, 34–43, 46–49, 51–53, 55, 57–59, 61, 63, 65, 67, 69, 71–73, 78, 81, 83, 88, 94, 96, 99, 101–102, 130, 191 Markowitz, 21, 37, 40 martingale, 3–8, 12, 19–20, 25, 28, 37, 83, 126, 141, 178 mean-variance, 21–23, 25, 27, 29–31, 35, 37, 39–45, 47 MLE, 25–26 Monte-Carlo, 79, 89 MSCI, 21, 30–31, 183, 185 MSIAH, 23–24, 28, 31, 33–34, 37 multivariate, 21, 23–25, 27–30, 37, 46, 79, 93, 101, 123

neural, 161, 163–170 nonparametric, 20, 78, 89, 102–103, 175 nonstationary, 11, 48, 72, 77–79, 97, 100–101, 110, 141, 172 Nyblom test, 114 NYSE, 49, 145

parity, 63, 77 Pearson, 114, 116, 118, 120, 183 Perron, P., 140–141 persistence, 11, 35, 52, 58, 61–62, 71–72, 75, 77–79, 81–89, 91, 93–97, 99–104, 108, 122–125,


196 Index 127, 129–131, 133, 135, 137, 139, 141, 147, 152, 173, 176 Phillips curve, 77 polynomial, 84, 89, 91–92, 106, 108, 111, 166, 170, 178, 187–188, 190 Portmanteau, 28, 31, 37, 56 premium, 12–13, 18 price dynamics, 147, 150, 160, 181, 188 probability, 5–7, 20, 25, 28–30, 35, 39, 42, 51–54, 58, 61, 63, 102, 144 Radon-Nikodym, 12 Ross, S., 19 Schwert, G., 72, 193 SDF, 4, 10, 12 Shiller, R., 19–20 Sims, C., 193 sinusoidal, 161 skewness, 13, 31–32, 56, 123, 151, 184, 187 sparse-pattern, 161, 163–167, 169–170 spillover, 50–51, 66–69, 72, 144, 158, 168 stationary, 11, 23, 48, 72, 77–82, 84, 86–89, 93–98, 100–101, 104, 106–110, 122, 128, 130–131, 133, 141, 154, 161–162, 171–172, 176–177, 183–184

sterling, 180, 191 stochastic, 3, 6, 9–12, 19–20, 51, 54, 59, 70, 85, 102, 152, 174, 183, 185 sub-prime crisis, 157 switching, 1, 21–23, 25, 27–31, 34–43, 46–49, 51–53, 55, 57–65, 67, 69–73, 130, 144–146, 152, 165, 178, 180 Taylor-MacLaurin, 104 Terasvirta, T., 103, 159–160, 192–193 threshold, 31, 78, 102, 112, 115, 122, 147–148, 154, 156, 159, 178–179, 188, 190–192 Tong, H., 103, 159 transient, 173–174, 190 transition function, 99, 101, 147–148, 152–153, 156–157, 179, 186, 189–190 unit root, 87, 129, 140–141, 150, 153, 157, 180, 191, 193 Van Dijk, D., 160, 192–193 VARMA, 27, 35 wavelet, 161–165, 167–170 Whittle, 125, 129, 132


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