Business Cycle Fluctuations and Economic Policy

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BUSINESS CYCLE FLUCTUATIONS AND ECONOMIC POLICY

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

BUSINESS CYCLE FLUCTUATIONS AND ECONOMIC POLICY

KHURSHID M. KIANI

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Kiani, Khurshid M. Business cycle fluctuations and economic policy / Khurshid M. Kiani. p. cm. ISBN 978-1-60741-406-3 (E-Book) 1. Business cycles--Kazakhstan. 2. Kazakhstan--Economic policy. I. Title. HB3711.K484 2009 338.5'42--dc22 2008046176

Published by Nova Science Publishers, Inc.   New York

CONTENTS

Preface

...................................................................................................vii

Chapter 1

Asymmetric Business Cycle Fluctuations via Time Series Models and Neural Network Linearity Tests ................... 1

Chapter 2

Neural Network Forecasts Evaluations and Business Cycle Fluctuations ..................................................................... 27

Chapter 3

A Robust Evidence of Business Cycle Asymmetries in G7 Countries.............................................................................. 45

Chapter 4

Asymmetric Business Cycle Fluctuations and Contagion Effects in G7 Countries ............................................................. 63

Chapter 5

Business Cycle Asymmetries in Asian Economies via Nonlinear Time Series Models and Neural Networks ............... 85

Index

................................................................................................. 109

PREFACE The primary purpose of this book entitled Business Cycle Fluctuations and Economic Policy is to disseminate knowledge and understanding of the techniques that can be employed for testing possible existence of similarity and asymmetries in business cycle fluctuations in macroeconomic time series. In doing so, the book assembles a number of papers that employ various parametric and non-parametric time series techniques for modeling macroeconomic time series in addition to modeling these series using the other non-parametric techniques such as artificial neural networks to study whether business cycle fluctuations in all the countries studied are alike. The question whether business cycle fluctuations in G7 countries are alike was unsuccessfully explored by previous research which poses a serious challenge to macroeconomic theorist for building new theories on business cycle fluctuations unless empirical evidence warrants that business cycle fluctuations in all the countries are alike. Therefore, to answer this question of prime empirical importance the present work employs a number of nonlinear augmented and switching time series models and artificial neural networks that are considered to be highly flexible functional form of nonlinear models to investigate whether business cycle fluctuations in G7 countries are alike. Additionally, forecast performance of the neural network vis-à-vis traditional time series models are also compared to conclude that in-sample forecast performance of neural networks is superior to the in-sample forecasts from linear and other nonlinear models but inferior to the out-of-sample performance from linear models. After establishing that business cycle fluctuations are asymmetrical in G7 countries, thus alike, the analysis is extended to a number of additional selected countries that include, developed, developing, transition and other economics particularly from the Asian region to study whether business cycle fluctuations in

viii

Khurshid M. Kiani

this heterogeneous group of countries of the world are alike. Finally, based on the results obtained from all the macroeconomic time series employed in different spells of time using various nonlinear and switching time series models, and artificial neural networks it can be concluded that business cycles in all the countries studied are asymmetrical, thus alike. The target audience of this book is graduate students, researchers, and policymakers who would particularly be interested to know the impact of a unit shock on the output in any of the countries studied that would destabilize the economy. In such situations, the policymakers would be interested to know the impact of a unit monetary shock on output so that appropriate stabilization policies e.g. monetary policy is put into action to stabilize the economy. The analyses of the book reveals that business cycles in all the countries studied (G7 plus 10 Asian economies) are asymmetrical, therefore, given all this I end up concluding that linear model forecasts including those derived from linear vector auto-regressions cannot be employed for forecasting the impact of any shock on output when the underlying data generating process is nonlinear. Indeed, the policymaker would need to employ appropriate nonlinear forecasting models to anticipate the impact of one unit monetary policy or any other shocks on the output in any of the economies studied. I hope you, the reader; find benefit in my having done so.

Chapter 1

ASYMMETRIC BUSINESS CYCLE FLUCTUATIONS VIA TIME SERIES MODELS AND

NEURAL NETWORK LINEARITY TESTS ABSTRACT The present work employs highly flexible functional form of nonlinear models that are also known as artificial neural network (ANN), and a few nonlinear time series models for constructing linearity tests that are employed for testing possible existence of business cycle asymmetries in Canada, France, Japan, UK, and USA real GDP growth rates. While results from each of the nonlinear time series models employed are used to construct one type of linearity test, linearity tests constructed from artificial neural networks approximations are of two types. One is neural network test for neglected nonlinearities, and the other type is neural network test for possible existence of nonlinearities in Canada, France, Japan, UK, and USA real GDP growth rates. Neural network tests for neglected nonlinearities show statistically significant evidence of nonlinearities in all the series. Likewise, neural network tests for possible existence of nonlinearities also show statistically significant evidence of business cycle asymmetries in all the series. Similarly, most linearity tests constructed from time series models also show statistically significant evidence of business cycle asymmetries in all of the series. Therefore, policymakers in these countries would not be able to anticipate the impact of one unit monetary policy or any other shock on the

2

Khurshid M. Kiani output based on forecasts obtained from linear models including those derived from linear vector autoregressions.

Keywords: asymmetries; business cycles; neural network; nonlinearities; principal components; real GDP

1. INTRODUCTION People were not familiar with business cycles until nineteenth century when economists thought of using new economic tools to learn the consequences of shifts in aggregate supply and demand. However, starting early twentieth century, economists started working on business cycle fluctuations and the factors affecting business cycles. However, the quest to understand business cycles gotten strengthened after the great depression of 1930s, whereupon the Keynesian macroeconomics remained in tact for some time until 70s when Lucas (1976) pioneered rational expectation hypothesis, and thereafter Kydland and Prescott (1982), and Long and Plosser (1983) showed that business cycle models do not encompass monetary factors and money management. Business cycles were classified into a number of types according to their peak-to-peak and trough-to-trough duration. Schumpeter (1939) divided business cycles into four phases i.e. recession, depression, recovery and boom. While many macroeconomic variables follow these phases, the span and the amplitude of a business cycle would depend on its type, and in general a business cycle lasts for about ten years (Sims, 1980). Possible existence of business cycle asymmetries is imperative to detect for several reasons. For instance nonlinearities imply that the effects of various monetary policy shocks on the output are not symmetric; therefore, any asymmetry would invalidate the measures of the persistence of monetary policy or any other shock on output that is based on linear models as well as linear vector autoregressions. Therefore, to validate the theories of business cycles such as real business cycles (RBC) one would need to go beyond merely matching the first and second moment of data with the moments implied by these theories. Most of the empirical research on business cycles focuses on univeriate econometric modeling techniques. For example, Neftci (1984), Brunner (1992, 1997), Beaudry and Koop (1993), Potter (1995), and Ramsey and Rothman (1996) demonstrated business cycle asymmetries in macroeconomic time series. Contrary to that Falk (1986), Sichel (1989), Delong and Summers (1986), and

Asymmetric Business Cycle Fluctuations via Time Series Models …

3

Diebold and Rudebusch (1990) have either failed to conclude that business cycles are asymmetric or found weak evidences in this prospect. Granger (1995) recommended testing linearity using heteroskedasticity-robust test. French and Sichel (1993), and Brunner (1992, 1997) demonstrated the existence of the conditional heteroskedasticity in real GNP data. Similarly, Schienkman and Lebaron (1989) reported a weakening evidence of linearity after accounting for conditional heteroskedasticity in GNP data. Blanchard and Watson (1986) demonstrated presence of outliers in GNP data. Tsay (1988) concluded that linearity could be rejected by the presence of outliers in the data. Balke and Fomby (1994), and Scheinkman and LeBaron (1989) reported weakened evidence against linearity in US real GNP data once outliers were taken into account. Thus, there is a growing perception that the evidence of nonlinearity in macroeconomic data reported in many studies so far could be because of the presence of outliers. Beaudry and Koop (1993) used an ad hoc nonlinear term in autoregressive moving average framework to find nonlinearities in macroeconomic time series, Terasvirta and Anderson (1992) used smooth transition autoregressive model to study business cycles, Eithreim and Terasvirta (1996) tested adequacy of smooth transition autoregressive model, and Luukkonen et al. (1988) used smooth transition autoregressive model to study nonlinearities. Bidarkota (1999-2000) found a robust evidence of nonlinearities even after accounting for time varying volatility, outliers and long memory in the data series. Likewise, Kiani and Bidarkota (2004) analyzed data from the group of seven (G 7) industrialized countries i.e. Canada, France, Italy, Germany, Japan, UK, and USA real GDP growth rates to test possible existence of nonlinearities in these countries and concluded that strong evidence of asymmetries does exists in Canada, Italy, Germany, Japan, and USA, but did not find any evidence of asymmetries in France and UK series. This necessitated undertaking another study to detect possible existence of business cycle asymmetries in all the series and possible existence of neglected nonlinearities in France and UK real GDP growth rates. The present work employs neural network test for neglected nonlinearities (nnw1) due to Lee et al. (1993), and neural network test for possible existence of neglected nonlinearities (nnw2) that was proposed by Terasvirta et al. (1993) for testing possible existence of nonlinearities in Canada, France, Japan, UK, and USA real GDP growth rates. In addition, time series linearity tests e.g. Keenan (1985), Tsay (1986), and Ramsey (1969) and some of its improved versions (RESET1, and RESET2) are also employed to test possible existence of asymmetric business cycle fluctuations in all the series. Finally, results obtained from both the neural network tests are compared with that of the entire set of

4

Khurshid M. Kiani

linearity test that are constructed from time series linearity tests for each of the series. The remaining work is split into the following sections. Section 2 incorporates the structure of neural network models and neural network tests. Section 3 consists of a number of time series models and nonlinearity tests based on forecasts from these models. In section 4 hypotheses tests, and nonlinearity tests results are discussed, and finally section 5 incorporates brief conclusion.

2. ARTIFICIAL NEURAL NETWORKS An artificial neural network (ANN) is an artificial intelligence technology that imitates human brain’s learning, and its executive procedure. While any individual neuron or node would be worthless, a combination of a number of neuron would make an efficient network. ANN does possess an efficient information processing capacity that enables them to learn from examples and generalize these learnings to solve complex problems never seen before (Reilly and Cooper, 1990). The traditional statistical modeling approach requires distribution of the underlying data generating processes, ANN are data driven, which can easily be implemented to solve obscure problems. That is the reason why ANN modeling approach is useful for forecasters, and researchers especially in problems where data is available but the data generating process is unknown. Therefore, ANN are treated as highly flexible functional form of nonlinear models that are independent of the distribution of the underlying data generating processes (White 1989). While a number of ANN have been proposed so far, one of the most influential neural network models is the multilayer perceptrons (MLP), which consists of several layers of nodes where information from sources external to ANN is included, whereas the output layer is the highest node, where the solution of the problem is realized. Both input and output layers are separated by a number of layers called hidden nodes (Zhang et. al. (1998). ANN can be used in time series modeling because these are highly nonlinear functional form of models, and can be structured to fit any nonlinear time series model as well as these can approximate any continuous function with desired level of precision (Hornik et al.1990). The functional relationship of a typical ANN model can be shown in following Equations:

yt = f ( yt −1, yt − 2 , yt −3 , ......, yt − k )

(1.1)


5

A typical single layer feed forward neural network as of Lee, et al. (1993) can be written as follows.

f ( x, ξ ) = θ 0 +

k

∑θ {ψ (γ w )}, '

j

t

k∈N

(1.2)

i =1

~ ' )' , where, wt = (1, w t

and

~ = ( y , y , y , ......, y )' . Equation 1.2 w t t −1 t −2 t −3 t −k

shows flexible functional forms (Lee et al. (1993), and White (1989) where ψ is a transfer function. The transfer function can be either sigmoid (logistic) or hyperbolic (tangent) cumulative distribution function. Here sigmoid function is used as transfer function, which is proposed by Lee, et al. (1993). The inputs ( y t −1 , y t − 2 , y t −3 , ......, y t − k ) are fed to hidden / signal-processing units that augment/diminish the signal by some factor γ ij where, γ i = γ i 0 , γ i1 , γ i 2 ,......, γ ik .

The

processed

signals

ψ ( ~x 'γ i ), i = 1,......., k are

fed

to

the

transfer

−x

function (ψ ( x) = (1 + e )) , which passes them to output. A single layer feed forward neural networks model is estimated which is sufficient to approximate any complex nonlinear function with desired level of accuracy (Zhang et. al. (1998)). Most researchers use one single hidden layer for forecasting purposes. However, one hidden layer may require a large number of nodes, which is not desirable because it would worsen the network generalization and increase the training time unnecessarily. Therefore, neural networks with fewer hidden nodes are preferable for forecasting as they usually have better generalization, and less over fitting problems. Likewise, a few nodes would be preferred for neural network linearity tests because the test will have less power to reject linearity when more nodes would be used. Neural networks literature contains a number of rules of thumb to select the number of nodes in a neural networks model. For example Lippmann (1987) suggests 2n + 1 nodes, Wong (1991) proposes 2n , Tang and Fishwich (1993) suggest n , Kastens et al. (1995) suggests 2 , and Kang (1991) recommends n / 2 nodes and Kiani (2005), and Kiani and Kastens (2008) recommended as low as two nodes to be added in a neural network, where n is the number of the input nodes. However, it is hard to say if any or all of these rules of thumb would be adequate for real life problems. Following Zhang et al. (1998), using trial and error approach, five nodes are selected in the neural network architecture employed which appears to be

Khurshid M. Kiani

6

adequate when using neural network linearity test for test statistic that is proposed by Lee et al. (1993). Contrary to the evidence from the empirical research discussed in the preceding paragraphs, more biased nodes are used to approximate neural networks for constructing neural network test 1 to calculate principal components of ψ t matrix based on the selection of maximum q * < q , where q is the number of maximum nodes in a neural networks model. However, for approximating nnw2 model the number of nodes is restricted to as low as two. These tests are established in the following paragraphs. Before performing neural network tests, one lag for Canada and USA, two lags for France and UK, and four lags for Japan are chosen, using Schwarz Bayesian Criterion (SBC) for the lag order selection. These lags are employed in restricted as well as unrestricted models for constructing neural network tests. yt = π ' wt + ut

(1.3)

where u t ~ N (0, σ 2 ),

and

~ ' )' , wt = (1, w t

~ = ( y ,..., y )' w t t −1 t− p

π = ( π 0 , π 1 ........, π p )' .

2.1. Neural Network Test 1 (nnw1) This test is based on in-sample forecasts obtained from nnw1 model that was proposed by Lee et al. (1993). This test is designed to test possible existence of business cycle nonlinearities in time series data. The test compares a linear model forecasts with approximations from a neural networks model. Both neural network model and its linear counterpart can be described by the following two equations:

yt = π ' wt + ut

(1.4)

where, ~ ' )' , u t ~ N (0, σ 2 ), wt = (1, w t

and

~ = ( y ,..., y )' w t t −1 t− p


7

and π = (π 0 , π 1 ,........,π p )' yt = π ' wt +

k

∑θ

0 j {ψ (γ

'

wt )} + vt

(1.5)

j =1

where, ψ (γ ' wt ) = (1 + exp{−γ ' wt )) −1 and π 0 is intercept. Equation 1.5 is a neural network model that nests a linear model of the form of Equation 1.4 . A slightly modified version of Equation 1.5 is employed to recover model predictions. The test consists of the following four-step procedure. First of all, the contemporaneous variable ( yt ) is regressed on an intercept and its lags ( y t −1 ,.......... yt −k ) using the regression equation of the form of Equation 1.4 to recover residuals ( uˆ t ), and predictions. In the second part of this test ANN of the form of Equation 1.5 are approximated, where contemporaneous variable ( yt ) is employed as dependent variable, and its lags ( y t −1 ,.......... y t − k ) are used as independent variables in the model. From the ANN approximations, ψ t matrix is recovered for principal component analyses1. This is done because the test statistics prepared by Lee et al. (1993) is hard to implement, therefore the present work employs an alternate test statistics proposed by Lee et al. However, the elements of the ψ t matrix might be collinear with themselves or with the data matrix X t . Therefore, principal component analysis of theψ t undertaken

to

eliminate

collinearity

between

its

elements

matrix is selecting

maximum q < q to obtain a matrix of principal components i.e. ψ t* matrix. *

Finally, residuals ( uˆ t ) are regressed on the matrix of principal components (ψ t* ), and the data matrix ( X t ) to recover R 2 that is used to construct the test statistics of the form of Equation 1.6 .

TS = nR 2

(1.6)

The results obtained from neural network linearity test 1 (nnw1) are shown in Table 2 . In this Table column 2 rows 1 through 5 show test statistics for

1

See Dunteman (1989) for principal component explanation

Khurshid M. Kiani

8

Canada, France, Japan, UK and USA respectively. p-values for each of the test statistic are juxtaposed in the next column of the Table.

2.2. Neural Network Test 2 (nnw2) This test is based on forecasts obtained from neural network test for possible existence of nonlinearities (nnw2) that was proposed by Tersvirta et al. (1993). The test compares a linear model forecasts with approximations from a neural networks model. Both the neural networks and its linear counterpart can be described by the following two equations: yt = π ' wt + ut

(1.7)

where, ~ ' )' , wt = (1, w t

u t ~ Nid (0, σ 2 ),

and

~ = ( y ,..., y )' w t t −1 t− p

π = (π 0 , π 1 ,........,π p )'

uˆt = π ' wt +

k

∑θ

0 j {ψ

(γ j wt )} + vt '

(1.8)

j =1

where, ψ (γ ' wt ) = (1 + exp{−γ ' wt )) −1 and π 0 is intercept. Equation 1.7 shows a linear model whereas the Equation 1.8 shows a neural networks model that nests the linear model represented by Equation 1.7 . In the first step yt is regressed on an intercept and, y t −1 ,.......... y t − k as per regression Equation 1.7 . From this regression, residual ( uˆ t ) are recovered that are subsequently employed with yt −1 ,.......... y t −k to approximate a neural network model as per Equation 1.8 . From this model, residuals ( vˆt ) are recovered for constructing the test statistics in conjunction with residual ( uˆ t ) recovered from the earlier part as per Equation 1.9 .

TS = {( SSE1 − SSE 2) / m} /{SSE 2 /(n − p − m − 1)}

(1.9)

where, m denotes the number of restrictions in the unrestricted model, n is the number of observations, and p is the number of lags in the regression equations.


9

The test statistics (TS) is distributed F [m, (n − p − m − 1)] under normality hypothesis. Table 2 presents results from the test that is constructed form neural network 2 approximations for Canada, France, Japan, UK, and USA. In this Table row 6 to 10 in column 2 show test statistics respectively for Canada, France, Japan, UK, and USA real GDP growth rates. p-values for each test are juxtaposed in parentheses in the next column.

2.3. Estimation Issues Neural network models are employed to test nonlinearities in all the series, which is a nonparametric approach. While estimating neural network models convergence appeared to be difficult because of the additive linear term in the neural network models. To avoid the likelihood of obtaining local optima, genetic algorithm (GA) was employed with a couple of random starts to obtain the best parameter vector for neural network approximations. GA is considered to be a reliable estimation algorithm but it appeared to be very slow, therefore, a nonlinear optimization routine was combined with the GA, which worked satisfactorily although it was still slower than any other optimization routine. Following De Jong (1975), GA was used in biology, engineering and operation research (Goldberg 1989). The first ever, economic application of GA was implemented by Axelord (1987), and thereafter by Marimon, McGartten, and Sargent (1990), and Dorsey and Walter (1995).

2.4. Data Sources Quarterly data on real GDP growth rates series for Canada, France, Japan, UK and USA is obtained from the International Financial Statistic (IFS)’s CDROM for the month of September 2001. Canada, Japan, UK and USA data range from 1957:1 to 2000:4, whereas the France data range from 1970:1 to 2000:4. The analysis started with the data on real GDP growth rates for G7 countries, i.e. Canada, France, Japan, UK and USA, but there was a jump of 35.43 percent in Germany real GDP growth rates in 1991:1. This likely would have been because of its reunification. Italy showed an inexplicable spike of 87.5 percent in 1970:1. Therefore, series for Italy and Germany were excluded from the analysis. Table 1.1 describes the data, and Figure 1 shows plots of all the data series employed.

Khurshid M. Kiani

10 Canada

F ra n c e

0.04

0.02 0.01

0.02 0 0 -0 . 0 1 -0 . 0 2 1950

1960

1970

1980

1990

2 0 00

2010

-0 . 0 2 1970

1 9 75

1980

Ja p a n

1985

1990

1995

2000

2005

UK

0.15

0.06 0.04

0.1

0.02 0.05 0 0 -0 . 0 5 1950

-0 . 0 2 1960

1970

1980

1990

2 0 00

2010

1990

2 0 00

2010

-0 . 0 4 1950

1 96 0

1970

1980

19 9 0

2000

2010

US A

0.04 0.02 0 -0 . 0 2 -0 . 0 4 1950

G D P G ro w t h R a t e s O ve r Tim e 1960

1970

1980

Figure 1. Plot of Real GDP Growth Rates over Time.

2.5. Preliminary Data Analysis Table 1.2 provides some useful statistics on the raw data and Table 1.2 summarizes results of routine statistical tests for some interesting data hypotheses. The average annualized quarterly GDP growth rates range from 2.44 to 8.84 percent, with the average growth rates for all countries being significantly positive. UK has the lowest, and Japan has the highest growth rate. The quarterly (annualized) standard deviations range from 2.40 percent for France to 7.80 percent for Japan. Skewness measure ranges from –0.54 for France to 0.64 for Japan, being statistically significant for all countries except Canada. Excess kurtosis measures range from 0.14 for Canada to 2.85 for the UK, with significant fat tails found in UK, and USA, and marginally in Japan. The Jarque-Bera test rejects normality for all countries except Canada. These preliminary results are subject to some qualification because the tests are based on the assumption that the growth rates are independently and identically distributed (iid) normal. As will be evident later, this assumption is not appropriate. The Augmented Dicky Fuller (ADF) test indicates unit roots in levels (with constant and time trend) for all countries but not in growth rates (with constant only). The only exception is Japan, for which the test fails to reject unit roots in growth rates with a constant term only but does reject with constant and time trend. A Goldfeld-Quandt test


11

fails to reject homoskedasticity in all countries, and the Lagrange Multiplier (LM) test detects autoregressive conditional heteroskedasticity (ARCH) only in Japan. Table 1.1. Data Description

Data Series Sample Period

Canada Quarterly Real GDP 1957:12000:4 176

France Quarterly Real GDP 1970:12000:4 124

Japan Quarterly Nominal GDP 1957:12000:4 176

UK Quarterly Real GDP 1957:12000:4 176

USA Quarterly Real GDP 1957:12000:4 176

Sample Length Notes: 1. The quarterly seasonally adjusted real GDP growth rates for Canada, France, Japan, UK, and USA were obtained from the September 2001 edition of the International Financial Statistics (IFS) CD-ROM. 2. The nominal GDP series are employed for Japan because seasonally adjusted data were only available for the nominal series and not for the real series on the IFS CD-ROM.

Table 1.2. Summary Statistics Canada 0.91 Mean (p-value for mean = 0) (0.00) Standard Deviation 0.97 0.17 Skewness (p-value for skewness = 0) (0.49) 0.14 Excess Kurtosis (p-value for excess kurtosis = 0) (0.32) 0.21 Jarque-Bera Test (p-value for normality) (0.89) ADF Test for Unit Roots Levels (constant + trend) -2.24 First Differences (constant only) -4.59 Orders of Significant 1-3 Autocorrelations (at 5% level) Orders of Significant 1,11 Partial Autocorrelations (at 5% level) Goldfeld-Quandt Test 24.81 (p-value for homoskedasticity) (1.00)

France 0.63 (0.00) 0.60 -0.54 (0.01) 0.48 (0.14) 7.16 (0.03)

Japan 2.21 (0.00) 1.95 0.64 (0.00) 0.57 (0.06) 14.15 (0.00)

UK 0.61 (0.00) 1.05 0.36 (0.03) 2.85 (0.00) 62.8 (0.00)

USA 0.83 (0.00) 0.95 -0.44 (0.01) 1.26 (0.00) 17.16 (0.00)

-2.25 -3.99 1,2

-1.39 -4.64 3

0.14 -4.83 1,2

1,2

-1.83 -1.80 1-10, 12, 13, 16 1-5,8,9

3,8,16,31 1,2

20.1 (1.00)

36.21 (0.99)

14.72 (1.00)

13.67 (1.00)

Khurshid M. Kiani

12

Table 1.2. (Continued) Canada France 0.01 0.00 (0.91) (0.95)

LM Test for ARCH (p-value for no ARCH effects)

Japan 24.51 (0.00)

UK 0.01 (0.91)

USA 0.03 (0.85)

Notes: 1. All statistics reported in this Table are for quarterly growth rates defined as 100Δ log yt , except the ADF test in levels for which the test statistics are for log levels log y t . 2. p-values for statistics pertaining to skewness, excess kurtosis, Jarque-Bera, GoldfeldQuandt, and the LM test for ARCH are reported in parentheses beneath each test statistic. 3. Optimal lag order for the ADF tests is chosen using the minimum SBC criterion. Asymptotic critical values at the 10% significance level for the model with constant, and with constant and time trend, are -2.57 and -3.13, respectively. 4. Optimal lag order for the LM test is chosen using the minimum SBC criterion.

Table 1.3. Summaries of Statistical Inferences

Nonzero Mean Skewness Fat Tails Non-Normality Unit Roots Levels (constant + trend) First Differences (constant only) Homoskedasticity ARCH Effects

Canada Y N N N

France Y N N Y

Japan Y Y N Y

UK Y Y Y Y

USA Y N Y Y

Y N

Y N

Y Y

Y N

Y N

Y N

Y N

Y Y

Y N

Y N

Notes: 1. ‘ Y ’ means that a given hypothesis is true. 2. ‘ X ’ means that a given hypothesis is false.

Table 2. Neural Network Tests Results Tests Neural Network Test 1

Test Statistics 124.83 31.36 85.58

p-values (0.00) (0.00) (0.00)


13

Table 2. (Continued) Tests

Test Statistics 35.79 26.31 342.69 178.35 73.04 268.47 503.72

Neural Network Test 2

p-values (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)

Notes: 1. Neural network test compares neural network approximations to forecasts from linear model that is nested in neural network model. 2. Both neural network and linear models are shown in the following Equations:

yt = π ' wt + ut

(2a)

where,

~ ' )' , u t ~ N (0, σ 2 ), wt = (1, w t

and

~ = ( y ,..., y )' w t t −1 t− p

and π = (π 0 , π 1 ,........,π p )' yt = π ' wt +

k

∑θ

0 j {ψ (γ

'

wt )} + vt

(2b)

j =1

~ ' )' , where, wt = (1, w t

~ = ( y , y , y , ......, y )' , and w t t −1 t −2 t −3 t −k

ψ

is a transfer

function. 3. Test statistics for neural network test 1 are presented in column 2 . In this column the first row denotes test statistic constructed from neural network test 1 for Canada, second France, third Japan, fourth UK, and fifth USA. The test statistics are constructed using Equation shown below:

TS = nR 2 where,

uˆ t on

(2c)

n is the number of observation, and R X t'

and

ψ t* (

2

is obtained from regressing regress

a matrix of principal components).

2 * 4. ‘‘TS” denotes calculated values of ‘Test Statistics” that is distributed χ ( p ) under

the assumption of normality. 5. p-values for test statistics for Canada, France, Japan, UK, and USA are juxtaposed with each of the test statistic in parenthesis. 6. The neural network and linear models for neural network test 2 are shown in the following Equations:

Khurshid M. Kiani

14 yt = π ' wt + ut

(2d)

where,

~ ' )' , wt = (1, w t

u t ~ Nid (0, σ 2 ),

and

~ = ( y ,..., y )' w t t −1 t− p

π = (π 0 , π 1 ,........,π p )'

uˆt = π ' wt +

k

∑θ

0 j {ψ

(γ j ' wt )} + vt

(2e)

j =1

where, ψ (γ ' wt ) = (1 + exp{−γ ' wt )) −1 and

π0

is intercept.

7. Column 2 rows 5 to 10 show test statistics for neural network test 2 for Canada, France, Japan, UK, and USA respectively. The test statistic is constructed using following Equation: TS = {( SSE1 − SSE 2) / m} /{SSE 2 /(n − p − m − 1)} (2f) where in Equation 2 f , model,

m denotes

the number of restrictions in the unrestricted

n is the number of observations, and p is the number of lags in the regression

equations selected by SBC criterion. The test statistics (TS) is distributed F [m, (n − p − m − 1)] under normality hypothesis. 8. p-values for test statistics for Canada, France, Japan, UK and USA are juxtaposed with the test statistics in parentheses.

3. TIME SERIES LINEARITY TESTS 3.1. Keenan Test Keenan test is a linearity test that was proposed by Keenan (1985). This test is designed to test if squared forecasts have any additional forecasting power. The test consists of the following four steps. In the first step of this test yt is regressed on an intercept and lags ( ys −1 , ys −2 ,..................., ys − M ) of yt to recover residuals ( u t ) for s = M + 1,.........., n . Thereafter, squared predictions ( f t 2 ) are regressed on an intercept and lags ( y s −1 , y s − 2 ,..................., y s − M ) of yt . Thereafter, residuals from linear regression ( uˆ ) are regressed on residuals from nonlinear regression ( vˆ ) for obtaining predictions ηˆ and estimates of η 0 ( ηˆ0 ) using Equation 1.10 .


ηˆ = ηˆ0 (

15

n

∑v )

2 1\ 2 t

(1.10)

i = M +1

Finally, the test statistics ( TS ) is calculated using SSE1 and ηˆ as per Equation 1.11 .

TS = (ηˆ 2 (2 − 2M − 2)) /( SSE1 − ηˆ 2 )

(1.11)

where, TS is distributed F [1, (n-2M-2)] under the assumptions of normality. Test results based on Keenan model predictions are presented in Table 3 . In this Table rows1 to 5 in column 2 show test statistics for Keenan Test for Canada, France, Japan, UK, and USA real GDP growth rates. p-values for each test statistic are juxtaposed in parenthesis in the subsequent column. Table 3. Other Nonlinearity Test Results Test Keenan Test

Tsay Test

RESET

RESET1

RESET2

Test Statistics 6.78 6.86 6.79 6.79 6.78 163.60 116.98 163.10 167.47 169.37 6.00e+29 5.00e+30 1.42e+00 6.00e+24 2.00e+30 7.35 0.53 0.00 0.17 8.89 19.26 29.41

p-values (0.99) (0.98) (0.99) (0.99) (0.99) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.32) (0.00) (0.00) (0.01) (0.59) (1.00) (0.95) (0.99) (0.00) (0.00)

Khurshid M. Kiani

16

Table 3. (Continued) Test

Test Statistics 17.50 174.91 26.61

p-values (0.00) (0.00) (0.00)

Notes: 1. Model 1 is the most general nonlinear model that nests linear model to construct linearity test. 2. All test statistics for each of the series employed for all the tests are shown in column 2 of the Table. 3. In this Table, rows 1 to 5 show test statistics for Keenan Test, 6 to 10 for Tsay Test, 11 to 15 for RESET, 16 to 20 for RESET1, and finally rows 21 to 25 for RESET2 respectively for Canada, France, Japan, UK, and USA. 4. Test Statistic for Keenan Test is distributed F [1, n-2p-2] under the assumption of

normality which is calculated using Equation 3a .

TS = (ηˆ 2 (2 − 2M − 2)) /( SSE1 − ηˆ 2 )

(3a)

where, ηˆ = ηˆ 0 ( SSE )1 / 2 ηˆ = η 0^ (SSE2)1/2. ηˆ 0 is the regression coefficient, which is obtained from regressing uˆ (errors from the linear model) on vˆ (errors from the nonlinear models). 5. Test statistic for Tsay test is distributed as χ2 with p ( p + 1) / 2 degrees of freedom. The test statistic for Tsay test is calculated using Equation 3b .

TS = T ( SSE0 − SSE

(3b)

where, SSE1 are the residual sum of squares from unrestricted model, SSE 0 from the restricted model, and T is the number of the observations. 6. Test statistic for RESET is approximately distributed F [k-1, n-k] when Ho is true. This test statistic is calculated using Equation 3c .

TS = ((SSE0 − SSE1 ) /(k − 1)) /( SSE

(3c)

where SSE1 are the residual sum of squares from unrestricted model, and

n is

the

number of the observations in the model, and k is the number of the parameter estimated in the model. 7. The test statistic for RESET1 is calculated using Equation 3d .

TS = ((SSE2 − SSE1 ) /( p* )) /( SSE2 /(n − k ))

(3d)

where, SSE1 is constructed from the residual sum of squares from unrestricted model and

n is

the number of the observations in the model,

k is

the number of the

parameter estimated in the model, and p * is chosen in such a way that p* < (k − 1) . The test statistic (TS) is approximately distributed F [p*, n-k] when Ho is true.


17

8. The test statistic for RESET2 is approximately distributed χ ( p * ) when Ho is true which can be calculated from the following Equation.

TS = nR 2

(3e)

where, R 2 is obtained from the second part of the test and, n is the total number of observation in the data series being tested. 9. p-values for each test statistics for Canada, France, Japan, UK and USA are juxtaposed in the next column in parentheses.

3.2. Tsay Test Tsay (1986) proposed this test for possible existence of nonlinearities in time series data, and is also being employed in the present work for testing possible existence of business cycle asymmetries in all the series. This test is an improvement over the Keenan test because additional cross terms p( p + 1) / 2 are added in the nonlinear model. This test can be performed in three steps. In the first step the contemporaneous variable y t is regressed on an intercept and lags ( y t −1 ,.........., y t − k ) of y t for k = 1,..........., p to recover residuals ( uˆt ) for t = 1,......., T . In the second step, residuals ( uˆt ) obtained from the first step are

regressed on the cross products ( y t − j y t − k ) for k ≥ j , j , k = 1,..........., p to recover residuals ( vˆt ) for t = 1,......., T . Finally, in the third step, the test statistics (TS) is computed using Equation 1.12 . The test statistic is distributed χ2 with p( p + 1) / 2 degrees of freedom.

TS = T ( SSE0 − SSE1 ) / SSE0

(1.12)

Test statistics for linearity tests based on Tsay model for all the series are presented in Table 3 . In this Table rows 6 to 10 in column 2 show test statistics for Tsay Test for Canada, France, Japan, UK, and USA real GDP growth rates. pvalues for each test statistic are juxtaposed in parentheses in the subsequent column.

Khurshid M. Kiani

18

3.3. Ramsey Test RESET Another test for testing possible existence of business cycle asymmetries due to Ramsey (1969) is also employed for testing possible existence of business cycle asymmetries in all the series. This test employs polynomial of the forecast ( Yhat1 = f t ), for an alternate formulation of nonlinear models that nests linear model. This test can be completed in three steps. In the first step of this test, the contemporaneous variable y t is regressed on an intercept, and lags ( y t −1 ,.........., y t − k ) of y t for k = 1,..........., p for recovering residuals ( uˆ t ) for t = 1,......., T . In the second step, y t is regressed on forecasts ( f t ), and

(

)

polynomials of forecasts f t , f t1 , f t 2 , f t3 ,.... using Equation 1.13 for any k ≥ 2 to recover residuals ( vˆt ), for t = 1,......., T .

yt = f t + β1 f t 2 + β 2 f t3 + ........ + β k f t k +

(1.13)

In the last step of this test, the test statistics (TS) is computed using Equation 1.14 .

TS = ((SSE0 − SSE1 ) /(k − 1)) /( SSE0 /(n − k ))

(1.14)

where, in Equation 1.14 , k is the number of polynomials of forecasts, and n is the total number of observation in the data series being tested. The test statistic is approximately distributed F [k-1, n-k] assuming that the null hypothesis of linearity is true. Table 3 shows test statistics for linearity tests for all the series that are based on RESET model predictions. Rows11 to 15 in column 2 show test statistics for RESET Test for Canada, France, Japan, UK, and USA real GDP growth rates. pvalues for each of the test statistic are juxtaposed in parentheses in the subsequent column.

3.4. RESET1 This linearity test incorporates principal component analysis of the polynomial of forecasts ( f t1 ,.............., f t k ) for the largest p * < (k − 1) , and regressing y t on these principal components and linear term. In the first part of


19

this test the contemporaneous variable y t is regressed on an intercept, and its lags [ y t −1 ,.........., y t − k ] for k = 1,..........., p to recover residuals ( uˆt ) for t = 1,......., T . In the second step of this test y t is regressed on f t , and principal components of the polynomials of f t ( f t1 ,.............., f t k ) as per regression Equation 1.15 to recover residuals ( vˆt ).

yt = f t + β1 f t 2 + β 2 f t3 + ........ + β k

(1.15)

Finally, the test statistics are computed using Equation 1.16 .

TS = ((SSE2 − SSE1 ) /( p* )) /( SSE2 /(n − k

(1.16)

where, in Equation 1.16 , k is the number of polynomials of forecast, n is the total number of observation in the data series being tested, and p * is chosen in such a way that p * < (k − 1) . The test statistic (TS) is approximately distributed * F [p , n-k] when Ho is true. Test statistics for linearity tests based on RESET1 predictions for all the series are presented in Table 3 , where, rows 16 to 20 in column 2 show test statistics for RESET1 Test respectively for Canada, France, Japan, UK, and USA real GDP growth rates. p-values for each test are juxtaposed in parentheses in the next column.

3.5. RESET2 RESET2 is a different version of the RESET linearity test that also encompasses principal components of the polynomials of the forecast ( f t1 ,.............., f t k ) from the regression in the first part of the test. This test consists of the following three-step procedure. In the first step of this test the contemporaneous variable y t is regressed on an intercept and lags

[y t −1 ,.........., y t −k ]

of y t for k = 1,..........., p in order to recover residuals ( uˆt )

for t = 1,......., T . In the second step, residuals ( uˆt ) are regressed on an intercept,

Khurshid M. Kiani

20

and lagged dependent variables [ y t −1 ,.........., y t − k ] of y t for j = 1,..........., p , and principal components of polynomials of f t using Equation 1.17 for the largest p * < (k − 1) to recover residuals ( vˆt ) for t = 1,......., T and R 2 , where both k and p * are arbitrarily selected.

yt = f t + β1 f t 2 + β 2 f t3 +,........,+ β k f t k + vt

(1.17)

where, in Equation 1.17 f t = Xˆ t'θ , and finally the test statistic is computed using Equation 1.18.

TS = nR 2

(1.18)

where R 2 is obtained from the second step of the test, n is the total number of observations in the data series being tested. The test statistic is approximately distributed χ 2 ( p * ) assuming that the null hypothesis is true. Test statistic for linearity test for each of the series based on RESET2 predictions is shown in Table 3 , where rows 21 to 25 in column 2 show test statistics for RESET2 tests respectively for Canada, France, Japan, UK, and USA real GDP growth rates. pvalues for each test are juxtaposed in parentheses in the next column.

4. EMPIRICAL RESULTS 4.1. Hypotheses Tests Two types of neural networks tests are employed to find possible existence of business cycle asymmetries in real GDP growth rates for Canada, France, Japan, UK, and USA. Results from neural network linearity tests are compared with the results for linearity tests that are constructed from other time series models employed. Comparing linearity tests results that are constructed from forecast from time series models with that of neural network linearity tests, it transpires that the neural network approximations appeared to be better for constructing linearity tests for all the series.


21

The null hypothesis both for neural network and other linearity tests2 is linearity, against the alternative hypotheses of nonlinearity. Nonlinearities would prevail if the null is not true, alternately nonlinearities would not prevail when the null would be true.

4.2. Empirical Results on Hypotheses Tests Empirical results from various linearity tests based on neural network tests are shown in Table 2 . Likewise, empirical results based on linearity test constructed from nonlinear time series models i.e. Keenan Test, Tsay Test, and Ramsey Test (RESET), RESET1, and RESET2 are presented in Table 3 . All these tests are based on test statistics proposed by different researchers, which are constructed in most cases from the residuals from a linear model, and nonlinear model that nests the linear model. Neural networks test for neglected nonlinearities (nnw1) accepts the alternative hypothesis against the null of linearity for Canada, France, UK and USA at 5 percent level of significance. Likewise, a neural network test for possible existence of nonlinearities (nnw2) also accepts the alternative hypothesis against the null of linearity for Canada, France, UK and USA at 5 percent level of significance. The inferences do not change when significance level is changed from 5 to 10 percent. The Keenan test does not reject the null of linearity for Canada, France, Japan, UK and USA. Alternately, the Tsay test rejects linearity hypothesis in Canada, France, Japan, UK and USA real GDP growth rates. Similarly, RESET rejects the null hypothesis of linearity against alternative of nonlinearities for Canada, France, UK and USA. However, the test fails to rejects null hypothesis for Japan only. On the other hands the RESET1 fails to rejects the null hypothesis of linearity against alternative for France, Japan, UK and USA. Test rejects linearity hypothesis for Canada at 5 percent level of significance. However, the RESET2 rejects the null hypothesis against alternative for Canada, France, Japan, UK and USA. All the inferences do not change when significance level is changed from 5 to 10 percent.

2

Keenan Test, Tsay Test, and Ramsey Test (RESET), RESET1, and RESET2

22

Khurshid M. Kiani

5. CONCLUSION Linearity tests constructed from various time series models i.e. Keenan test due to Keenan (1985), Tsay test due to Tsay (1986), Ramsay test due to Ramsay (1969), and some of its improved versions i.e. RESET1 and RESET2 are employed to test possible existence of business cycle asymmetries in Canada, France, Japan, UK, and USA real GDP growth rate series. Additionally, neural network test for neglected nonlinearities (nnw1) due to Lee et al (1993), and neural network test for possible existence of nonlinearities (nnw2) proposed by Terasvirta et al. (1993) that are constructed from in-sample approximations from neural networks are employed to test possible existence of business cycle asymmetries in Canada, France, Japan, UK, and USA series. The results show that linearity hypothesis is rejected for linearity tests constructed from neural network test for neglected nonlinearities (nnw1), and neural network test for possible existence of nonlinearities (nnw2) for Canada, France, Japan, UK, and USA real GDP growth rates. Likewise, the null hypothesis is easily rejected for Tsay and RESET. RESET1 rejects the null hypothesis in five whereas RESET accepts the null in four out of five series studies. These results show that both the neural network linearity tests (nnw1 and nnw2) outperform the linearity hypotheses in all the series when compared to the linearity test constructed from all the time series models employed. The results on linearity tests are in line with previous studies including Lee, et al. (1993) and Terasvirta, et al. (1993). The results on linearity in the conditional mean for USA are in line with Bidarkota (2000) and Kiani and Bidarkota (2004). This shows that evidence against linearity for USA is robust to changes in sample as well as change in testing approach. Similarly evidence against linearity is also robust for Canada, and Japan based on testing approach. Neural Networks models outperformed nonlinear time series models for testing possible existence of nonlinearities, which is in line with the previous studies due to Lee, et al. (1993), and Terasvirta, et al. (1993). Compared to Kiani and Bidarkota (2004), the results on nonlinearity show robust evidence of neglected nonlinearities in France and UK. Thus, the results show statistically significant evidence of business cycle asymmetries in Canada, France, Japan, UK, and USA. However, compared to Kiani and Bidarkota (2004), there exists an evidence of neglected nonlinearities in France, and UK real GDP growth rates.


23

6. REFERENCES Axelrod, R. (1987). The evolution of the strategies in the iterated prisoner’s dilemma: Algorithm and Simulated Annealing. Morgan Kaufmann, Los Altos: CA. Balke, N. & Fomby, T. (1994). Large shocks, small shocks and economic fluctuations: outliers in macroeconomic time series. Journal of Applied Econometrics,9, 181-200. Beaudry, P., & Koop, G. (1993). Do recessions permanently change output? Journal of Monetary Economics, 31, 149-163. Bidarkota, P. (1999). Sectoral investigation of asymmetries in the conditional mean dynamics of the real US GDP. Studies in Nonlinear Dynamics and Econometrics, 3, 191-200. Bidarkota, P. (2000). Asymmetries in the conditional mean dynamics of real GNP: robust evidence. The Review of Economics and Statistics, 82, 153-157. Blanchard, O., & Watson, M. (1986). Are Business Cycles All Alike? In Gordon, R.J., (Eds.), The American Business Cycle: Continuity and Change, University of Chicago Press, Chicago. Brunner, A. (1992). Conditional asymmetries in real GNP: a semi-nonparametric approach. Journal of Business & Economics Statistics, 10, 65-72. Brunner, A. (1997). On the dynamic properties of asymmetric models of real GNP. The Review of Economics and Statistics, 79, 321-326. DeLong, J., & Summers, L. (1986). Are business cycles symmetrical? in Gordon R (ed.), The American Business Cycle: Continuity and Change. Chicago, IL: University of Chicago Press. De Jong, K. (1975). An analysis of the behavior of a class of Genitive Adaptive System. Unpublished Ph.D. dissertation, University of Michigan, Department of Computer Science. Dorsey, R., & Walter, J. (1995). Algorithm for Estimation Problems with Multiple Optima, Nondifferentiability, and other Irregular Features. Journal of Business Economics and Statistics, 13, 53-66. Diebold, F., & Rudebusch, G. (1990). A nonparametric investigation of duration dependence in the American business cycle. Journal of Political Economy, 98, 596-616. Dunteman, G. (1989). Principal components analysis. Sage: Newbury Park. Eitrheim, O., & Terasvirta, T. (1996). Testing adequacy of smooth transition autoregressive models. Journal of Econometrics, 74, 59 -75.

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Falk, B. (1986). Further evidence on the asymmetric behavior of economic time series over the business cycle. Journal of Political Economy, 94, 1096-1109. French, M., & Sichel, D. (1993). Cyclical patterns in the variance of economic activity. Journal of Business and Economic Statistics, 11, 113-119. Granger, C. (1995). Modeling nonlinear relationships between extended-memory variables. Econometrica, 63, 265-279. Goldberg, D. (1989). Genetic Algorithm in Search, Optimization and Machine Learnings. Addison Wisely: Readings, MA. Hornik, K., Stinchcombe, M., and White, H. (1990). Universal approximation of an unknown mapping and its derivatives using multilayer feed forward neural networks. Neural networks, 3, 551-560. Kang, S. (1991). An introduction of the Use of Feed Forward Neural Networks for Forecasting. Ph.D. Thesis: Kent State University. Kastens T., Featherstone, A., and Biere, W. (1995). A neural network premier for agricultural economists. Agricultural Finance Review, 55, 54-73. Keenan, D. (1985). A Tucky nonlinear type test for time series nonlinearities. Biometrika, 72, 39- 44. Kiani, K.M., & Bidarkota, P. V. (2004). On business cycle asymmetries in G7 countries. Oxford Bulletin of Economics and Statistics, 66, 333-353. Kiani, K. M. (2005). Detecting business cycle asymmetries using artificial neural networks and time series models. Computational Economics, 2, 65-89. Kiani, K. M., & Kastens, T.L. (2008). Testing forecast accuracy of foreign exchange rates: predictions from feed forward and various recurrent neural network architectures,” Computational Economics, (forthcoming). Kydland, F., Prescott, E. (1982). Time to build and aggregate fluctuations. Econometrica, 50, 1345-70. Lee, T., White, H., & Granger, C. (1993). Testing for neglected nonlinearity. Journal of Econometrics, 56, 269-290. Lippmann, R. (1987). An introduction to computing with neural nets. IEEE ASSP magazine, April, 4-22. Long, J., & Plosser, C. (1983). Real business cycles. Journal of Political Economy, 91, 39-69. Lucas, R. (1976). Econometric policy evaluation: a critique. Carnegie-Rochester Conference Series in Public Policy, 1, 19-46. Luukkonen, R., Saikkonen, P., & Terasvirta, T. (1988). Testing linearity against smooth transition autoregressive models. Biometrika, 75, 491- 499. Marimon, R., McGratten, E., & Sargent, T. (1990). Money as medium of exchange in an economy with artificially intelligent agents. Journal of Economics Dynamics and Control, 14, 329- 373.


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Neftci, S. (1984). Are Economic time series asymmetric over the business cycle? Journal of Political Economy, 92, 307-328. Potter S. (1995). A non-linear approach to U.S. GNP. Journal of Applied Econometrics, 10, 109-125. Ramsey, J. (1969). Tests for specification errors in classical linear least-squares regression analysis. Journal of the Royal Statistical Society, 31, 350-371. Ramsey, J., & Rothman, P. (1996). Time irreversibility and business cycle asymmetry. Journal of Money Credit and Banking, 28, 1-21. Reilly, D., & Cooper, L. (1990). An overview of neural networks: early models to real world systems. In Zornetzer, S. F., Davis, J. L., & Lau, C. (Eds.), An Introduction to Neural and Electronic Networks, Academic Press, New York. Scheinkman, J., and LeBaron, B. (1989). Non-linear dynamics and GNP data. In Barnett, W. et al. (Eds.), Economic Complexity: Chaos, Sunspots, Bubbles, and Non-linearity, Cambridge University Press, Cambridge. Schumpeter, J. (1939). Business cycles: a theoretical, historical, and statistical analysis of the capitalist process. McGraw-Hill:New York. Sichel, D. (1989). Are business cycles asymmetric? a correction. Journal of Political Economy, 97, 1255-1260. Sims, K. (1980). Macroeconomic reality. Econometric, 48, 1-49. Tang, Z., & Fishwick, A. (1993). Feed forward neural networks as models for time series forecasting. ORSA Journal of Computing, 5, 374-385. Terasvirta, T., Lin, C., & Granger, C. (1993). Power of neural network test. Journal of Time Series Analysis, 14, 209-220. Terasvirta, T., & Anderson, H. (1992). Characterizing nonlinearities in business cycles using smooth transition autoregressive models. Journal of Applied Econometrics, 7, S119-S136. Tsay, R. (1986) Non-linearity tests for time series. Biometrika, 73, 461-466. Tsay, R. (1988). Nonlinear time series analysis of blowfly population. Journal of Time Series Analysis, 9, 247-263. White, H. (1989). Learning in artificial neural networks: a statistical perspective. Neural Computations, 1, 425-464. Wong, F.W. (1991). Time series forecasting using back propagation using neural networks. Neurocomputing, 2, 147-149. Zhang, G., Patuwo, B., & Hu, M. (1998). Forecasting with artificial neural networks: the state of arts. International Journal of Forecasting, 14, 35-62.

Chapter 2

NEURAL NETWORK FORECASTS EVALUATIONS AND BUSINESS CYCLE FLUCTUATIONS ABSTRACT The present work employs in-sample forecasts from linear models in conjunction with neural network approximations to construct neural network linearity tests for testing possible existence of business cycle asymmetries in Canada, France, Japan, UK, and USA, real GDP growth rates. Additionally, neural network linearity tests are constructed from jackknife out-of-sample forecasts from linear models as well as neural networks approximations for testing possible existence of business cycle asymmetries in Canada, France, Japan, UK, and USA real GDP growth rates. Finally in-sample as well as jackknife out-of-sample forecasts from linear models are compared with those of neural network approximations using a pair-wise test of forecast evaluation due to Ashley, Granger, and Schamalensee (1980) to determine suitable forecasting model for a particular situation. These results show that statistically significant evidence of business cycle asymmetries does exist in all the series. Moreover, in-sample forecast performance of neural networks is superior to linear models, however, when using out-of-sample forecasts linear models would be a better choice.

Keywords: AGS test; asymmetries; business cycles; in-sample forecasts; neural network tests; out-of-sample forecasts JEL codes: C32, C45, E37

28

Khurshid M. Kiani

1. INTRODUCTION Most studies on business cycle research are focused on univariate nonlinear econometric modeling. Using univariate nonlinear time series models Neftci (1984), Brunner (1992, 1997), Beaudry and Koop (1993), Potter (1995), and Ramsey and Rothman (1996) concluded that business cycles are characterized by asymmetric fluctuations. However, Falk (1986), Sichel (1989), Delong and Summers (1989), and Diebold and Rudebusch (1990), either found a weak evidence of asymmetries in business cycles fluctuations or failed to conclude that business cycles are asymmetric. Studies by Bidarkota (1999, 2000) provided strong evidence of asymmetries in USA macroeconomic time series data. However, when Kiani and Bidarkota (2004) conducted similar analyses on Canada, France, Germany, Italy, Japan, UK, and USA, they found strong evidence of asymmetries in real GDP growth rates for Canada, Germany, Italy, Japan, and USA, but they were not able to find asymmetries in France and UK real GDP growth rates. Therefore, the basic research question whether business cycle asymmetries in all the countries of the world particularly in G7 countries are alike remained unanswered. Therefore, using artificial neural networks (ANN), the present work focuses on two issues. One, detecting business cycle asymmetries in Canada, Germany, Japan, and USA, via in-sample testing as well as jackknife out-of-sample testing, and two, testing in-sample as well as jackknife out-of –sample forecasts performance of linear models versus artificial neural networks. A number of nonlinear time series models were developed in the recent past, this includes models due to Auerbach (1982), Beaudry and Koop (1993), Gordon (1986), Kling (1987), Koch and Rasche (1988), Diebold and Rudebusch (1989), Hamilton (1989), and Estrella and Mishkin (1998). However, because of ample need for additional nonlinear time series models in this area, Kuan and White (1994) including others specifically discussed neural networks and their applications in economics. Neural networks have been used successfully in science, engineering, medical, business and economics. Although neural networks have demonstrated some success in financial and economics applications, only a few studies, for example Vishwakarma (1995), and Qi (2001) have focused on business cycles. Therefore a study of business cycles might benefit from additional nonlinear models, especially neural networks. In particular, the present work uses neural network models to find possible existence of business cycle asymmetries in Canada, France, Japan, UK, and USA real GDP growth rates, using in-sample as

Neural Networks Forecasts Evaluations …

29

well as jackknife out-of-sample forecasts from linear models and artificial neural networks approximations. Moreover, the present work also seeks to test in-sample as well as jackknife out-of-sample forecast performance of neural networks versus linear models for all the series. ANN are able to estimate any continuous function with a desired level of precision (Hornik et al. (1989)). The ability of information processing makes ANNs powerful computational devices that can learn from examples and generalize these learning to solve problems that are never seen before (Reilly and Cooper (1990)). Performance of ANNs in biology, engineering, and finance is well known, and this should not be surprising given that ANNs are typically highly flexible nonlinear models that can easily fit any data series. On the other hand in terms of generalizing important predictive relationship, ANNs might actually tend to over fit data, finding relationships where there really are none. Thus, studies using ANNs would benefit from more out-of-sample testing. Then, ANN superiority would be less of foregone conclusion. Using a test of pair-wise forecast evaluation due to Ashley, Granger, and Schamalensee (1980), henceforth referred to as the AGS test, the present work employs in-sample forecasts from linear models, and neural networks. Thereafter, in-sample and jackknife out-of-sample forecasts are employed to test possible existence of business cycle asymmetries in Canada, France, Japan, UK, and USA real GDP growth rates. Finally, forecast performance of neural networks is tested via in-sample as well as jackknife out-of-sample forecast from linear models versus approximations from neural networks. The remaining paper is organized as follows. Section 2 incorporates description on neural networks, estimation issues, neural network linearity tests, jackknife re-sampling, AGS test, data sources and preliminary data analysis, whereas section 3 presents empirical results, and forecast performance of neural networks. Finally, section 4 presents conclusion.

2. NEURAL NETWORKS An artificial neural network (ANN) is said to be an advanced artificial technology that mimics human-brain's learning, and decision-making process. ANN can learn through trial and experience, and use these learnings to solve problems never seen before. The ability of information processing makes ANNs powerful computational devices that can learn from examples, and generalize learning to solve the problems (Reilly and Cooper (1990).

Khurshid M. Kiani

30

ANN are treated as nonlinear, nonparametric statistical methods that are independent of the distributions of the underlying data generating processes (White 1989). Therefore, the ANN modeling approach is useful for forecasters and researchers, especially in problems where data are available but the data generating processes as well the underlying laws for data generating processes are unknown. The general form of a neural network model employed in the present work is presented in equation 2.1 . ⎡ f (x ) = sig ⎢α 0 + ⎢⎣

n

∑α j =1

⎛ k ⎞⎤ ⎜ β ij xi + β 0 j ⎟⎥ + ε ⎟⎥ ⎝ i =1 ⎠⎦

j sig ⎜

∑

(2.1)

where, n is the number of the hidden nodes in the network, k is the number of explanatory variables in the network, sig (x) = 1/(1+e-x), α j represents a vector of parameters or weights linking the hidden to the output layers’ units, β ij (i = 1, … , k; j = 1 , … , n) denotes a matrix of parameters linking the input to the hidden layers’ units, and ε is the error term. The error term ε can be made arbitrarily small if a sufficient number of explanatory variables is included, and if n is chosen to be large enough. However, the present work considers up to only 2 explanatory variables (k = 1, 2), so that n is fixed to be 2. The parameters in Equation 2.1 are chosen to minimize the sum of squared errors (

∑ε

2

) , referred

to as SSE.

2.1. Neural Network Linearity Tests Neural network linearity test tests the null hypothesis of linearity against the alternative hypothesis of nonlinearity. The test is employed to test possible existence of business cycle asymmetries in Canada, France, Japan, UK, and USA real GDP growth rates. The test statistic for this test is constructed using forecasts from a linear model of the form of the Equation 2.2 , and approximations from an unrestricted nonlinear model i.e. a neural network model of the form shown in Equation 2.3 . yt = π ' wt + ut

(2.2)


31

where, u t ~ N (0, σ 2 ),

and

~ ' )' , wt = (1, w t

π = ( π 0 , π 1 ........, π

uˆt = π ' wt +

k

∑θ

0 j {ψ

~ = ( y ,..., y )' w t t −1 t− p p

)'

(γ j ' wt )} + vt

(2.3)

j =1

where u t ~ Nid (0, σ 2 ),

and

~ ' )' , wt = (1, w t

~ = ( y ,..., y )' , w t t −1 t− p

π = (π 0 , π 1 ,........, π p )' ,

ψ (γ ' wt ) = (1 + exp{−γ ' wt )) −1 , and π 0 is intercept. Equation 2.3 shows a nonlinear neural network model that nests the linear model (Equation 2.2 ). The test consists of three-step procedure. In the first step real GDP growth rates ( y t ) are regressed on an intercept and lags ( y t −1 ,.......... y t − k ) of yt to recover residual ( uˆ t ) and squared sum of errors ( SSE1 ). Likewise, ANN is approximated using Equation 2.3 to recover residuals ( vˆt ), and squared sum of errors ( SSE 2 ). Finally, the test statistic is computed using Equation 2.4 , which is distributed F [m, (n − p − m − 1)] under normality hypothesis.

TS = {( SSE1 − SSE 2) / m} /{SSE 2 /(n −

(2.4)

where, in Equation 2.4 , m denotes the number of restrictions in the unrestricted model, n is the number of observations, and p is the number of lags.

2.2. Estimation Issues Due to relatively large number of parameters and the nonlinearity inherent in the ANN specification, the objective function is unlikely to be globally convex, and thus can have many local minima. Thus, for the required SSE minimization we started by using a genetic algorithm (GA), which often is considered among the most reliable (does not get “hung up”) algorithms for estimating highly nonlinear functional forms, but which typically is much slower than others. To

32

Khurshid M. Kiani

further increase the probability of obtaining a global minimum, 4 independent runs of the GA were considered, each with a fixed number of iterations. The parameter vector (of the 4) that had the smallest SSE was used as starting conditions for Matlab’s fminsearch algorithm, which is a Nelder-Mead simplex algorithm that worked well for closing in on the final optimum.

2.3. Jackknife Out-of-sample Forecasts Researchers often use jackknife re-sampling techniques when the distribution of the parameters under review is either unknown, when it cannot be characterized by a mathematical function, or when the mathematical function is especially difficult to estimate. Quenouille (1949) used this technique to reduce bias in estimators, and thereafter Tuckey (1958) employed it for estimating variances. The standard jackknife estimate is calculated deleting one observation and estimating parameters using n − 1 observations of the data. The model and estimated observations are than used to conditionally predict the dependent value for the deleted observation. This process continues, detecting and predicting a different observation for each model, until each of the observations in the data has been predicted out-of-sample. Sub-sample jackknife technique was initially proposed by Wu (1990), and extended by Politis and Romano (1994). Politis et al. (1997) and Ziari el.al (1997) also used these re-sampling techniques for their empirical work. Compared to standard jackknife, in sub-sample jackknife more than one observation is dropped to estimate out-of-sample forecasts of the remaining m = n − d observations, where n is the total number of observations, and d = 2, 3,........., n − 1 . The model and the estimated observations are than used to conditionally predict the dependent value for the deleted observation, and this process continues until each observation in the data is predicted or until all possible sub-samples are considered, depending on the statistical assumptions and the application.

2.4. AGS Test The test of pair-wise forecast evaluation by Ashley, Granger, and Schmalensee (1980), henceforth called as AGS test provides a test for statistical significance of the difference between root mean square errors of the two competing forecasts. The procedure for the AGS test is also described in Bessler


33

and Brandt (1992), Bradshaw and Orden (1990), and Kastens, & Brester (1996). The AGS test statistics are obtained by estimating the regression shown in Equation 2.5 . However, if the sample mean of the forecast error from either model is negative, that forecast error series must be multiplied by –1 before estimating the regressions.

dt = β1 + β 2 ( st − smean) + et

(2.5)

where, d t = difference between forecast errors (lower RMSE forecasts are subtracted

from higher RMSE forecasts). st = sum of forecast errors

smean = sample mean of s t

et =

white noise

The test statistic for the AGS test is calculated from the residuals obtained from estimating an unrestricted model represented by Equation 2.5 , and its restricted version. The restricted model is obtained restricting β1 = β 2 = 0 in Equation 2.5 . The test statistic is distributed F but the comparable value of this test statistic is one fourth of the tabulated value of this test statistic because it does not take into account the sign of the coefficient estimates. The test statistic is calculated using Equation 2.6 given below.

TS = ((SSER − SSEUR ) /(k − 1)) /( SSEUR /(n where, SSE R = Sum of squared residuals from restricted model SSEUR = Sum of squared residuals from un-restricted model n = Number of observations in the sample k = Number of variables in the regression model

(2.6)

Khurshid M. Kiani

34

2.5. Data Sources Quarterly data on real GDP growth rates series for Canada, France, Japan, UK and USA is obtained from the International Financial Statistic (IFS)’s CDROM for the month of September 2001. Canada, Japan, UK and USA data range from 1957:1 to 2000:4 whereas the France data range from 1970:1 to 2000:4. Table 1. Data Description

Data Series Sample Period

Canada Quarterly Real GDP

France Quarterly Real GDP

Japan UK Quarterly Quarterly Nominal GDP Real GDP

USA Quarterly Real GDP

1957:12000:4 176

1970:12000:4 124

1957:12000:4 176

1957:12000:4 176

1957:12000:4 176

Sample Length Notes: 1. The quarterly seasonally adjusted real GDP growth rates for Canada, France, UK, and USA were obtained from the September 2001 edition of the International Financial Statistics (IFS) CD-ROM. 2. The nominal GDP series are employed for Japan because seasonally adjusted data were only available for the nominal series and not for the real series on the IFS CD-ROM.

Table 2.1. In-sample Performance of Neural Network vs. Linear Models

STD 0.126 0.070 0.226 0.139 0.119

LIN RMSE 0.123 0.070 0.228 0.134 0.119

ANN2 RMSE 0.121 0.066 0.170 0.134 0.118

ANN5 RMSE 0.121 0.062 0.170 0.127 0.118

ANN10 RMSE 0.120 0.062 0.152 0.123 0.116

Canada France Japan UK USA Notes: 1. The results presented in this Table are based on in-sample forecasts from linear models and Jackknife out-of-sample approximations from artificial neural networks. 2. In this Table ‘STD’ represents standard deviation of the series, ‘Lin” represents linear model, ANN2 represents, artificial neural network with two hidden nodes, ANN5, artificial neural network with five hidden nodes, and ANN10 represents an artificial neural network models with ten hidden nodes. 3. This Table presents, standard deviation, in-sample forecasts from linear models and insample approximations from artificial neural networks (ANN), and root mean squared


35

errors (RMSE) for linear models ANN2, ANN5, and ANN10 for Canada, France, Japan, UK, and USA series. 4. Column 2 shows standard deviation (STD), column 3 shows RMSE for linear models, column4 RMSE for ANN2, column5 RMSE for ANN5, and column5 shows RMSE for ANN10 model for each of the series.

The analysis started with the data on real GDP growth rates for G7 countries but there was a jump of 35.43 percent in Germany GDP in 1991:1. This likely would have been because of its reunification. Italy showed an inexplicable spike of 87.5 percent in 1970:1. Therefore, series for Italy and Germany were excluded from the analysis. Table 1 presents additional information on Canada, France, Japan, UK, and USA real GDP growth rate series. Table 2.2. Out-of-sample Performances of Neural Network vs. Linear Models LIN RMSE 0.125 0.070 0.223 0.134 0.118

ANN2 RMSE 0.257 0.168 0.586 0.344 0.279

ANN5 RMSE 0.287 0.179 0.688 0.349 0.305

ANN10 RMSE 0.335 0.195 0.718 0.434 0.377

STD Canada 0.126 France 0.070 Japan 0.226 UK 0.139 USA 0.118 Notes: 1. See notes on table 2.1. 2. Results presented in this Table are based on Jackknife out-of-sample forecasts from linear models and jackknife out-of-sample approximations from artificial neural network for Canada, France, Japan, UK, and USA real GDP growth rates.

3. EMPIRICAL RESULTS In-sample forecasts from linear model and neural network approximations for Canada, France, Japan, UK, and USA real GDP growth rates are presented in Table 2.1 . This Table shows root mean squared errors for linear model estimates (RMSE1) as well as for neural network model approximations (RMSE2) with two (ANN2), five (ANN5), and ten (ANN10) hidden nodes. Similarly, Table 3.1 shows mean of the residuals from the linear models (M1), and neural network models (M2), and root mean squared errors from linear (RMSE1), as well as neural network models (RMSE2).

Khurshid M. Kiani

36

In addition to in-sample forecasts/approximations from linear models and ANN, the present work employs jackknife re-sampling technique for estimating out-of-sample forecasts from linear models for all the series. Root mean errors calculated from linear modes for all the series are shown in Table 2.2. Finally, the Table also show root mean square errors from jackknife out-of-sample approximations from neural networks with two (ANN2), five (ANN5), and ten (ANN10) hidden nodes. Likewise, using out-of-sample forecasts, mean of the residuals from linear models (M1), and neural network models (M2) as well as root mean square errors from linear models (RMSE1), and neural networks (RMSE2) are presented in Table 3.2 for all the series. Table 3.1. In-sample Forecast Results: Neural Network vs. Linear Models M1 M2 RMSE1 RMSE2 Canada 0.000 0.000 0.123 0.121 France 0.000 0.000 0.070 0.066 Japan 0.000 0.000 0.224 0.170 UK 0.000 0.000 0.134 0.134 USA 0.000 0.000 0.119 0.118 Notes: 1. The results presented in this Table are based on in-sample forecasts from linear models and Jackknife out-of-sample approximations from artificial neural networks. 2. Row 1 shows M1, M2, RMSE1, RMSE2 for Canada real GDP growth rates which are respectively the mean of residuals series recovered from linear model, mean of residuals from NNW model, RMSE from linear model and RMSE from NNW model residuals. 3. Row 2 shows M1, M2, RMSE1, and RMSE2 for France real GDP growth rates, row 3 for Japan, row 4 for UK, and row 5 for USA residual series. 4. These statistics are the basis for the hypothesis if an artificial neural network is an improvement over the linear models.

Table 3.2. Out-of-sample Forecast Results: Neural Network vs. Linear Models

Canada France Japan UK USA

M1 -1.80e-4 -1.80e-4 -1.50e-4 -5.00e-5 -7.00e-5

M2 -4.00e-4 -3.00e-4 -0.003 0.004 -0.003

RMSE1 0.125 0.069 0.223 0.134 0.118

RMSE2 0.257 0.168 0.586 0.344 0.279


37

Notes: 1. See notes on Table 3.1. 2. Results presented in this Table are based on Jackknife out-of-sample forecasts from linear models and jackknife out-of-sample approximations from artificial neural network for Canada, France, Japan, UK, and USA real GDP growth rates.

Table 4. Neural Network Nonlinearity Tests

Canada France Japan UK USA

In-sample Forecasts RMSE Test Statistics 0.121 274.235 (0.000) 0.069 270.082 (0.000) 0.211 184.470 (0.000) 0.132 400.694 (0.000) 0.117 507.968 (0.000)

Jackknife Out-of-sample Forecasts RMSE Test Statistics 0.169

123.339 (0.000) 121.633 (0.000) 44.967 (0.000) 268.614 (0.000) 267.132 (0.000)

0.096 0.362 0.156 0.150

Notes: 1. The results presented in this Table are based on in-sample as well as jackknife out-ofsample forecasts from linear models and in-sample and jackknife out-of-sample approximations from artificial neural networks. 2. Column 1 in this Table show names of the countries included in the study. 3. Test statistics presented in the Table are based on linear as well as neural network models represented primarily by the following two Equations:

y t = π ' wt + u t

(4.1a)

where,

u t ~ N (0, σ 2 ),

~ ' )' , wt = (1, w t

~ = ( y ,..., y )' w t t −1 t− p

and yt = π ' wt +

k

∑θ

0 j {ψ

π = ( π 0 , π 1 ........, π p )'

(γ j ' wt )} + vt

(4.1b)

j =1

4. The test statistics are F distributed with

m numerator

and

(n − p − m − 1)

denominator degrees of freedom under normality hypothesis. The test statistic for this distribution is calculated using following Equation: TS = {( SSE1 − SSE 2) / m} /{SSE 2 /(n − p − m − 1)} (4.1c)

38

Khurshid M. Kiani where, in Equation 4.1c , m denotes the number of restrictions in the unrestricted model, n is the number of observations, and p is number of lags both in the

restricted (linear) and unrestricted (neural network) models. 5. Column 3 shows test statistics constructed from in-sample forecasts whereas column 6 shows test statistics constructed from jackknife out-of-sample forecasts. p-values for each test statistic are shown beneath each test statistic in parenthesis. 6. In column 2 , we present root mean squared error (RMSE) for in-sample forecasts whereas in column 5 , we show RMSE calculated from jackknife out-of-sample forecasts.

3.1. Results on Linearity Tests The present work tests the null hypothesis of linearity versus the alternative hypothesis of nonlinearity. This hypothesis is tested on Canada, France, Japan, UK, and USA real GDP growth rates based on nonlinearity test statistic constructed from in-sample forecast approximated from neural network in conjunction with forecasts from its linear counterparts. These hypotheses are repeated using jackknife out-of sample forecasts approximated from neural network models, and jackknife out-of-sample forecasts from linear models. If null hypothesis of linearity is true, nonlinearities do not exist, however, if the null is false, nonlinearities do exist in the data series being tested. Under linearity hypothesis, the test statistic is distributed F [m, (n-p-m-1)], where, m is the number of restrictions in the nonlinear model, n is the total number of observations in the series being tested, and p is the numbers of lags used in each regression. Table 4 show results from in-sample as well as jackknife out-of-sample forecast from linear model. In this Table column 2 rows 1 − 5 show in-sample RMSE respectively for Canada, France, Japan, UK, and USA. The test statistic constructed from in-sample forecasts are shown in column 3 rows 1 − 5 respectively for Canada, France, Japan, UK, and USA real GDP growth rates. Likewise, RMSE as well as the test statistics obtained from the jackknife out-ofsample forecast for all the series are shown respectively in column 4 and 5 . The results based on in-sample forecasts from linear model as well as jackknife out-of-sample approximations show that the hypothesis of linearity is rejected in Canada, France, Japan, UK, and USA real GDP growth rates. Likewise, neural network linearity test constructed from jackknife out-of-sample forecast, overwhelmingly rejects linearity hypothesis in all the series. However,


39

test statistics for the tests based on jackknife out-of-sample forecasts though significant have lower magnitude than that of the linearity tests that are constructed from in-sample forecasts.

3.2. Forecast Performance of Neural Networks The AGS test that is a test of pair-wise forecast evaluation between any two competing models is employed for testing forecast performance of ANN versus linear models for all the series. For doing so, in the present work, in-sample as well as jackknife out-of-sample forecast from linear models as well as in-sample and jackknife out-of-sample approximations from neural networks are employed. If the coefficients β 1 and β 2 shown in Equation 2.5 are both positive, the appropriate test is an F-test of the joint hypothesis that β 1 = β 2 = 0 . Because the F-test does not consider the sign of the coefficient estimates, the actual significance levels are only one fourth of that is reported in a standard Fdistribution Table. That means that the probability of obtaining F-Statistic is greater than the critical value and having both estimates positive is equal to one fourth of the significance level normally associated with the critical value. If one of the estimated coefficients is significantly negative then one cannot conclude that the neural network model is a significant improvement over the linear model. However, if one of the estimated coefficients is negatively insignificant then a 1 − tail t-test is used to judge the significance of the positive coefficient. Contrary to that, if the null hypothesis is not accepted against the alternative that at least one of the coefficients is significant then one might conclude that the neural networks model is an improvement over the linear model. The null hypothesis for in-sample forecast performance is that the neural networks model is not a significant improvement over the linear model (RMSEs of the competing forecasts are equal) versus the alternative hypothesis that the neural networks model is an improvement over the linear model. If the null is rejected, the neural networks model forecast performance is superior to that of the linear model. However, if the null is accepted, then the two model forecasts cannot be distinguished, meaning that the evidence does not suggest that the neural network model is better than the linear model. The results from in-sample AGS test statistics that are presented in Table 5.1 for all the series reveal that neural network models perform better than linear models when in-sample forecasts are employed. However, the results based on jackknife out-of-sample forecasts shown in Table 5.2 demonstrate that linear

Khurshid M. Kiani

40

model performance is superior to neural network models. Therefore, linear models appear to offer better forecasts when out-of-sample forecast are to be worked with. While neural network forecasts are superior in an in-sample framework, it might not be the only reason for being chosen for out-of-sample forecasts. Table 5.1. AGS TEST: In-sample Performance of Linear vs. ANN Model Parameters

β1

Canada 2.00e-6

France 1.70e-5

Japan 3.88e-4

UK 7.60e-5

USA 4.00e-6

β2

(0.999) 0.807

(0.929) 0.032

(0.621) 0.154

(0.691) 0.015

(0.960) 0.003

(0.323) 0.492 (0.419)

(0.048) 1.946 (0.466)

(0.000) 13.224 (0.419)

(0.114) 1.329 (0.419)

(0.395) 0.365 (0.419)

Test Statistic

Notes: 1. The results presented in this Table are based on in-sample forecasts from linear models, and in-sample approximations from artificial neural networks. 2. In AGS test is the test of the joint significance of the parameter estimates of β 1 and

β 2 i.e. β 1 = β 2 = 0 is tested to conclude that the neural network models are improvement over the linear models. The AGS test is contemplated using the model shown in Equation 5.1 a :

dt = β1 + β 2 b + et

(5.1a)

3. β1 is the coefficient on the intercept term, and β 2 is the slope coefficient of the AGS regression model. 4. p-values for parameter estimates of the coefficients of intercept ( β 1 ) , and the slope coefficients for each of the series are shown in parentheses below each parameter estimate for all the series. 5. Test statistics for AGS tests are presented in last row of the Table, and the relevant pvalues for the joint significance of the parameter estimates are presented below each statistic in parentheses.

Table 5.2. AGS TEST: Out-of-sample Performance of Linear vs. ANN Models Parameters

β1

Canada 2.00e-4

France -4.00e-4

Japan 0.003

UK 0.004

USA 0.003

(0.852)

(0.694)

(0.182)

(0.024)

(0.001)


41

Table 5.2. (Continued) Parameters

β2 Test Stat

Canada 0.565

France 0.616

Japan 0.604

UK 1.033

USA 0.521

(0.000) 32.862 (0.000)

(0.000) 30.458 (0.000)

(0.000) 49.151 (0.000)

(0.000) 49.572 (0.000)

(0.000) 46.111 (0.000)

Notes: 1. See note on Table 5.1 2. The results presented on this Table are based on jackknife out-of-sample forecasts from linear models, and jackknife out-of-sample approximation from artificial neural network.

4. CONCLUSION The present work employs artificial neural networks (ANN) as well as linear models to compare in-sample, and jackknife out-of-sample approximations from neural networks and forecasts from linear models. Neural network linearity test is constructed from in-sample forecasts/approximations from linear models and ANN that nests the linear model for testing possible existence of nonlinearities in Canada, France, Japan, UK, and USA real GDP growth rates. Likewise, neural network linearity tests are constructed based on jackknife out-of-sample forecasts from linear models and neural networks for testing possible existence of business cycle asymmetries in Canada, France, Japan, UK, and USA real GDP growth rates. For testing pair-wise forecast evaluation between two competing models the present work employs a test of forecast evaluation due to Ashley, Granger and Schmalensee (1980) using in-sample as well as jackknife out-of-sample forecasts from the two competing models i.e. linear model and ANN for Canada, France, Japan, UK, and USA real GDP growth rates. The results show that in-sample forecast performance of ANN is superior to that of linear models. These results are in line with the previous studies due to Lee, et al. (1993), and Terasvirta, et al. (1993). However, out-of-sample performance of linear models is better than neural networks models for real GDP growth rates for all series. This means that a superior in-sample forecast performance of neural networks models should not be the only reason for employing them for out-of-sample forecasting.

42

Khurshid M. Kiani

The results based on in-sample forecasts provide a robust evidence of asymmetries in Canada, France, Japan, UK, and USA. In addition, the results based on jackknife out-of-sample forecast strengthen the evidence for existence of business cycle asymmetries in Canada, France, Japan, UK, and USA real GDP growth rates. Therefore, the policymakers might not able to evaluate the impact of monetary policy or any other shock on real GDP series in these countries based on forecast from linear models. The results on nonlinearity for USA are in line with Bidarkota (2000) and Andreano and Savio (2002). Similarly the results on nonlinearity for Canada, Japan, and USA are in line with Kiani and Bidarkota (2004). This shows that evidence against linearity for USA is robust to change in samples as well as change in testing approach. Neural network models outperform the traditional statistical tests for remaining nonlinearities, which is in line with the previous studies (Terasvirta, et al. 1993).

5. REFERENCES Andreano, M., and G. Savio, (2002). Further evidence on business cycle asymmetries in G7 countries, Applied Economics, 34, 895-904. Ashley, R, Granger, C.W.J. & Schmalensee, R. (1980). Advertising and aggregate consumption: an analysis of causality, Econometrica, 48, 1149-1167. Auerbach, A. J., 1982, The index of leading indicators: measurement without theory, thirty-five year later, Review of Economics and Statistics, 64, 589595. Beaudry, P. & Koop, G. (1993). Do recessions permanently change output? Journal of Monetary Economics, 31, 149-163. Bessler, D.A.& Brandt, J.A. (1992). An analysis of forecasts of livestock prices, Journal of Economic Behavior and Organization, 18, 249-63. Bidarkota, P.V. (1999). Sectoral investigation of asymmetries in the conditional mean dynamics of the real US GDP, Studies in Nonlinear Dynamics and Econometrics, 3, 191-200. Bidarkota, P.V. (2000). Asymmetries in the conditional mean dynamics of real GNP: robust evidence, The Review of Economics and Statistics, 82, 153-157. Bradshaw, G.W. & Orden, D. (1990). Granger causality from the exchange rates to agricultural prices and export sales, W. J. Agricultural Economics, July, 100-110.


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Brunner, A.D. (1992). Conditional asymmetries in real GNP: a seminonparametric approach, Journal of Business & Economics Statistics, 10, 6572. Brunner, A. (1997). On the dynamic properties of asymmetric models of real GNP, The Review of Economics and Statistics, 79, 321-326. DeLong, J.B. & Summers, L.H. (1986). Are business cycles symmetrical? In R.J. Gordon (Eds.), The American Business Cycle: Continuity and Change. Chicago, IL: University of Chicago Press, pp. 166-179. Diebold, F.X. Rudebusch, G.D. (1990). A nonparametric investigation of duration dependence in the American business cycle, Journal of Political Economy, 98, 596-616. Estrella, A., and Mishkin, F. S., (1998). Predicting U.S. business cycle regimes, Journal of Business and Economic Statistics, 17, 313-323. Falk, B. (1986). Further evidence on the asymmetric behavior of economic time series over the business cycle, Journal of Political Economy, 94, 1096-1109. Gordon, R. J., (1986). The American Business Cycle : continuity and change, NBER studies in business cycles, Vol. No. 25, University of Chicago Press, Chicago. Hamilton, J.D., (1989). A new approach to the economic analysis of nonstationary time series and business cycle, Econometrica, 57, 357-384. Hornik, K., Stinchcombe, M. & White, H. (1989). Multilayer feed forward neural networks are universal approximations, Neuralnetworeks, 2, 359-356. Kastens, T. L., & Brester, G. W. (1996). Model selection and forecasting ability of theory-constrained food demand system, American Journal of Agricultural Economics, 78, 301-312. Kiani, K. M., & Bidarkota, P.V. (2004). On business cycle asymmetries in ‘G7” countries,” Oxford Bulletin of Economics and Statistics, 66, 333-353. Kling, J.L., (1987). Predicting the turning points of business and economic time series, Journal of Business, 60, 201-238. Koch, P.D. & Rasche, R. H., (1988). An examination of the commerce department leading indicator approach, Journal of Business and Economic Statistics, 6(2), 167-187. Kuan, C., and White, H., (1994). Artificial neural networks: an econometric perspective. Econometric Review, 13, 1–91. Lee, T.H., White, H. & Granger, C.W.J. (1993). Testing for neglected nonlinearity, Journal of Econometrics, 56, 269-290. Neftci, S.N. (1984). Are economic time series asymmetric over the business cycle? Journal of Political Economy, 92, 307-328.

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Politis, D.N., & Romano, J.P. (1994). Large sample confidence region based on sub-samples under minimal assumptions, Annals of Statistics, 22, 2031-2052. Politis, D.N., Romano, J.P.& Wolf, M. (1997). Subsampling for heteroskedastic time series, Journal of Econometrics, 81, 281-317. Potter, S.M. (1995). A non-linear approach to U.S. GNP, Journal of Applied Econometrics, 10, 109-125. Qi, M., (2001). Predicting US recessions via leading indicators via neural network models, International Journal of Forecasting, 17, 383-401. Quenouille, M.H. (1949). A note on bias in estimation, Biometrika, 43, 353-60. Ramsey, J.B. & Rothman, P. (1996). Time irreversibility and business cycle asymmetry, Journal of Money Credit and Banking, 28, 21. Reilly, D.L., & Cooper, L.N. (1990). An overview of neural networks: early models to real world systems. In S.F. Zornetzer, J.L. Davis, and C. Lau (Eds.), An Introduction to Neural and Electronic Networks, New York: Academic Press, pp. 227-248. Sichel, D.E. (1989). Are business cycles asymmetric? a correction, Journal of Political Economy, 97, 1255-1260. Terasvirta, T., Lin, C.F. & Granger, C.W.J. (1993). Power of neural network test, Journal of Time Series Analysis, 14, 209-220. Tuckey, J.W. (1958). A bias and confidence in not-quite large samples, Annals of Mathematical Statistics (abstracts), 29, 614-623. Vishwakarma, K., (1995). A neural networks to forecast business cycle indicators, Elsevier Science, Mathematics, and Computers in Simulations, 39, 287-291. White, H. (1989). Learning in artificial neural networks: a statistical perspective, Neural Computations 1, 425-464. Wu, C.F.J. (1990). On the asymptotic properties of the jackknife histogram, Annals of Statistics, 18,1438-1452. Ziari, H.A., Leatham, D.J. & Ellinger, P.N. (1997). Development of statistical discriminant mathematical programming model via resampling estimation techniques, American Journal of Agricultural Economics, 79, 1352-1362.

Chapter 3

A ROBUST EVIDENCE OF BUSINESS CYCLE ASYMMETRIES IN G7 COUNTRIES ABSTRACT Artificial neural networks are employed for possible existence of business cycle asymmetries in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. The relationship between real GDP growth rates and business cycle fluctuations in these countries is modeled using artificial neural networks. Finally, neural network linearity tests are constructed from linear model predictions as well as neural network approximations for all the series. The results from neural network linearity tests based on in-sample forecasts show statistically significant evidence of business cycle asymmetries in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. Similarly, neural network linearity test results based on jackknife out-of-sample forecasts are not much different. However, compared to Andrano and Savio (2002), there exists statistically significant evidence of neglected nonlinearities in France, Germany, and UK, and France, and UK when compared to Kiani and Bidarkota (2004). Therefore, due to underlying nonlinear data generating processes all types of linear models cannot be employed to forecast the impact of monetary policy or any other shock on output in Canada, France, Germany, Italy, Japan, UK, and USA series.

Keywords: asymmetries; business cycles; neural networks; nonlinearities; vector autoregressions

46

Khurshid M. Kiani

JEL codes:C32, C45, E32

1. INTRODUCTION The empirical research on business cycle asymmetries that employed USA macroeconomic time series data showed that asymmetries do exist in business cycle fluctuations. Beaudry and Koop (1993), Brunner (1992-1997), and Bidarkota (1999-2000) including many others who investigated USA gross national product (GNP) concluded that business cycles are asymmetric. Similarly, Neftci (1984), Ramsey and Rothman (1996) including others investigated USA unemployment rates, and concluded that business cycle asymmetries are prevalent in these series. Likewise studies by Potter (1995), Anderson and Vahid (1998) and Anderson and Ramsey (2002) also showed that business cycle asymmetries do exist in time series data. Alternately, Falk (1995), Sichel (1989), Delong and Summers (1986), and Diebold and Rudebusch (1990) were unable to show statistically significant evidence of business cycle asymmetries in the series they studied. A number of studies including Auerbach (1982), Gordon (1986), Kling (1987), Koch and Rasche (1988), Diebold and Rudebusch (1990), Hamilton (1989), Klein (1990), Estrella and Mishkin (1998) including others focused on business cycle research, however, only a few studies investigated possible existence of asymmetries in business cycle fluctuations using international data. For example, Andreano and Savio (2002) investigated business cycle asymmetries in G7 countries using Markov Switching models but were not able to detect business cycle asymmetries in France, Germany, and UK series. Similarly, Kiani and Bidarkota (2004) investigated possible existence of business cycle asymmetries in G7 countries but despite using nonlinear and switching time series models with stable distributions, and long memory they were not able to detect asymmetries in France, and UK real GDP growth rates. While this study made contribution over its other counterparts including Andreano and Savio (2002) who also studied business cycle asymmetries in G7 countries, the basic question whether “business cycles in G7 countries are alike” remained unanswered. This posed a challenge for the macroeconomic theorists to develop new theories of business cycles if empirical research is unable to show an evidence of business cycle asymmetries in France, and UK that are two prominent members of G7 countries, and the European Union.

A Robust Evidence of Business Cycle Asymmetries …

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It is important to detect business cycle asymmetries in macroeconomic time series because asymmetries imply that the effects of contractionary and expansionary monetary policy, and other shocks on output are asymmetric. Therefore, any nonlinearity would invalidate the measures of the persistence of monetary policy, and other shocks on output that are based on linear models including those observed from linear vector autoregressions. Therefore, it is felt that the present study can contribute to fill this gap adequately. Therefore, to further this work, and contrary to Andreano and Savio (2002) who employed Markov switching models, and Kiani and Bidarkota (2004) who employed alternative regime switching models, we prefer to use artificial neural networks for modeling asymmetries in Canada, France, Germany, Italy, Japan, UK, USA real GDP growth rates, that is approximated using genetic algorithm, because neural networks are flexible form of nonlinear models which can fit data well even when distribution of the data generating process as well underlying laws pertaining the data generating process are unknown (White 1989b). Artificial neural networks (ANN) have been applied successfully in many disciplines including business and economics. For example, Kuan and White (1994) and Swanson and White (1995, 1997a, 1997b), Hutchinson, Lo, and Poggio (1994), Garcia and Gencay (2000), and Qi and Madala (1999), Gencay (1999), Vishwakarma (1995), Qi (2001), and Kiani, Bidarkota, and Kastens (2005) employed neural networks in economics and finance. However, ANN have been under heavy criticism because of their tendency to overfit the data although this tendency can be mitigated with careful construction of neural network architecture (Kiani 2005). Therefore, artificial neural networks (ANN) are employed to approximate in-sample as well as jackknife out-of-sample forecasts from linear models as well neural networks to construct neural network linearity tests to investigate possible existence of business cycle asymmetries in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. The remaining work is split into the following sections. Section 2 gives a brief description of theoretical justification, and description of the nonlinear empirical model i.e. neural network models and underlying tests, and section 3 shows data sources, hypotheses tests, empirical results on hypotheses tests, and forecast performance of neural network models. Section 4 incorporates brief conclusions.

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Khurshid M. Kiani

2. EMPIRICAL MODEL 2.1. Theoretical Justification of Asymmetric Models With the advent of the Great Depression in the United States of America (USA), and its impact on other nations in the world, economists started of thinking on its causes, and there is a voluminous literature that discusses the causes of the great depression (GD) of 1930s in the USA. Monetarists’ believed that the Federal Reserve System (FED) of USA was responsible for it, and that FED’s adequate action would have averted the recessions but the inapt decision of the FED was solely responsible for pushing the USA economy into the Great Depression. Keynesians opposed the monetarist’s viewpoint saying that the FED was responsible for the GD. The debate between Keynesians and monetarists started soon after GD of 1930s, which lasted for some time, and this was the time when people started thinking clearly about the business cycles. In this context, Schumpeter (1939) classified business cycles into four phases i.e. peak, expansion, recession, depression or trough based on their peak-to-peak and trough-to-trough duration. Likewise, Mitchell and Burns (1938), and Burns and Mitchell (1946) developed techniques for analysis of the classical business cycles. National Bureau of Economic Research (NBER) constituted business cycle chronology on retrospective identification of business cycle turning points. NBER’s business cycle dating committee used data on a number of macroeconomic variables including output, income, employment and trade etc. to identify the business cycles dates when they occur. This type of business cycle tells about an absolute decline in output (Stock and Watson 1999). However, historically, studies on forecasting recessions remained quite sparse until lately. For business cycle research, post 1930s Great Depression, Keynesian macroeconomics remained dominant until the 70s when Lucas (1976) pioneered the rational expectation revolutions. In 80s economists such as Kydland and Prescott (1982), and Long and Plosser (1983) put forth the idea of real business cycles without monetary factors and monetary management. This was the time when Bernanke (1983) in a seminal paper discussed the non-monetary effects that were the cause of the propagation of the Great Depression. Bernanke was pioneer to unveil nonmonetary affects that propagated the great depression. Beaudry and Koop (1993) including others concluded that recessions are short lived whereas expansions are of longer duration showing that business cycle fluctuations are asymmetric. Thus, assuming that data-generating process is linear, the impulse responses function generated from one unit positive shock will have


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span and amplitude equal to the impulse response function generated by negative shock of equal magnitude though in an opposite direction. On the other hand, an impulse response function generated from one unit positive shock is not equal to an impulse response generated by one unit negative shock of equal magnitude when the underlying data generating process in nonlinear. Therefore, business cycles fluctuations associated with linear models are symmetric, and those associated with nonlinear model are asymmetric. Policymakers would be interested to know the impact of monetary policy or other shocks on output. However, one may not be able to forecast the impact of such shocks based on linear models when the underlying data generating process is nonlinear. Therefore, it is imperative to investigate possible existence of nonlinearities in data series so that appropriate linear or nonlinear forecasting models are employed to anticipate the impact of monetary policy or other shocks on output. Moreover, it is of interest to know whether business cycles in different countries are alike. If they are not, then this poses a serious challenge to macroeconomic theorists to develop new theories of business cycles that can explain fluctuations in economic activity without considering country specific institutional factors. That is the reason why the present work considers artificial neural networks that are considered to be highly flexible functional form of nonlinear models that are data driven and can fit to any time series data adequately without taking into account the distribution of the data generating process. These models are elaborated in the following sub-sections in detail.

2.2. Nonlinear Empirical Model Artificial neural network (ANN), which pertains to artificial intelligence technology, imitates human brain's learning and decision-making processes. The ability of information processing makes ANN powerful computational devices, which can learn from examples and generalizes these learning’s to solve problems never seen before (Reilly and Cooper (1990). ANN are nonlinear, and nonparametric statistical methods which are independent of the distributions of underlying data generating processes (White 1989b). This work is intended to find possible existence of nonlinearities in Canada, France, Germany, Italy, Japan, UK, and USA real GDP using in-sample approximations from neural networks, which is an extension to Kiani and Bidarkota (2004). Further, these analyses are extended to jackknife out-of sample framework to observe forecast performance of these models in out-of-sample

Khurshid M. Kiani

50

framework when compared to the in-sample forecast as well as to investigate the performance of the models with respect to previous studies (Kiani el al. 2005). Artificial neural networks (ANN) of the form shown in Equation 3.2 is employed which nests a linear model of the form shown in Equation 3.1 to construct neural network linearity tests for testing possible existence of asymmetries in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. An ANN linearity test comprises of three parts. In the first part a linear model of the form of Equation 3.1 is estimated to recover model forecasts and residuals for Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. y t = π ' wt + u t

(3.1)

where u t ~ N (0, σ 2 ),

and

~ ' )' , wt = (1, w t

~ = ( y ,..., y )' w t t −1 t− p

π = ( π 0 , π 1 ........, π p )'

In the second part forecasts from a neural network model are approximated using residual obtained from the first part as endogenous variables and lagged real GDP growth rates as exogenous variables for all the series using Equation 3.2 to recover residuals and forecasts. ut = π ' wt +

k

∑θ

0 j {ψ

(γ j ' wt )} + vt

j =1

(3.2)

where, ψ (γ ' wt ) = (1 + exp{−γ ' wt }) −1 and π 0 is intercept. Finally, in the third part using residuals from linear model as well as neural networks, a test statistic is computed using Equation 3.3 , which is distributed approximately1 F [m, (n − p − m − 1)] under normality hypothesis. Terasvirta et al. (1993) and Kiani et al. (2005) also implemented this type of test. TS = {(SSE1 − SSE 2) / m} /{SSE 2 / (n − p − m − 1)} 1

(3.3)

This test statistic is approximate because of the nuisance parameter that appears under the alternative hypothesis (Davies, 1977; Andrews, 2001).


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where, in Equation 3.3 , m denotes the number of restrictions, n is number of observations, and p is the number of lags in each of the models estimated/approximated. Neural networks linearity tests are constructed using in-sample forecasts from linear models and neural networks approximations. In addition, neural networks linearity tests using jackknife2 out-of-sample forecasts are also constructed using linear model forecasts, and neural networks approximations. Jackknife out-ofsample forecasts are employed to study the behavior of these models in out-ofsample testing, and to compare these results with those of previous studies (White 1989a; Kiani at al 2005).

2.3. Estimation Issues To avoid the likelihood of obtaining local optima, genetic algorithm is employed with a couple of random starts to obtain best parameter vector for neural network approximations which is considered to be a reliable estimation algorithm but it appeared to be very slow. Therefore, a nonlinear optimization routine is combined with genetic algorithm3, which worked satisfactorily although it was still slower than any other optimization routine.

3. EMPIRICAL RESULTS 3.1. Data Sources Quarterly real GDP data for Canada, France, Germany, Italy, Japan, the 2

3

Jackknife re-sampling technique is used when the distribution of the parameters under review is either unknown, when it cannot be characterized by a mathematical function, or when the mathematical function is especially difficult to estimate. Standard jackknife out-of-sample forecasts were used by Quenouille (1949), and thereafter, Tuckey (1958). However, sub-sample jackknife technique was initially proposed by Wu (1990) and thereafter by Politis and Romeo (1994), Politis et al. (1997), and Ziari et al. (1997). Compared to the standard jackknife, subsample jackknife drops more than one observation to estimate out-of-sample forecast of the remaining m = n − d observations, where, n is the total number of observations and d = 2,3,......... ....., n − 1 . Following De Jong (1975) genetic algorithm was used in biology, engineering and operation research (Goldberg 1989). The first ever, economic application of genetic algorithm was implemented by Axelord (1987), and thereafter by Marimon, McGartten, and Sargent (1990), and Dorsey and Walter (1995).

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United Kingdom (UK), and the United States of America (USA) were obtained from the November 2006 version of the International Financial Statistics’ CDROM. The dataset spans from 1957:1 to 2006:2 for all countries except for France for which the data spans from 1965:1 to 2006:2, and for Germany, and for Italy, the data series span from 1960:1 to 2006:2. Table 1 presents additional information on the data series employed, and Figure 1 plots real GDP growth rates for Canada, France, Germany, Italy, Japan, UK, and USA Table 1. Data Description Canada Data Series Sample Period Sample Length

France

Germany Italy

Japan

UK

USA

Quarterly Quarterly Quarterly Quarterly Real GDP Real Real GDP Real GDP GDP 1957:1- 1970:1- 1960:1- 1970:22006:2 2006:2 2006:2 2006:2

Quarterly Quarterly Nominal Real GDP GDP 1957:1- 1957:12006:2 2006:2

Quarterly Real GDP 1957:12006:2

197

197

197

165

185

185

197

Notes 1. Quarterly seasonally adjusted GDP data is used for all countries from the November 2006 edition of the International Financial Statistics (IFS) CD-ROM. 2. Nominal GDP is employed for Japan because seasonally adjusted data was only available for the nominal series and not for the real series on the IFS CD-ROM.

3.2. Hypotheses Tests The chief hypothesis of this chapter is linearity versus the alternative hypothesis of nonlinearity for Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. The hypothesis is based on the test statistics constructed from in-sample approximations from neural networks, and forecasts obtained from linear models. These hypotheses are repeated for Canada, France, Germany, Italy, Japan, UK, and USA based on jackknife out-of sample forecasts from linear model as well as neural network approximations. When the null hypothesis of linearity is true, nonlinearities do not prevail in the series being studied, however, when the null is false, nonlinearities do exist in the data series being tested.


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Figure 1. Annualized Real GDP Growth Rates.

3.3. Results on Hypothesis Test Table 2 shows neural network linearity test results for Canada, France, Germany, Italy, Japan, UK and USA real GDP growth rates. In this Table for example, the neural network test statistic for Canada that is constructed from insample forecasts from linear models and neural networks is shown in column 2 row 1 of the Table. The relevant critical values from F distribution are shown beneath each of the test statistic in brackets, and p-value are presented in parenthesis. Test statistics for France, Germany, Italy, Japan, UK, and USA are shown in column 2 , row 2 through 7 respectively, and relevant critical values are shown in brackets, and p-values in parentheses beneath each test statistics.

Khurshid M. Kiani

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Table 2. Neural Network Tests with Linear Models

Canada

France

Germany

Italy

Japan

UK

USA

In-Sample Forecasts Test Statistics RMSE 280.8325 0.1404 [2.0586] (0.0001) 2428.0671 0.1568 [2.0687] (0.0000) 380.9606 0.2169 [2.0613] (0.0001) 672.838 0.2687 [2.0613] (0.0001) 209.5664 0.1793 [2.0586] (0.0001) 202.0907 0.2201 [2.0586] (0.000) 346.0786 0.1247 [2.0586] (0.000)

Jackknife Out-of-Sample Forecasts Test Statistics RMSE 41.4805 0.2918 [2.0586] (0.000) 520.4057 0.3247 [2.0687] (0.000) 191.1076 0.2905 [2.0613] (0.000) 218.7765 0.4444 [2.0613] (0.000) 36.6850 0.3405 [2.0586] (0.000) 59.7420 0. 3512 [2.0586] (0.000) 52.9753 0. 2641 [2.0586] (0.000)

Notes: 1. Column 1 in this Table show name of the countries included in the study. 2. Test statistics presented in the Table are based on linear as well as neural network models that are shown in the following two Equations:

yt = π ' wt + ut

2.1a

where,

u t ~ N (0, σ 2 ),

and

~ ' )' , wt = (1, w t

π = ( π 0 , π 1 ........, π

ut = π ' wt +

k

∑θ

0 j {ψ

~ = ( y ,..., y )' w t t −1 t− p p

)'

(γ j ' wt )} + vt

j =1

2.1b 3. The test statistic is distributed F [m, (n − p − m − 1)] under normality hypothesis are calculated using following Equation:


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TS = {(SSE1 − SSE 2) / m} /{SSE 2 / (n − p − m − 1)}

2.1c where, in Equation 2.1c , m denotes the number of restrictions in the unrestricted model, n is the number of observations, and p is number of lags. 4. Column 2 shows test statistics constructed from in-sample forecasts whereas column 4 shows test statistics constructed from jackknife out-of-sample forecasts. 5. In this Table, for each test statistic constructed from in-sample as well as jackknife outof-sample forecasts, the relevant critical values from F distribution are shown beneath each test statistic presented in this Table in addition to the p-values that are shown beneath each test statistic in parenthesis. 6. In column 3 , root mean squared errors (RMSE) for in-sample forecasts are presented whereas in column 5 , RMSE calculated from jackknife out-of-sample forecasts are shown.

Neural network test statistic for Canada is statistically significant at 5 percent level of significance based on relevant p-values shown in the Table. Inferences do not change when significance level is switched from 5 percent to 10 percent level. Neural network linearity test statistics for the remaining countries i.e. France, Germany, Italy, Japan, UK, and USA are also significant at 5 as well as 10 percent level of significance. These results reveal that business cycle asymmetries do prevail in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. For the purpose of detecting possible existence of nonlinearities in all the series, these results would suffice, however, the analysis is extended to jackknife out-of-sample framework to detect possible existence of nonlinearities in all the series in out-of-sample framework, and to study out-of-sample performance of the models compared to their in-sample performance and with the results from the previous studies in this area. In this context, for example, column 4 rows 1 through 7 in Table 2 presents neural network tests constructed from jackknife out-of-sample forecasts for Canada, France, Germany, Italy, Japan, UK, and USA series. The relevant critical values from F distribution are shown beneath each test statistic in brackets, and p-value in parenthesis. Neural network linearity test for possible existence of business cycle nonlinearities, using in-sample forecasts from neural networks with univariate linear models, show statistically significant evidence of asymmetries in all the series. Similarly, neural network nonlinearity tests constructed from jackknife outof-sample forecasts from neural networks also show statistically significant evidence of business cycle asymmetries in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates.

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3.4. Forecast Performance of Neural Network Models Table 2 shows in-sample root mean square error (RMSE) computed from insample as well as jackknife out-of-sample forecasts from neural network models for all the series. In this Table, RMSE for in-sample forecasts are shown in column 3 rows 1 to 7 and RMSE for jackknife out-of-sample forecasts are shown in column 5 rows 1 to 7 respectively for Canada, France, Germany, Italy, Japan, UK, and USA. Comparing RMSE from in-sample to jackknife out-ofsample forecasts, for example for Canada, it transpires that in-sample forecast performance of neural network models for Canada is superior to jackknife out-ofsample forecasts from neural networks because RMSE from in-sample forecasts is lower than what is computed from jackknife-out-of-sample forecasts. These results hold for the remaining series, and are in line with previous studies including Terasvirta, et al. (1993), and Kiani et al. (2005). The results on nonlinearity for USA are in line with Bidarkota (2000) and Andreano and Savio (2002). Similarly the results on nonlinearity for Canada, Italy, Japan, and USA are in line with Andreano and Savio (2002). Moreover, the results for Canada, Italy, Germany, Japan, and USA are in line with Kiani and Bidarkota (2004). This shows that evidence against linearity for Canada, Italy, Japan, and USA is robust to change in-samples as well as change in testing approach. Neural network models outperform the traditional statistical tests for remaining nonlinearities, which is in line with the previous studies including Terasvirta, et al. (1993). Similarly, in-sample forecast performance of neural network models is superior to linear model which is in line with Kiani et al. (2005).

4. CONCLUSION In the present work artificial neural networks (ANN) are employed with univariate linear models for possible existence on business cycle asymmetries in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates based on in-sample and jackknife out-of-sample forecasts from linear models and those approximated from neural networks. Neural network linearity tests are constructed from in-sample as well as jackknife out-of-sample forecasts for all the series.


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The study results based on in-sample forecasts provide a robust evidence of asymmetries in business cycle fluctuations in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. In addition, the results based on jackknife out-of-sample forecasts strengthen the evidence for existence of business cycle asymmetries in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. Therefore, the impact of monetary policy or any other shock on output in these countries cannot be determined using linear models because the underlying data generating processes are nonlinear. These findings also reveal that in-sample forecasts from neural networks are superior to the jackknife out-of-sample forecasts from neural networks that is in line with previous studies (Kiani at al. 2005). The results show statistically significant evidence of business cycle asymmetries in all the countries studied. Therefore, the impact of monetary policy or any other shock on output in Canada, France, Germany, Italy, Japan, UK, and USA cannot be determined using linear model and those derived from vector autoregressions.

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Bernanke, B. (1983). Nonmonetary Effects of the Financial Crisis in Propagation of the Great Depression. American Economic Review, 73(3), 257-76. Bidarkota, P. (1999). Sectoral investigation of asymmetries in the conditional mean dynamics of the real US GDP. Studies in Nonlinear Dynamics and Econometrics, 3, 191-200. Bidarkota, P. (2000). Asymmetries in the conditional mean dynamics of real GNP: robust evidence. The Review of Economics and Statistics, 82, 153-157. Brunner, A. (1992). Conditional asymmetries in real GNP: a semi-nonparametric approach. Journal of Business & Economics Statistics, 10, 65-72. Brunner, A. (1997). On the dynamic properties of asymmetric models of real GNP. The Review of Economics and Statistics, 79, 321-326. Burns, A., & Michell, W. (1946). Measuring business cycles. NBER, New York. Davies, R. (1977). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, 64, 247–254. De Jong, K. (1975). An analysis of the behavior of a class of Genitive Adaptive System unpublished Ph.D. dissertation. University of Michigan, Department of Computer Science. DeLong, B., & Summers, L. (1986). Are business cycles symmetrical? In Gordon, R.J. (Eds.), The American Business Cycle: Continuity and Change, Chicago, IL, Chicago University Press. Diebold, F., & Rudebusch, G. (1990). A nonparametric investigation of duration dependence in the American business cycle. Journal of Political Economy, 98, 596-616. Dorsey, R., & Walter, J. (1995). Algorithm for Estimation Problems with Multiple Optima, Nondifferentiability, and other Irregular Features. Journal of Business Economics and Statistics, 13, 53-66. Estrella, A., & Mishkin, F. (1998). Predicting U.S. Business Cycle Regimes. Journal of Business and Economic Statistics, 17, 313-323. Falk, B. (1986). Further evidence on the asymmetric behavior of economic time series over the business cycle. Journal of Political Economy, 94, 1096-1109. Garcia, R., &Gencay, R. (2000). Pricing and hedging derivative securities with neural networks and a homogeneity hint. Journal of Econometrics, 94, 93115. Gencay, R. (1999). Linear, nonlinear and essential foreign exchange prediction with simple technical trading rules. Journal of International Economics, 47, 91-107. Goldberg, D. (1989). Genetic Algorithm in Search, Optimization and Machine Learning’s. Addison Wisely, Readings, MA.


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Politis, D., Romeo, J., & Wolf, M. (1997). Sub-sampling for heteroskedastic time series. Journal of econometrics, 81, 81-317. Potter, S. (1995). A non-linear approach to U.S. GNP. Journal of Applied Econometrics, 10, 109-125. Qi, M. (2001). Predicting US recessions via leading indicators via neural network models. International Journal of Forecasting, 17, 383-401. Qi, M., & Madala, G. (1999). Economic factors and the stock market: a new perspective. Journal of Forecasting, 18, 151-166. Quenouille, M. (1949). A note on bias in estimation. Biometrika, 43, 353-60. Ramsey, J., & Rothman, P. (1996). Time irreversibility and business cycle asymmetry. Journal of Money Credit and Banking, 28, 1-21. Reilly, D., & Cooper, L. (1990). An overview of neural networks: early models to real world systems. In Zornetzer, J., Davis, L., & Lau, C. (Eds.), An Introduction to Neural and Electronic Networks, New York: NY. Schumpeter, J. (1939). Business cycles: a theoretical, historical, and statistical analysis of the capitalist process. McGraw-Hill:New York. Sichel, D. (1989). Are business cycles asymmetric? a correction. Journal of Political Economy, 97, 1255-1260. Swanson, N., & White, H. (1995). A model selection approach to assessing the information in the term structure using linear model and artificial neural networks. Journal of Business Economics and Statistics, 13, 265-275. Swanson, N., & White, H. (1997a). A model selection approach for real time macroeconomic forecasting using linear models and artificial neural networks. The Review of Economics and Statistics, 79, 540-550. Swanson, N., & White, H. (1997b). Forecasting economic time series using flexible vs. fixed and linear vs. nonlinear economic models. International journal of Forecasting, 13, 439-461. Stock, J., & Watson, M. (1999). Business cycle fluctuations in US macroeconomic time series. In Taylor, J.B., & Woodford, M. (Eds.), Handbook of macroeconomics, Volume 1, Elsevier Science: B.V. Terasvirta, T., Lin, C., & Granger, C. (1993). Power of neural networks test. Journal of Time Series Analysis, 14, 209-220. Tuckey, J. (1958). A bias and confidence in not-quite large samples. Annals of Mathematical Statistics (abstracts), 29, 614-623. Vishwakarma, K. (1995). A neural networks to forecast business cycle indicators. Mathematics and Computers in Simulations, 39, 287-291. White, H. (1989a). Some asymptotic results for learning in single hidden-layer feed forward network models. American Statistical Association, 84, 10031013.


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White, H. (1989b). Learning in artificial neural networks: A statistical perspective. Neural Computations, 1, 425-464. Wu, C. (1990). On the asymptotic properties of the jackknife histogram. Annals of Statistics, 18, 1438-1452. Ziari,H., Leatham, D., & Ellinger, P. (1997). Developing of statistical discriminant mathematical programming model via re-sampling estimation techniques. American Journal of Economics, 79, 1352-1362.

Chapter 4

ASYMMETRIC BUSINESS CYCLE FLUCTUATIONS AND CONTAGION EFFECTS IN G7 COUNTRIES ABSTRACT Real gross domestic product (GDP) growth rates are employed for studying possible existence of business cycle asymmetries in the group of seven (G7) industrialized countries, i.e. Canada, France, Germany, Italy, Japan, UK and USA. Asymmetries in these countries are modeled using artificial neural networks (ANN) via in-sample as well as jackknife out-ofsample forecasts. In addition, asymmetric business cycles and contagion effects are explored in Canada, France, Germany, Italy, Japan, UK and USA real GDP growth rates using bivariate framework. The results show statistically significant evidence of asymmetries in business cycle fluctuations in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rate series. The results on business cycle asymmetries for Canada Italy, Japan, UK, and USA are in line with Andreano and Savio (2002), and Kiani and Bidarkota (2004), in exception of France and UK that show statistically significant evidence of neglected nonlinearities. Therefore, the impact of monetary policy or any other shock on the output in these countries cannot be determined using linear models including those derived from linear vector autoregressions. In addition, using bivariate framework, similar results are obtained on business cycle asymmetries. In addition existence of statistically significant evidence of contagion effects does exists in all the series.

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Khurshid M. Kiani

Keywords: asymmetries; business cycles; neural networks; nonlinearities; vector autoregressions JEL codes: C32, C45, E32

1. INTRODUCTION A wide body of empirical research that focused on detecting business cycle asymmetries in economic fluctuations employed USA macroeconomic time series and concluded that asymmetries in business cycle fluctuations do exist in macroeconomic time series. In this context studies by Beaudry and Koop (1993), Brunner (1992-1997), and Bidarkota (1999-2000) including others investigated USA gross national product (GDP), and showed existenceof business cycles asymmetries in the series. Likewise, Neftci (1984), Ramsey and Rothman (1996) including others studied USA unemployment rates, and concluded for predominance of asymmetric business cycle fluctuations in the series. Similarly, Potter (1995), Anderson and Vahid (1998), and Anderson and Ramsey (2002) also showed existence of business cycle asymmetries in macroeconomic time series. Contrary to that Falk (1986), Sichel (1989), Delong and summers (1986), and Diebold and Rudebusch (1990) either failed to reveal asymmetries in business cycle fluctuations or were unable to find significant evidence of business cycle asymmetries in the series they studied. Nonlinearities imply that the effects of contractionary and expansionary monetary policy and other shocks on output are asymmetric. Therefore, any nonlinearity would invalidate the measures of the persistence of monetary policy or any other shock on output that is based on linear models including those derived from linear vector autoregressions when the underlying data generating process is nonlinear. Policymakers would be interested to know the impact of monetary policy or any other shock on output; however, one might not be able to foretell the impact of such shocks on output that would be based on linear models when the underlying data generating process is nonlinear. Therefore, it is imperative to detect possible existence of nonlinearities in data series so that appropriate forecasting models (linear or nonlinear) are employed to anticipate the impact of monetary policy or other shocks on output. Moreover, it would be of interest for macro theorists if business cycle fluctuations in all the countries of the world are alike. If they were not, the macro-theorists would need to come up with

Asymmetric Business Cycle Fluctuations and Contagion Effects…

65

new theories of business cycles without taking into account the underlying country specific institutional factors. While studies by Auerbach (1982), Gordon (1986), Kling (1987), Koch and Rasche (1988), Diebold and Rudebusch (1990), Hamilton (1989), Klein (1990), Estrella and Mishkin (1998) focused on business cycle research, only a few studies investigated possible existence of asymmetries in business cycle fluctuations using international data. For example, Andreano and Savio (2002) investigated business cycle asymmetries in G7 countries using Markov Switching models but they were not able to detect asymmetries in France, Germany, and UK series. Similarly, Kiani and Bidarkota (2004) studied possible existence of business cycle asymmetries in real GDP series in G7 countries but despite using nonlinear, and switching time series models with stable distributions, and long memory, Kiani and Bidarkota were not able to find possible existence of business cycle asymmetries in France and U.K series. This necessitated undertaking another study to address the basic question whether business cycles in all the countries of the world starting with the G7 countries are alike is still unanswered. The present study fills this gap adequately modeling real GDP growth rates in G7 countries using artificial neural networks (ANN) that are considered to be highly flexible functional form of nonlinear models that can fit to any data series without taking into consideration the distribution of the underlying data generating process. Neural Networks have been applied successfully in many disciplines including business and economics. For example, Kuan and White (1994), and Swanson and White (1995, 1997a, 1997b) employed ANN in economics. Hutchinson, Lo, and Poggio (1994), Garcia and Gencay (2000), and Qi and Madala (1999), and Gencay (1999) employed ANN in finance. However, only Vishwakarma (1995), Qi (2001), Kiani (2005), and Kiani, Bidarkota, and Kastens (2005) focused on business cycles using neural networks. Therefore, in-sample as well as jackknife out-of-sample forecasts are approximated from neural networks, and linear models to construct neural network linearity tests that were originally proposed by Terasvirta, Lin and Granger (1993) for possible existence of business cycle asymmetries in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. Moreover, the present analysis is extended to bivariate framework to reveal further evidence of asymmetric fluctuations, linkages, and spillover effects within these countries. The remaining study is split into the following sections. Section 2 shows a brief description of empirical models, and underlying tests, and section 3 shows data sources, preliminary data analysis, hypotheses tests, empirical results on

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hypotheses tests, and forecast performance of neural network models. Section 4 incorporates brief conclusions.

2. EMPIRICAL MODEL: ARTIFICIAL NEURAL NETWORK Artificial neural networks (ANN) pertain to artificial intelligence technology that mimics human brain's learning and decision-making process. The ability of information processing makes ANN powerful computational devices that can learn from examples and generalize these learning to solve problems never seen before (Reilly and Cooper 1990). ANN are nonlinear, nonparametric statistical methods which are independent of the distributions of underlying data generating processes (White 1989b). The present research employs ANN to investigate possible existence of business cycle nonlinearities in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates using in-sample forecasts approximated from ANN that is an extension to Kiani and Bidarkota (2004). Likewise, the analysis pertaining to testing business cycle asymmetries is extended to neural network linearity tests using bivariate framework that are constructed from in-sample as well as jackknife out-of sample approximations from neural networks to investigate behavior of these models out-of-sample, and to observe spillover and contagion effects within G7 countries in bivariate framework. Quenouille (1949) used jackknife re-sampling to reduce the bias in estimators, and thereafter Tuckey (1958) employed jackknife re-sampling for estimating variances. However, Wu (1990) introduced sub-sample jackknife technique, which was also used by Politis and Romeo (1994). Later, Politis et al. (1997), and Ziari et al. (1997) also used this re-sampling technique. In the subsample jackknife, more than one observation is dropped to estimate out-of-sample forecast of the remaining m = n − d observations, where, n is the total number of observations, and d = 2,3,......... ....., n − 1 when compared to the standard jackknife.

2.1. Neural Network Linearity Test The model for constructing neural network linearity test due to Terasvirta et al. (1993) comprise of a linear model shown in Equations 4.1 as well as a neural network model as of Equation 4.2 . Although this model is constructed to work with lagged exogenous variables from more than one data series, it can be


67

restricted to contemporaneous independent and lagged dependent variables from one series only. A general form neural network linearity test can be shown in the following two Equations. yt = π ' wt + ut

4.1.

where, u t ~ N (0, σ 2 ),

and

~ ' )' , wt = (1, w t

~ = ( y ,..., y , x ,..., x )' w t t −1 t− p t −1 t− p

π = ( π 0 , π 1 ........, π p )'

ut = π ' wt +

k

∑θ

0 j {ψ

(γ j ' wt )}

j =1

where, in Equation 4.2 ψ (γ ' wt ) = (1 + exp{−γ ' wt )) −1 and

4.2.

π 0 is intercept. Equation

4.2 shows a nonlinear neural network model that nests linear model shown in Equation 4.1 . Under normality conditions, the test statistics for this test is distributed approximately F [ m, ( n − p − m − 1)] , which is computed using Equation 4.3 . The distribution of this test statistic is approximate because of the nuisance parameter appearing under the alternative hypothesis (Davies, R. 1977; and Andrews, W. 2001). TS = {(SSE1 − SSE 2) / m} /{SSE 2 / (n − p − m − 1)}

4.3.

where, in Equation 4.3 , m denotes the number of restrictions, n the number of observations, and p the number of lags in the linear, and ANN models. Quarterly real GDP series for Canada, France, Germany, Italy, Japan, UK, and USA were obtained from November 2006 version of the International Financial Statistic’s CD-ROM. The dataset spans from 1957:1 to 2006:2 for all countries except for France, for which the data spans from 1965Q1-2006Q2, and Germany, and Italy for which the data ranges from 1960:1 to 2006Q2. Figure 1 plots real GDP growth rates for Canada, France, Germany, Italy, Japan, UK, and USA series, and Table 1 shows additional information for all the series.

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Table 1. Data Description

Data Series Sample Period Sample Length

Canada

France

Germany Italy

Japan

UK

USA





Quarterly Nominal GDP 1957:12006:2


Quarterly Real GDP

197

165

185

185

197

197

197

1957:12006:2

Notes: 1. The quarterly seasonally adjusted GDP data is obtained all countries from the November 2006 edition of the International Financial Statistics (IFS) CD-ROM. 2. Nominal GDP is used for Japan because seasonally adjusted data was available only for the nominal series and not for the real series in the IFS CD-ROM.

Figure 1. Real GDP Growth.

In in-sample approximations are obtained from ANN in conjunction with linear models to construct neural network linearity tests using test statistic of the


69

form shown in the Equation 4.3 for each of the series i.e. Canada, France, Germany, Italy, Japan, UK, and USA. Likewise, jackknife out-of-sample forecasts from ANN are also approximated to construct neural network linearity test for each of the series studied. While neural network linearity test are constructed from in-sample as well as jackknife out-of-sample approximations from ANN in conjunction with univariate linear model forecasts, the present work also considers constructing neural network linearity tests using in-sample as well as jackknife out-of-sample approximations from neural networks in conjunction with forecast from bivartiae linear models. The rational for this extension is to find possible existence of additional evidence of business cycle asymmetries in addition to considering linkages or spillover, and contagion effects between these countries. These types of linkages were also discussed in Anderson and Ramsay (2002) in a bivariate framework for Canada, and USA time series. A general bivariate vector autoregression (VAR) model can be represented by the following Equations: p

p

yt = α 1 +

∑β

1i y t − i

+

i =1

p

xt = δ 1 +

∑ j =1

∑β

2 i xt − i

+

i =1

4.4.

p

γ 1 j yt − j +

∑γ j =1

2 j xt − j

+e

4.5.

where, y t and xt are contemporaneous whereas y t − p and x t − p are lagged real GDP growth rates for example for a bivariate vector autoregression model CAFR that comprises of two series i.e. Canada and France for all p ≥ 1 . The CAFR bivariate model consist of two equations, i.e. CAFR1 which is of the form of Equation 4.4 , and CAFR2 as of Equation 4.5 . These equations are employed to construct two separate neural network linearity tests. In the first part of neural network linearity test, a VAR model CAFR for Canada and France is estimated to recover residuals ( μˆ i for i = 1, 2 ), and residual sum of squares (RSS i for i = 1,2) for each of the equations of this VAR are

recovered. This process is repeated for each of the twenty one bivariate models i.e. CAFR, CAGR, CAIT, CAJP, CAUK, CAUS, FRGR, FRIT, FRJP, FRUK, FRUS, GRIT1, GRJP, GRUK, GRUS, ITJP, ITUK, ITUS, JPUK, JPUS, and UKUS that are estimated using Canada, France, Germany, Italy, Japan, UK, and

Khurshid M. Kiani

70

USA real GDP growth rates for recovering residuals, and (RSS i

) that are finally

employed for constructing neural network linearity tests. In the second part of the bivariate neural network linearity test, in-sample forecasts from ANN are approximated using Equation 4.2 wherein residuals ( uˆ i for i = 1, 2 ) from each of the VAR equation are employed as endogenous variables, and lagged real GDP growth rates ( y t −1 ,.......... y t − k

xt −1 ,.....xt − k

(for k ≥ 0) ) as exogenous variables. For example, for the bivariate VAR model CAFR, two separate neural network linearity test are constructed using its equations CAFR1 and CAFR2 . From this part of the test, the residuals ( vˆt for i = 1, 2 ), and sum of squared residuals SSE i (for i = 1, 2 ) are recovered from each of the neural network model approximated. Finally, test statistic is calculated using Equation 4.6 .

TS = {( RSSi − SSE) / m} /{SSEi /(n − p − m −

4.6.

where in Equation 4.6 , m denotes the number of restrictions, n is the number of observations, and p the number of lags in the linear, and ANN the models. In Equation 4.6 RSS i

( for i = 1,2)

are residual sum of squares from the first

equation whereas SSEi show squared residual sum from the second part of the neural network linearity test. The process employed for constructing neural network linearity tests using bivariate CAFR model is repeated for all the equations of the remaining VAR models estimated for each of the series. In addition to using in-sample forecast from linear models in conjunction with ANNs, jackknife out-of-sample forecasts from linear models and ANN approximations are also employed for constructing neural network linearity tests for Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. To avoid the likelihood of obtaining local optima, genetic algorithm (GA) is employed with a couple of random starts to obtain best parameter vector for neural network approximations which is considered to be the most reliable estimation algorithm for estimating any type of nonlinear functional form including ANN but it appeared to be very slow. Therefore, GA is employed in conjunction with fminsearch routine which is a canned optimization routine from MATLAB which is simplex algorithm that worked satisfactorily although it was still slower than any other optimization algorithm that could have been used for


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estimating nonlinear models. After its applications in biology, and engineering, GA was employed in operation research by Goldberg (1989). However, its economic application was implemented by Axelord (1987), Marimon, McGratten, and Sargent (1990), and Dorsey and Mayer (1995).

3. EMPIRICAL RESULTS 3.1. Preliminary Data Analysis Preliminary data analysis (detailed results not shown for brevity) reveals that skewness is statistically significant, and excess kurtosis show significant fat tails in Canada, France, Germany, Italy, UK, and USA series including Japan that show marginal fat tails. The Jarque-Bera test rejects normality for all countries except Canada. Augmented Dicky Fuller (ADF) test indicates unit roots in levels (with constant and time trend) for all countries but not in growth rates (with constant only). The only exception is Japan for which the test fails to reject unit roots in growth rates with a constant term only but does reject with constant and time trend. Goldfeld-Quandt test fails to reject homoskedasticity in all countries, and Lagrange Multiplier (LM) test detects autoregressive conditional heteroskedasticity (ARCH) in Japan only.

3.2. Estimation Results Table 2 shows test statistics for neural network linearity tests that are constructed from in-sample, and jackknife out-of-sample forecasts from univariate linear models as well as neural networks. In this Table column 2 rows 1 − 7 show test statistics for neural network linearity tests respectively for Canada, France, Germany, Italy, Japan, UK, and USA series. The relevant p-values for each of the test statistic are juxtaposed in the subsequent columns in the same row. Column 5 , rows 1 − 7 show test statistics for neural network linearity tests that are constructed using jackknife out-of-sample forecasts from linear models as well as neural networks. Again the relevant p-values are juxtaposed in the subsequent column in the same row. This Table also show root mean squared errors (RMSEs) for in-sample as well as jack out-of-sample forecasts that are shown in columns 4 and 7 in rows 1 − 7 respectively for Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates.

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Table 2. Neural Network Tests with Univariate Linear Models

Model Canada France Germany Italy Japan UK USA

In-Sample Forecasts Test Statistic p-Values

RMSE

Jackknife Out-of-Sample Forecast Test Statistic p-value RMSE

280.8325 2428.0671 380.9606 672.8380 209.5664 202.0907 346.0786

0.1404 0.1568 0.2169 0.2687 0.1793 0.2201 0.1247

41.4805 520.4057 191.1076 218.7765 36.6850 59.7420 52.9753

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0003 0.0226 0.0000 0.0000 0.0080

0.2918 0.3247 0.2905 0.4444 0.3405 0. 3512 0. 2641

Notes: 1. Column 1 in this Table shows the names of the countries included in the study. 2. Test statistics presented in this Table are based on in-sample as well as jackknife out-ofsample forecasts from univariate linear and neural network. These models are shown in the following two Equations: 3. yt = π ' wt + ut

(2.1a)

where,

u t ~ N (0, σ 2 ),

and

~ ' )' , wt = (1, w t

π = ( π 0 , π 1 ........, π

ut = π ' wt +

k

∑θ

0 j {ψ

~ = ( y ,..., y )' w t t −1 t− p p

)'

(γ j ' wt )} +

j =1

(2.1b)

4. Test statistic for this test is distributed F [ m, ( n − p − m − 1)] under normality hypothesis. The test statistic can be shown using following Equation:

TS = {(SSE1 − SSE 2) / m} /{SSE 2 / (n − p − m − 1)}

(2.1c) where in Equation 2.1 c m denotes the number of restrictions in the unrestricted model,

n is the number of observations, and p is the number of the lags.

5. Column 2 shows test statistics constructed from in-sample forecasts whereas column 5 shows test statistics constructed from jackknife out-of-sample forecasts. The pvalues for each test statistic are juxtaposed in the subsequent column in the same row. 6. In column 4 , root mean squared errors (RMSE) for in-sample forecasts are presented whereas in column 7 , RMSE calculated from jackknife out-of-sample forecasts are shown.


73

Table 3 shows test statistics for neural network linearity tests that are based on in-sample as well as jackknife out-of-sample forecasts from bivariate linear models, and neural networks. In this Table column 2 and 5 show test statistics computed from all the bivariate linear models in conjunction with the relevant neural networks for Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. Relevant p-values for each of the test statistic are juxtaposed in the subsequent column in the same row. The Table also show root mean squared errors (RMSE) for in-sample and jackknife out-of-sample forecasts that are shown in columns 4 and 7 respectively for all the models estimated for Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. Table 3. Neural Network Tests with Bivariate Linear Models In-Sample Forecasts Jackknife Out-of-Sample Forecast Model Test Statistic p-Values RMSE Test Statistic p-Value RMSE CAFR1 270.1986 0.0000 1.0292 16.4824 0.0000 0.3735 CAFR2 2332.0594 0.0000 0.4591 33.2566 0.0000 1.2228 CAGR1 265.5669 0.0000 1.1298 5.6772 0.0003 0.4257 CAGR2 307.3003 0.0000 0.8458 2.9238 0.0226 0.8345 CAIT1 260.8549 0.0000 0.1218 8.4628 0.0000 0.4107 CAIT2 736.0613 0.0000 0.2289 15.2931 0.0000 1.2066 CAJP1 273.4044 0.0000 1.1353 3.5625 0.0080 0.4504 CAJP2 205.0926 0.0000 1.1700 3.3339 0.0117 0.5027 CAUK1 280.1213 0.0000 1.1268 21.8905 0.0000 0.3857 CAUK2 242.9319 0.0000 1.1096 6.2060 0.0001 0.5817 CAUS1 286.7682 0.0000 0.1182 15.4751 0.0000 0.4037 CAUS2 316.0419 0.0000 1.0988 8.2575 0.0001 0.4279 FRGR1 2017.3321 0.0000 0.4850 39.5745 0.0000 1.1219 FRGR2 290.9946 0.0000 0.7832 16.2750 0.9964 0.8074 FRIT1 983.0594 0.0000 0.6914 35.3386 0.0000 1.0014 FRIT2 530.1332 0.0000 0.1829 20.6363 0.0000 1.2042 FRJP1 2160.8583 0.0000 0.4770 48.6139 0.0000 1.1151 FRJP2 199.4415 0.0000 1.0736 0.0185 0.9993 0.4730 FRUK1 2139.9850 0.0000 0.4794 56.9604 0.0000 1.0577 FRUK2 255.1859 0.0000 0.9347 2.0882 0.0843 0.5425 FRUS1 2221.8645 0.0000 0.4704 54.8573 0.0000 1.0666 FRUS2 339.6425 0.0000 0.1197 6.3342 0.0843 0.3880 GRIT1 301.3993 0.0000 0.8540 6.9836 0.0000 0.7881 GRIT2 735.5024 0.0000 0.6535 16.3525 0.0000 1.1293 GRJP1 301.7980 0.0000 0.8540 4.1894 0.5239 0.7702

Khurshid M. Kiani

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Table 3. (Continued)

Model GRJP2 GRUK1 GRUK2 GRUS1 GRUS2 ITJP1 ITJP2 ITUK1 ITUK2 ITUS1 ITUS2 JPUK1 JPUK2 JPUS1 JPUS2 UKUS1 UKUS2

In-Sample Forecasts Test Statistic p-Values RMSE 193.2506 0.0000 1.1915 313.5724 0.0000 0.8372 241.2715 0.0000 1.0500 377.9493 0.0000 0.7678 307.2726 0.0000 1.0590 788.0186 0.0000 0.6316 199.9694 0.0000 1.1673 661.9456 0.0000 0.6871 250.8894 0.0000 3.7961 667.8976 0.0000 0.6848 332.7416 0.0000 1.0189 169.5096 0.0000 1.3045 184.0335 0.0000 1.2437 185.8725 0.0000 1.2511 306.3379 0.0000 1.0978 180.6804 0.0000 1.2559 294.7611 0.0000 1.1199

Jackknife Out-of-Sample Forecast Test Statistic p-Value RMSE 2.9310 0.0221 0.4795 7.3175 0.0000 0.7536 18.1193 0.0000 0.4684 15.6002 0.0000 0.7434 19.0447 0.0000 0.1139 29.3709 0.0000 1.0908 1.8838 0.1153 0.4927 17.9637 0.0000 0.5019 17.9637 0.0000 0.5019 19.7459 0.0000 1.1672 12.6557 0.0000 0.3922 3.4933 0.0090 0.5052 5.4035 0.0004 0.5915 13.8287 0.0000 0.5219 7.2141 0.0000 0.4275 10.3327 0.0000 0.5658 4.8548 0.0010 0.4380

Notes: 1. Column 1 in this Table shows the names of the neural network models employed for approximating in-sample as well as jackknife out-of-sample forecasts with bivariate linear models. 2. Test statistics presented in this Table are constructed using in-sample and jackknife outof-sample forecasts from bivariate linear models and those approximated from the relevant neural networks using the following two Equations:

yt = π ' wt + ut

(3.1a)

where,

u t ~ N (0, σ 2 ),

and

~ ' )' , wt = (1, w t

π = ( π 0 , π 1 ........, π

ut = π ' wt +

k

∑θ j =1

0 j {ψ

(γ j ' wt )} + vt

~ = ( y ,..., y , x ,..., x )' w t t −1 t− p t −1 t− p p

)' (3.1b)

Test statistics shown in columns 2 and 5 are constructed respectively using insample and jackknife out-of-sample forecasts that are distributed


75

F [m, (n − p − m − 1)] under normality hypothesis. The test statistic is calculated using Equation 3.1 c .

TS = {(SSE1 − SSE 2) / m} /{SSE 2 / (n − p − m − 1)} where, in Equation 3.1 c , observations, and

m denotes the number of restrictions, n

(3.1c) the number of

p the number of lags in the linear and ANN models. In Equation

3.1 c RSS i (for i = 1, 2 ) are residual sum of squares from the first equation whereas SSE i show squared residual sum from the second part of the neural network linearity test. 3. Test statistics for bivariate linear models using in-sample forecasts are presented in column 3 and those with jackknife out-of-sample forecasts are shown in column 5 . The relevant p-values for each test statistics are juxtaposed with each test statistics in parenthesis. 4. Columns 4 and 6 show root mean squared errors (RMSE) respectively for in-sample and jackknife out or sample forecast for each of the neural network model approximated.

3.3. Hypotheses Tests The chief hypothesis of this study is linearity versus the alternative hypothesis of nonlinearity for testing possible existence of asymmetries in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth. The linearity hypothesis is based on the test statistic that is constructed from in-sample forecast approximated from neural network models with its linear as well bivariate linear counterparts for Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. The linearity hypothesis is also tested using jackknife out-of-sample forecasts from linear models as well as ANN approximations for all the series. When the null hypothesis of linearity is true, nonlinearity does not prevail in the series being tested, alternately when the null is false, nonlinearities do prevail in the data series being tested. The test statistic for each of the series is calculated using Equation 2.3 . This test statistic is distributed F [m, (n-p-m-1)] under linearity hypothesis when considering neural network linearity test using univariate linear models. Where, in Equation 2.3 , m is the number of restrictions in the nonlinear model, n is the total number of observations in the series being tested, and p is the numbers of lags used in each regression. Likewise, neural network linearity tests are also constructed in conjunction with bivariate linear

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Khurshid M. Kiani

models using in-sample as well as jackknife out-of-sample forecasts from each of the model estimated using Equation 2.4 .

3.4.Results on Hypothesis Test Results on hypotheses tests for neural network linearity tests that are based on in-sample as well jackknife out-of-sample forecasts from univariate linear models, and neural networks are presented in Table 2 . Similarly, the results on hypotheses tests for neural network linearity tests that are constructed from bivariate linear models in conjunction with neural network are shown in shown in Table 3 . Neural network test statistics presented in Table 2 , column 2 , row 1 for Canada is statistically significant at 5 percent level of significance based on pvalues that are shown in column 3 row 1 of this Table. Inferences do not change when significance level is changed from 5 percent to 10 percent level. Neural network test statistics for the remaining countries i.e. France, Germany, Italy, Japan, UK, and USA are also significant at 5 as well as 10 percent level of significance. These results reveal that a statistically significant evidence of business cycle asymmetries does exists in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. For the purpose of detecting possible existence of nonlinearities in all the series, these results would suffice, however, this analysis is extended to jackknife out-of-sample framework for possible existence of nonlinearities in all the series after accounting for linkages between these countries in the form of contagion, and spillovers that might be effecting business cycle asymmetries in these countries in bivariate framework. Test statistics for all the series are statistically significant at all levels based on the relevant p-values. Thus, the results on neural network linearity tests show existence of a robust evidence of asymmetries in business cycle fluctuations in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. While the objective of the present study is to detect possible existence of business cycle asymmetries in Canada, France, Germany, Italy, Japan, UK, and USA series, contagion effects in G7 countries in bivariate framework is also focused. Inclusion of such tests in this analysis is intended to test additional evidence of nonlinearities in all the series (if any) in addition to exploring possible existence of contagion effects and to observe how one country’s business cycle would affect the magnitude of business cycle fluctuation of the other country. For example, for the bivariate model CAUS, the neural network linearity test statistic


77

shown in Table 2 column 2 row 12 indicates that USA business cycle is not affected much when Canadian GDP series is included in the bivariate model, however, test statistic in the previous row (row11) in same column shows that inclusion of USA series in bivariate model for Canada reduces the magnitude of test statistic showing that USA economy has more pronounced impact on the Canadian business cycles. This can be attributable to the fact that despite the size of their economies, USA, and Canada being neighboring countries depend heavily on each other’s products. For example, about 84 percent of Canada’s export goods are sent to USA and 57 percent of Canada’s import come from USA. Similarly, USA imports about 13 percent of its total import from Canada, and 23 percent of USA’s total export go to Canada. Using test statistic for the remaining bivariate models shown in the Table, one can analyze the impact of France, Germany, Italy, Japan and UK series on Canadian business cycles, and impact of Canadian economy on these countries’ business cycles. Similarly, impact of business cycle fluctuations among other countries included in the study can also be analyzed in a similar manner. Neural network nonlinearity test for detecting possible existence of business cycle nonlinearities using in-sample forecasts from neural networks with univariate linear, and bivariate linear models show statistically significant evidence of asymmetries in all the series. Similarly, neural network nonlinearity tests constructed from jackknife out-of-sample forecasts from neural networks, and its linear counterparts also show statistically significant evidence of business cycle asymmetries in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. However, in four out of forty cases that are studied using jackknife out-of-sample forecasts, neural network linearity tests fail to reject linearity hypothesis at all level whereas in two out of forty two cases linearity could not be rejected at conventional (5 percent) level of significance.

3.5. Forecast Performance of Neural Networks Figure 2 shows plots of in-sample forecasts from linear model against ANN approximations for Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. Comparing linear versus ANN forecasts this Figure shows that neural network models explain data series better than those of linear models for Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates.

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Khurshid M. Kiani

Figure 2. Forecasts from Linear Models and Neural Networks.

Table 2 shows in-sample root mean square error (RMSE) computed from insample as well as jackknife out-of-sample forecasts from neural network models for all the series. In this Table, RMSE for in-sample forecasts are shown in column 4 rows 1 to 7 and RMSE for jackknife out-of-sample forecasts are shown in column 7 rows 1 to 7 respectively for Canada, France, Germany, Italy, Japan, UK, and USA. Comparing RMSE calculated from in-sample to jackknife out-of-sample approximations from ANN for Canada for example, it transpires that jackknife out-of-sample forecast performance of neural network models is not superior to their in-sample forecasts. Similar results (not discussed for brevity) are obtained for the remaining series studied.

4. CONCLUSION The present work considers the usefulness of artificial neural networks to find possible existence of business cycle asymmetries in Canada, France, Germany, Italy, Japan, UK, and USA real GDP growth rates. Possible existence of business cycle asymmetries in all the series are studied using in-sample, and jackknife out-


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of-sample forecasts that are estimated from univariate and bivariate linear models in conjunction with their relevant ANN models. The results based on in-sample approximations from artificial neural networks, and other models show robust evidence of asymmetries in business cycle fluctuations in Canada, France, Germany, Italy, Japan, UK, and USA series. Likewise, the results based on jackknife out-of-sample forecasts strengthens the evidence for possible existence of business cycle asymmetries in all the series. These results reveal that the impact of monetary policy or any other shock on output in these countries cannot be determined that are based on forecasts obtained from linear models, and linear vector autoregressions. The results reveal that in-sample forecasts from neural networks are superior to their jackknife out-sample forecasts. In addition, results from bivariate analysis show spillover and contagion effects among G7 countries in bivariate framework. While these linkages can be viewed as trade and other linkages, and trade linkages between USA, and Canada business cycles can easily be explained because these neighboring highly industrialized countries depend heavily on each other’s products. Moreover, North American Free Trade Agreement (NAFTA) that includes Mexico also facilitates trade between these countries. Similar type of linkages can also be noticed between other countries in bivariate framework. However, it might be expeditious to determine the linkages other than trade linkages between these countries that might be effecting business cycle fluctuations when using bivariate framework. Neural network models outperform the traditional statistical tests for remaining nonlinearities, which is in line with the previous studies (Terasvirta, et al. 1993). However, in-sample forecast performance of neural network models is superior to univariate linear models (Kiani et al. 2005). Additionally, the effects of one country’s business cycle fluctuations on the other country in bivariate framework can be viewed as business cycle linkages or contagion effects due to trade and other linkages between them. For example the size of an economy and magnitude of trade between any two countries appear to be two important factors to determine high or a low magnitude of business cycle linkages or contagion effects. It might be of interest if future research employs additional models, and tests using data from G7 countries to explore the nature of trade and other linkages and their share in these countries in addition to detecting the impact of these linkages on economic fluctuations within these economies.

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5. REFERENCES Anderson, H., & F. Vahid, (1998). Testing Multiple Equation Systems for Common Nonlinear Components. Journal of Econometrics, 84, 1-36. Anderson, H., & Ramsey, J. (2002). U.S. and Canadian Industrial Production Indices as Coupled Oscillators. Journal of Economic Dynamics and Control, 26, 33-67. Andreano, M., & Savio, G. (2002). Further Evidence on Business Cycle Asymmetries in G7 Countries. Applied Economics, 34, 895-904. Andrews, D. (2001). Testing When Parameter is Under the Boundary of the Maintained Hypothesis. Econometrica, 69, 683-734. Axelrod, R. (1987). The Evolution of the Strategies in the Iterated Prisoner’s Dilemma. In Algorithm and Simulated Annealing, Davis, L. D. (Eds.), Los Altos, CA: Morgan Kaufmann. 32 - 41. Auerbach, A. (1982). The Index of Leading Indicators: Measurement without Theory, Thirty-Five Year Later. Review of Economics and Statistics, 64, 589595. Beaudry, P., & Koop, G. (1993). Do Recessions Permanently Change Output? Journal of Monetary Economics, 31, 149-163. Bidarkota, P. (1999). Sectoral Investigation of Asymmetries in the Conditional Mean Dynamics of the real US GDP. Studies in Nonlinear Dynamics and Econometrics, 3, 191-200. Bidarkota, P. (2000). Asymmetries in the Conditional mean Dynamics of Real GNP: Robust Evidence. The Review of Economics and Statistics, 82, 153-157. Brunner, A. (1992). Conditional Asymmetries in Real GNP: A SemiNonparametric Approach. Journal of Business & Economics Statistics, 10, 65-72. Brunner, A. (1997). On the Dynamic Properties of Asymmetric Models of real GNP. The Review of Economics and Statistics, 79, 321-326. Davies, R. (1977). Hypothesis Testing When a Nuisance Parameter is Present under the Alternative. Econometrica, 69, 683-733. DeLong, B., & Summers, L. (1986). Are Business Cycles Symmetrical? In The American Business Cycle: Continuity and Change, Gordon, R.J. (Eds.), Chicago IL: Chicago University Press. 166-179. Diebold, F., & Rudebusch, G. (1990). A Nonparametric Investigation of Duration Dependence in the American Business Cycle. Journal of Political Economy, 98, 596-616.


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Dorsey, R., & Walter, J. (1995). Algorithm for Estimation Problems with Multiple Optima, Nondifferentiability, and other Irregular Features. Journal of Business Economics and Statistics, 13, 53-66. Estrella, A., & Mishkin, F. (1998). Predicting U.S. Business Cycle Regimes. Journal of Business and Economic Statistics, 17, 313-323. Falk, B. (1986). Further Evidence on the Asymmetric Behavior of Economic Time Series over the Business Cycle. Journal of Political Economy, 94, 10961109. Garcia, R. and R. Gencay, (2000). Pricing and Hedging Derivative Securities with Neural Networks and a Homogeneity Hint. Journal of Econometrics, 94, 93115. Gencay, R. (1999). Linear, Nonlinear and Essential Foreign Exchange Prediction with Simple Technical Trading Rules. Journal of International Economics, 47, 91-107. Goldberg, D. (1989). Genetic Algorithm in Search, Optimization and Machine Learnings. Readings, MA: Addison Wisely. Gordon, R. (1986). The American Business Cycle: Continuity and Change. NBER Studies in Business Cycles, Vol. No. 25, Chicago Illinois: University of Chicago Press. Hamilton, J. (1989). A New Approach to the Economic Analysis of Nonstationary Time Series and Business Cycle. Econometrica, 57, 357-384. Hutchinson, J., Lo, A., and Poggio, T. (1994). A Nonparametric Approach to Pricing and Hedging Derivative Securities via Learning Networks. Journal of Finance, 49, 851-889. Kiani, K. (2005). Detecting business cycle asymmetries using artificial neural network and time series models, Computational Economics, 26, 65-85. Kiani, K., Bidarkota, P., & Kastens, T. (2005). Forecast Performance of Neural Networks and Business Cycle Asymmetries. Applied Financial Economics Letters, 1, 205-210. Kiani, K., & Bidarkota, P. (2004). On Business Cycle Asymmetries in G7 Countries. Oxford Bulletin of Economics and Statistics, 66, 333-351. Klein, P. (1990). Analyzing Modern Business Cycles: Essays in Honor of Geoffery H. Moore. M.E. Sharpe, M.E. (Eds.). NY: Armonk. Kling, J. (1987). Predicting the Turning Points of Business and Economic Time Series. Journal of Business, 60, 201-238. Koch, P., & Rasch, R. (1988). An Examination of the Commerce Department Leading Indicator Approach. Journal of Business and Economic Statistics, 6, 167-187.

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Kuan, C., & White, H. (1994). Artificial Neural Networks: An Economic Perspective. Econometric Review, 13, 1-91. Marimon, R., McGratten, E., & Sargant, T. (1990). Money as Medium of Exchange in an Economy with Artificially Intelligent Agents. Journal of Economics Dynamics and Control, 14, 329-373. Neftci, S. (1984). Are Economic Time Series Asymmetric over the Business Cycle? Journal of Political Economy, 92, 307-328. Politis, D., & Romeo, J. (1994). Large Sample Confidence Region Based on SubSample under Minimal Assumption. Annals of Statistics, 22, 2031-2052. Politis, D., Romeo, J. & Wolf, M. (1997). Sub-Sampling for Heteroskedastic Time Series. Journal of econometrics, 81, 281-317. Potter, S. (1995). A Non-Linear Approach to U.S. GNP. Journal of Applied Econometrics, 10, 109-125. Qi, M. (2001). Predicting U.S. Recessions via Leading Indicators via Neural Network Models. International Journal of Forecasting, 17, 383-401. Qi, M., & Madala, G. (1999). Economic Factors and the Stock Market: A New Perspective. Journal of Forecasting, 18, 151-166. Quenouille, M. (1949). A note on Bias in Estimation. Biometrika, 43, 353-60. Ramsey, J., & Rothman, P. (1996). Time Irreversibility and Business Cycle Asymmetry. Journal of Money Credit and Banking, 28, 1-21. Reilly, D., & Cooper, L. (1990). An Overview of Neural Networks: Early Models to Real World Systems. In An Introduction to Neural and Electronic Networks, Zornetzer, J., Davis, L., & C. Lau. C. ed., New York: NY. Sichel, D. (1989). Are Business Cycles Asymmetric? A Correction. Journal of Political Economy, 97, 1255-1260. Swanson, N., & White, H. (1995). A Model Selection Approach to Assessing the Information in the Term Structure Using Linear Model and Artificial Neural Networks. Journal of Business Economics and Statistics, 13, 265-275. Swanson, N., & White, H. (1997a). A Model Selection Approach for Real Time Macroeconomic Forecasting Using Linear Models and Artificial Neural Networks. Review of Economics and Statistics, 79, 540-550. Swanson, N., & White, H. (1997b). Forecasting Economic Time Series Using Flexible vs. Fixed and Linear vs. Nonlinear Economic Models. International Journal of Forecasting, 13, 439 - 461. Terasvirta, T., Lin, C., & Granger, C. (1993). Power of Neural Networks Test. Journal of Time Series Analysis, 14, 209-220. Tuckey, J. (1958). A Bias and Confidence in Not-Quite Large Samples. Annals of Mathematical Statistics (abstracts), 29, 614-623.


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Vishwakarma, K. (1995). A Neural Networks to Forecast Business Cycle Indicators. Mathematics and Computers in Simulations, 39, 287-291. White, H. (1989a). Some Asymptotic Results for Learning in Single HiddenLayer Feed Forward Network Models. American Statistical Association, 84, 1003-1013. White, H. (1989b). Learning in Artificial Neural Networks: A Statistical Perspective. Neural Computations, 1, 425-464. Wu, C. (1990). On the Asymptotic Properties of the Jackknife Histogram. Annals of Statistics, 18, 1438-1452. Ziari, H., Leatham, D., & Ellinger, P. (1997). Developing of Statistical Discriminant Mathematical Programming Model via re-Sampling Estimation Techniques. American Journal of Economics, 79, 1352-1362.

Chapter 5

BUSINESS CYCLE ASYMMETRIES IN ASIAN ECONOMIES VIA NONLINEAR TIME SERIES MODELS AND NEURAL NETWORKS ABSTRACT The present work investigates possible existence of asymmetries in business cycle fluctuations in the selected Asian economies that include Armenia, Indonesia, Japan, Kazakhstan, Malaysia, Philippines, Russia, Singapore, Thailand, and Turkey real GDP growth rates. Asymmetries in these countries are modeled using nonlinear augmented and switching time series models. These models account for the presence of outlier, time varying volatility, and long memory that may be present in the series. Finally neural network linearity tests, and neural network tests for neglected nonlinearities are employed for testing existence of nonlinearities, and neglected nonlinearities (if any) in all the series. Results based on various tests constructed from nonlinear time series models show statistically significant evidence of business cycle asymmetries in Armenia, Indonesia, Japan, Malaysia, Philippines, Singapore, Thailand, and Turkey real GDP growth rates. Further, the neural network linearity tests overwhelmingly reject linearity hypothesis in all the series. Moreover, the results for Japan are in line with Kiani and Bidarkota (2004), and Kiani (2005). Finally, the results show that linear models cannot be employed to forecast the impact of monetary policy or any other shock on output in any of the countries studied.

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Key phrases: Real GDP growth rates; fat tails; stable distributions; neural networks; out-of-sample forecasts; long memory; nonlinearities; business cycles JEL Codes: C22, C32, C45, C53

1. INTRODUCTION Detecting possible existence of business cycle asymmetries in macroeconomic time series is important for several reasons. For instance nonlinearities imply that the effects of expansionary and contractionary monetary policy and other shocks on output are asymmetric. Therefore, any nonlinearity would invalidate the measures of the persistence of monetary policy or any other shocks on output that is based on linear models. The behavior of economic variables during different phases of business cycles is important for policymakers who might be interested in anticipating the impact of monetary policy or any other shocks on output for timely policy action that might help avoid downturns in the economy. Studies by Neftici (1984), Brunner (1992, 1997), Beaudry and Koop (1993), Potter (1995), and Ramsey and Rothman (1996), Anderson and Vahid (1998), Bidarkota (1999-2000), Anderson and Ramsey (2002), Kiani and Bidarkota (2004) provided evidence of business cycle asymmetries in real GDP growth rates in G7 countries. With the fair idea that business cycle fluctuations are asymmetric in developed countries it would be of interest to investigate the behavior of business cycle fluctuations in developing economies especially Asian economies. This is because Asian countries comprise of a group of countries1 that encompass highly developed economies like Japan, fastest growing economies including Malaysia, Singapore, and Indonesia, and transition economies like Kazakhstan and Russia, oil rich countries like Saudi Arabia, Kuwait, and Kazakhstan as well as a number of developing and less developed countries. However, the data limitations hampered the present empirical exercise to a small number of

1

Afghanistan, Armenia, Azerbaijan, Bahrain, Bangladesh, Bhutan, Brunei, Burma, Cambodia, China, Georgia, Hong Kong, India, Indonesia, Iran, Iraq, Israel, Japan, Jordan, Kazakhstan, Korea, North, Korea, South, Kuwait, Kyrgyzstan, Laos, Lebanon, Malaysia, Maldives, Mongolia, Myanmar, Nepal, Oman, Pakistan, Philippines, Qatar, Russia, Saudi Arabia, Singapore, Sri Lanka, Syria, Taiwan, Tajikistan, Thailand, Turkey, Turkmenistan, United Arab Emirates, Uzbekistan, Vietnam, and Yemen.

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countries2 because seasonally adjusted quarterly data for most of these countries was not available. However, despite data limitations the sample employed comprises of fastest growing economies, transition economies, highly developed economy and developing economies. Indeed, the present empirical exercise would help researchers, and policymakers to anticipate the impact of monetary policy or other shock(s) on output particularly in the countries studied using appropriate empirical models, which would help employing appropriate stabilizing policies when needed. For instance linear models would be adequate for forecasting output if the underlying data generating process is symmetric else nonlinear models would be considered adequate. Taking appropriate policy measures would help avoiding anticipated downturns, and likewise timely policy action might help avoiding anticipated future financial crises (if any). There is a growing perception that evidence of nonlinearity reported by several studies could be because of the presence of outliers in the data. In this context Tsay (1988) showed that linearity could be rejected due to the presence of outliers in the data, Blanchard and Watson (1986) concluded for the presence of outliers in GNP data, and Balke and Fomby (1994) and Scheinkman and LeBaron (1989) demonstrated a weak evidence of linearity in US real GNP data once outlier were taken into account. There is a strong consensus that non-normality as well as time varying volatility is present in macroeconomic time series data. However, most studies on business cycle asymmetries do not encompass these features in their models3. Inefficient estimation would result when such features are not included in the models that are employed to detect business cycle asymmetries in macroeconomic time series. The present chapter investigates whether asymmetries in business cycle fluctuations are present in Armenia, Indonesia, Iran, Japan, Kazakhstan, Malaysia, Philippines, Russia, Singapore, Thailand, and Turkey real GDP growth rates using nonlinear time series models that encompass long memory, time varying volatility, and stable distributions. Moreover, considering the difficulties faced by Andreano and Savio (2002), and Kiani and Bidarkota (2004) in detecting business cycle asymmetries in G7 countries, the present work employs artificial neural networks that are considered to be highly flexible functional forms of nonlinear models to study possible existence of business cycle asymmetries (if any) in real GDP growth rates in all the series. Kuan and White (1994) and Swanson and 2

Armenia, Indonesia, Iran, Japan, Kazakhstan, Malaysia, Philippines, Russia, Singapore, Thailand and Turkey 3 Exception is Bidarkota (2000), and Kiani and Bidarkota (2004).

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White (1995, 1997 a, 1997b) employed artificial neural networks in economic time series data. Likewise, Vishwakarma, (1995), Qi, (2001), Kiani (2005), and Kiani et al. (2005) used artificial neural networks in business cycles. The remaining chapter is organized as follows. Section 2 elaborates nonlinear time series models, and artificial neural networks. Estimation results and hypotheses of interest are discussed in section 3 . Section 4 presents conclusion.

2. EMPIRICAL MODELS In the present work nonlinear time series models and artificial neural networks are employed for possible existence of business cycle asymmetries in all the series. The model parameterizations for all the series are selected using Schwarz Bayesian Criterion (SBC) due to Schwarz (1978). Empirical models employed in this study are elaborated in the following sub-sections.

2.1. Time Series Models Two classes of time series models are employed to detect possible presence of business cycle asymmetries in Armenia, Indonesia, Iran, Japan, Kazakhstan, Malaysia, Philippines, Russia, Singapore, Thailand and Turkey real GDP growth rates. The models employed are termed as CDR-Augmented, and SETARSwitching models. Within each class of these models, three different versions of the models are entertained. Model 1 incorporates stable distributions, conditional heteroskedasticity, and fractional differencing parameters. Restricting fractional differencing in Model 1 yields Model 2. Imposing homoskedasticity on Model 2 gives Model 3. Model 1 is the most general model, which nests Models 2, and 3. Each of these classes of models is elaborated in the following three sub-sections.

2.1.1. CDR Augmented Models CDR-Augmented model was initially proposed by Beaudry and Koop (1993) who estimated an autoregressive moving average (ARMA) model, augmented with an ad hoc nonlinear term. The most general model (Model 1) in this class of models encompasses stable distributions, conditional heteroskedasticity, and long memory. Following Kiani and Bidarkota (2004), the model employed in the present work in shown in the following Equations:

Business Cycle Asymmetries in Asian Economies… Φ ( L)(1 − L) d (Δy t − μ ) = [Ω( L) − 1]CDRt + ε t

89 (5.1a)

ε t | I t −1 ~ z t ct , z t ~ i.i.d . S α (0,1) ctα = b1 + b2 ctα−1 + b3 | ε t −1 |α

(5.1b)

where, Δy t ≡ 100 * Δ(ln GDPt ) is the growth rate of GDP, μ is its unconditional mean, and d is the differencing parameter. The polynomial Ω(.) is of orders r , and polynomial Φ (.) of the order p in the lag operator L , with Ω(0) = Φ (0) = 1 . This ad-hoc nonlinear term is labeled as the current depth of a recession ( CDRt ) which measures the gap between the current level of output and the economy’s historical maximum level that is defined as CDRt = max{ y t − j } j ≥ 0 − y t . The term CDRt is designed to permit recessions to be less or more persistent than

expansions depending on the parameter estimates. A random variable X will have a symmetric stable distribution S α (δ , c ) if its log characteristic function can be expressed as ln E exp(iXt ) = iδt − | ct |α . Here c ∈ [0, ∞] is the scale parameter, δ ∈ [−∞, ∞] is the location parameter, and

α ∈ [0,2] is the characteristic exponent governing tail behavior, and its small values shows thicker tails. However, when α = 2 normal distribution results. When α = 2 in Equation 1b , normal GARCH (1,1) process prevails. However, when d = 0 there exists a unit root in yt , but with d = −1 , y t ends up being integrated of order zero I (0). ARFIMA models with long memory are defined in terms of the rate of decay of their autocovariances, so the extension of these models to infinite variance stable shocks is not immediate. A stationary casual and invertible solution to an ARFIMA model with Gaussian errors requires | d |< 0.5 (Brockwell and Davis (1991). However, according to Kokoszka and Taqqu (1995), MA (∞ ) representation to an ARFIMA model with stable shocks needs α (d − 1) < −1 . Therefore, d needs to be positive when α > 1 . In addition, α > 1 and | d |< (1 − 1 / α ) for the ARFIMA model to be a solution to an AR (∞ ) process. That is the reason why α and d is restricted in Equation 1 within these limits. Equation 1a reduces to an autoregressive (AR) model with non-integer differencing when Ω( L) = 1 . Because, it nests AR models, likelihood ratio (LR)

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statistic can be used to test statistical significance of the non-linear term governing the conditional mean dynamics. When autoregressive lag order p is 0 and r is 1, ω1 = 0 yields a random walk with drift. However, a positive ω1 implies that negative shocks are less persistent whereas a negative ω1 implies that positive shocks are less persistent. The existence of nonlinearities shows that innovations are symmetric but the impulse transmission mechanism is nonlinear, or the innovations are asymmetric but the impulse transmission mechanism is linear, or the innovations are asymmetric and the impulse transmission mechanism is also nonlinear. However, it is difficult to separate the asymmetric innovations from the nonlinear propagation mechanism, if they do exist in a data series. Although asymmetric α stable distributions exist and are well defined, the present work is merely investigating possible existence of asymmetries in the conditional mean regardless of how they can be characterized.

2.1.2. SETAR-Switching Models The SETAR-switching model or self-exciting threshold autoregressive (SETAR) model governs switching between the regimes that are defined in terms of the observed series yt . Potter (1995) used SETAR-Switching models with errors to come from a normal family, and thereafter Bidarkota (2000), and Kiani and Bidarkota (2004) estimated SETAR-Switching models with long memory, conditional heteroskedasticity, and stable distribution. The most general model estimated within this class of models is shown in the following Equations: In Regime 1:

(1 − φ1 L − φ2 L2 )(1 − L) d (Δyt − μ1 ) = ε t

ε t | I t −1 ~ zt ct

(5.2a)

z ~ iid Sα (0,1)

ctα = b1 + b2 ctα−1 + b3 | ε t −1 |α

(5.2b)

In Regime 2:

(1 − φ3 L − φ4 L2 )(1 − L) d (Δyt − μ 2 ) = ε t

(5.2c)


ε t | I t −1 ~ zt γct

z ~ iid

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Sα (0,1)

ctα = b1 + b2 ctα−1 + b3 | ε t −1 / γ |α

5.(2d)

The switch between the two regimes in the model presented in Equations 5.2 a − d is governed by the term Δy t − d > r , where y t is log real GDP, d is the delay, and r is the threshold parameter. Therefore, when, Δy t − 2 > 0 regime 1 prevails, but when Δy t − 2 ≤ 0 , regime 2 is obtained.

2.3. Artificial Neural Networks An artificial neural network (ANN) consists of a number of interconnected elements called neurons. The ANN are able to learn from examples that can be employed to solve unknown problems (Reilly and Cooper 1990). ANN are nonlinear, nonparametric models that are independent of the distributions of the underlying data generating processes (White 1989) which enables them to approximate any continuous function with desired level of precision (Hornick et al. 1989). Although ANN are criticized heavily for being “Black Boxes” because it is hard to know their exact functional form as well as overfitting issues associated with them. However, the present work is focused on final prediction ( Yhat ) rather than parameter estimates so the issue pertaining to the functional form is of least importance. Nevertheless, following Kiani (2005), overfitting can be mitigated using careful construction of neural network. A general form of neural network employed is shown in Equation 5.3. ⎡ f ( x) = sig ⎢α 0 + ⎣⎢

n

∑ j =1

⎛

k

∑β ⎝

α j sig ⎜⎜

i =1

ij xi

⎞⎤ + β 0 j ⎟⎥ + ε ⎟⎥ ⎠⎦

(5.3)

where, n is the number of hidden nodes in the network, k is the number of explanatory variables in the network, sig (x) = 1/(1+e-x) is a transfer function, that can either be sigmoid (logistic) or hyperbolic (tangent) cumulative distribution function, α j represents a vector of parameters or weights that link the hidden node to the output layers’ units, β ij (i = 1 , … , k; j = 1 , … , n) denotes a matrix of parameters linking the input to the hidden layers’ units, and ε is the error term.

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A neural network test for detecting possible existence of nonlinearities was originally proposed by Terasvista et al. (1993) which was also employed by Kiani (2005) including others. The test comprises of a linear model, and artificial neural networks. The linear model is shown in Equation 5.4 whereas artificial neural network is presented in Equation 5.5 . yt = π ' wt + u t

where,

(5.4)

~ ' )' , wt = (1, w t

~ = ( y ,..., y )' , w t t −1 t− p

π = (π 0 , π 1 ,........,π p )'

and

u t ~ Nid (0, σ 2 ) .

⎡ uˆt = π ' wt + sig ⎢α 0 + ⎣⎢ where in equation 5.5 ,

n

∑ j =1

⎛

k

∑β ⎝

α j sig ⎜⎜

i =1

ij xi

⎞⎤ + β 0 j ⎟⎥ + ε ⎟⎥ ⎠⎦

(5.5)

π 0 is intercept, ψ (γ ' wt ) = (1 + exp{−γ ' wt )) −1 is a transfer

function, uˆ t represents residuals from linear model, and the residuals from neural network are represented by vˆt . The test statistic of the form shown in Equation 5.6 is constructed using residuals and residual sum of squares from Equation 5.4 , and 5.5 .

TS = [(SSE1 − SSE2 ) / m] /[SSE2 / (n − p − m − 1)]

(5.6)

where, in Equation 5.6 , p is the number of lags in the series, m denotes the number of restrictions in the unrestricted model, and n is the number of observations. The test statistic is distributed approximately F4 under normality hypothesis with ( n − p − m − 1 ), and m degrees of freedom. Neural network linearity test are constructed using in-sample forecasts from linear models as well as neural networks. Likewise, neural network linearity test are also constructed using jackknife out-of-sample5 forecasts from linear models

4

See Davies, R. (1977) and Andrews, W. (2001) for discussions on the nuisance parameter that appears under the alternative hypothesis. 5 Sub-sample jackknife that is proposed by Wu (1990) and extended by Politis and Romeo (1994) are employed in the present work. Sub-sample jackknife was also used by Politis et al. (1997), and Ziari et al. (1997) including others.


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as well as neural networks. The test statistic for this test is constructed using insample as well as jackknife out-of-sample forecast from linear models, and artificial neural networks that nest the linear model shown in Equation 5.4 .

2.4. Estimation Issues To capture asymmetric fluctuations in the series Beaudry and Koop (1993) added an additive nonlinear term in an autoregressive moving average (ARMA) framework. For this purpose, Beaudry and Koop (1993) assumed that the shocks are identically, and independently distributed (iid) normal. Further, they confirmed that the additive ad hoc nonlinear term does not impose any estimation problems. Alternatively, Bidarkota (1999-2000), and Kiani and Bidarkota (2004) assumed that innovations follow more general stable distributions so that computational algorithm due to McCulloch (1997) could be employed to obtain stable densities for maximum likelihood estimation of the employed models which works well particularly when the errors are symmetric stable. However, the full information maximum likelihood (ML) method proposed by Sowell (1992a) can only be adopted if the errors are iid normal, but in the present work more complicated non-normal conditionally heteroskedastic models are used, therefore, conditional sum of squares (CSS) estimator is employed that is discussed in Box and Jenkins (1976) that was originally proposed by Hosking (1984). The CSS procedure is equivalent to the full information MLE asymptotically as was noted by Baillie et al. (1996), and it works better for complex models. Following Kiani and Bidarkota (2004) we do not consider any moving average (MA) terms in the specification of the model. Maximum likelihood estimation (MLE) of mixed ARMA models with stable errors poses a challenge; therefore, like Kiani and Bidarkota (2004), Whittle estimator due to Mikosch et al. (1995), and minimum dispersion estimators due to Brockwell and Davies (1991) are employed. The artificial neural networks (ANN) employed here encompass a nested linear model to construct neural network linearity tests, which poses difficulty in convergence. To surmount this difficulty, genetic algorithm (GA)6 is selected as an estimation algorithm for these complex models. The GA is considered to be the most reliable estimation algorithm for estimating any nonlinear functional form but it is notoriously slow. Therefore, to speed up the estimation process, and to 6

Initially, De Jong (1975) applied genetic algorithm, to mathematical optimization. Later, Goldberg, (1989) employed it in biology, engineering, and operation research. In economics, the genetic algorithm was employed by Axelord (1987), Marimon, McGratten and Sargent (1990), and Dorsey and Walter (1995).

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enhance the probability for obtaining global minima, estimation process is started with a couple of random starts using GA. Thereafter, the parameter vector encompassing the smallest sum of squares was chosen to run GA for upto 50,000 iteration, whereupon the parameter vector obtained from GA was used as the starting conditions for Matlab’s fminsearch algorithm, which is a Nelder-Mead simplex algorithm that worked well for closing on the final optimum.

3. EMPIRICAL RESULTS 3.1. Data Sources Quarterly real GDP7 growth rates for Armenia, Indonesia, Iran, Japan, Kazakhstan, Malaysia, Philippines, Russia, Singapore, Thailand, and Turkey were obtained from DataStream. Table 1 provides additional information about that data series on Asian economies that are included for testing asymmetries in business cycles fluctuations in these series.

3.2. Specification Search A wide-ranging specification search was performed for each country for all the models for each of the two classes of models i.e. CDR-Augmented and SETAR-Switching models, where the specification was done over all the parameterization with lag orders or the autoregressive, and CDR t terms of three or less to attain parsimony as was done by Beaudry and Koop (1993). However, for SETAR-Switching models the search was performed with the autoregressive lag polynomials in the two regimes restricted to be of orders (3,3), (2,2), (1,1), or (0,0) only.

3.3. Hypotheses Test Two types of hypotheses tests are employed in the present work. The first type of hypotheses pertains to linearity tests in conditional mean, which relates to augmented nonlinear time series models, and switching time series models. 7

Data limitations hampered our empirical exercise to eleven Asian Economies only.


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However, the second type of hypotheses tests relates to neural network linearity test. Both these tests are elaborated in the following paragraphs. The first type of hypothesis of interest is the test for linearity in the conditional mean, which pertains to SETAR-Switching models. Under the null hypothesis of the linearity, the unconditional means in the two regimes i.e. μ 1 and

μ 2 , and the corresponding autoregressive coefficients (if present in the models employed) in the two regimes are equal, and the scale ratio γ is equal to one. Failing to reject the null hypothesis would result in one regime only, however, a rejection of the null would result in two distinct regimes describing the GDP growth rates. On the other hand for CDR-augmented models the asymptotic distribution of t-test for significance of the nonlinear CDRt term, which is shown in the Equation, 1a is non-standard especially when the dependent variable is non-stationary (Hess and Iwata, 1977; Kiani and Bidarkota 2004). Therefore, the present work refrains from constructing tests for linearity in conditional mean based on these models. The second type of hypothesis pertains to neural network linearity tests that are constructed from in-sample forecasts from linear models as well as neural networks approximations. In addition to linearity tests constructed from in-sample forecasts, neural network linearity tests are also constructed using out-of-sample forecasts from linear models as well neural networks. In the second set of hypothesis, if the null hypothesis of linearity is rejected in favor of alternative hypothesis of nonlinearity, the series under investigation is said to have nonlinearity. However, if the null hypothesis is not rejected then nonlinearities do not prevail in the series being studied. Table 1. Data Description

Data Series Armenia Indonesia Iran Japan Kazakhstan Malaysia Philippines Russia Singapore

Quarterly Real GDP Quarterly Real GDP Quarterly Real GDP Quarterly Real GDP Quarterly Real GDP Quarterly Real GDP Quarterly Real GDP Quarterly Real GDP Quarterly Real GDP

Sample Period 1994Q4-2004Q1 1990Q1-2004Q1 1988Q1-2000Q1 1980Q1-2004Q2 1994Q1-2004Q1 1991Q1-2004Q1 1981Q1-2004Q2 1993Q3-2004Q1 1984Q3-2004Q2

Observations 37 56 48 97 40 52 93 42 79

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Table 1. (Continued)

Data Series Thailand Turkey

Quarterly Real GDP Quarterly Real GDP

Sample Period 1993Q1-2004Q2 1987Q1-2003Q4

Observations 45 67

Table 2. Specification Search/Log Likelihood: CDR Augmented Models

Armenia Indonesia Iran Japan Kazakhstan Malaysia Philippines Russia Singapore Thailand Turkey

Model 3

Model 2

Model 1

(3, 3) 172.682 (3, 3) 336.870 (3, 3) 288.465 (3, 3) 618.194 (3, 2) 231.659 (3, 3) 312.689 (3, 0) 482.579 (3, 3) 240.228 (3, 3) 488.767 (3, 1) 268.537 (3, 2) 291.728

(3, 2) 170.363 (3, 2) 333.238 (3, 3) 288.447 (3,3) 615.045 (3, 2) 231.611 (3, 3) 312.246 (3, 1) 481.979 (3, 2) 240.122 (3, 3) 472.286 (2, 3) 268.049 (3, 1) 290.820

(3, 3) 170.097 (3, 3) 328.901 (1, 3) 197.312 (3, 3) 610.483 (3, 2) 231.292 (3, 3) 312.175 (3, 1) 481.887 (3, 2) 170.176 (2, 3) 470.992 (3, 3) 266.273 (3, 0) 290.408

Notes: 1. For each country in columns 2, 3 , and 4 , the first row represents model parameterization, and the second row show log likelihood estimates for the model. For example for Armenia, the first row for each model (models 1-3) shows model parameterization and the second row shows log likelihood estimates for each of the models. The results for the remaining countries are presented in the similar manner. 2. The symbol “--“denotes that our nonlinear time series models failed to converge.


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Table 3. Specification Search/Log Likelihood: SETAR Switching Models


Model 3

Model 2

Model 1

(3, 3) 212.835 (2, 2) 333.599 (3, 3) 280.809 (1, 1) 614.997 (3, 3) 223.364 (3, 3) 302.221 (3, 3) 472.862 (3, 3) 233.799 (3, 3) 1096.431 (3, 3) 256.778 (2, 2) 349.057

(3, 3) 164.058 (1, 1) 333.852 (3, 3) 280.193 (2, 2) 613.895 (3, 3) 223.108 (3, 3) 299.004 (3, 3) 470.069 (3, 3) 234.601 (3, 3) 1077.185 (3, 3) 257.169 (2, 2) 340.519

(3, 3) 165.354 (1, 1 328.305 (3, 3) 279.308 (2, 2) 603.364 (3, 3) 222.850 (3, 3) 298.136 (3, 3) 470.023 (3, 3) 234.024 (3, 3) 1073.283 (3, 3) 256.499 (2, 2) 328.105

Notes: 1. For each country in columns 2, 3 , and 4 the first row represents model parameterization, and the second row shows log likelihood estimates for the model. For example for Armenia, the first row for each model (models 1-3) shows model parameterization and the second row shows log likelihood estimates for the model. The results for the remaining countries are presented in the similar manner.

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Table 4. Test for Linearity: LR Tests Based on SETAR-Switching Models Model 3

Model 2

Model 1

Armenia

--

Indonesia

8.2574 (0.016) 4.3048 (0.230) 12.83378 (0.000) 4.67282 (0.197) 6.62028 (0.085) 10.17062 (0.001) 2.09926 (0.552) --

0.33462 (0.953) 10.67372 (0.001) 5.31706 (0.149) --

80.27632 (0.000) 8.80742 (0.003) 6.9144 (0.074) 6.84474 (0.032) 5.13354 (0.162) 10.97686 (0.011) 15.47848 (0.001) 0.24182 (0.970) 1214.6804 (0.000) 3.41662 (0.331) --

Iran Japan Kazakhstan Malaysia Philippines Russia Singapore Thailand Turkey

8.57034 (0.035) 26.33814 (0.000)

5.18034 (0.159) 10.979 (0.011) 15.59288 (0.001) 0.80362 (0.848) -2.50884 (0.473) --

Notes: 1. For each country in columns 2, 3 , and 4 the number in the first row represents “test statistic” and the number in the second row in parenthesis shows p-values for the test statistics. For example for Armenia, the number in the first row for each of the models (models 1-3) shows test statistic for that model and the number in the second row in parenthesis shows p-values for the test statistic for that model. The results for the remaining countries are presented in the similar manner. 2. The symbol “--“denotes that our nonlinear time series models failed to converge.


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Table 5. Test for Linearity: Neural Network Tests In-Sample


Test Statistic

p-Values

Jackknife Out-of-sample p-Values Test

393.502 71.662 108.912 109.722 59.121 64.148 268.639 34.304 1049.931 63.222 186.245

(0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000)

22.237 0.517 67.164 55.676 36.524 46.217 138.723 19.021 705.312 44.707 24.992

Statistic (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000)

Notes: 1. Table presents neural network test results approximated from in-sample as well as jackknife out-of-sample forecasts from linear models and neural network models. 2. In columns 2 and 4 , starting rows 1 to 11 this Table presents neural network test statistics for Armenia, Indonesia, Iran, Japan, Kazakhstan, Malaysia, Philippines, Russia, Singapore, Thailand, and Turkey respectively. p-values for each test statistic are juxtaposed in parenthesis in next column.

3.4. Results on Hypotheses Tests The empirical results on hypotheses tests for linearity in conditional mean as well as neural network linarity tests for each of the series are performed as elaborated in the earlier sub-sections. All tests are based on the likelihood ratio (LR) test statistics that are reported in Table 4 . p-values for each test are juxtaposed in parentheses in the next column. Results from neural network linearity tests that are based on in-sample as well as jackknife out-of-sample forecasts from all the series are reported in Table 5. p-values for each test statistic are juxtaposed in the next column in parenthesis.

3.4.1. Results on Time Series Models The results based on linearity in conditional mean for Indonesia, Japan, Malaysia, and Philippines show strong evidence of asymmetries in business cycle

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fluctuations. However, results on Iran, Kazakhstan, and Russia are largely linear. On the other hand Armenia, Thailand, and Singapore show weak evidence of business cycle asymmetries in real GDP growth rates. All statistical inferences for these tests are drawn at 5 percent level of significance. However, the results do not change when we change the significance level from 5 percent to 10 percent level.

3.4.2. Results on Neural Network Tests The neural network test for possible existence of nonlinearities tests the null hypothesis of linearity against the alternative hypothesis of nonlinearities for Armenia, Indonesia, Iran, Japan, Kazakhstan, Malaysia, Philippines, Russia, Singapore, Thailand, and Turkey. Nonlinearity prevails in the data series being tested when the null hypothesis of linearity is rejected against the alternate hypothesis of nonlinearity. Neural networks linearity test results based on in-sample forecasts from linear models as well as neural network approximations show existence of statistically significant evidence nonlinearities in real GDP growth rates of Armenia, Indonesia, Iran, Japan, Kazakhstan, Malaysia, Philippines, Russia, Singapore, Thailand, and Turkey. Similarly, neural network linearity test results based on jackknife out-of-sample forecasts approximated from artificial neural networks also support these results. All statistical inferences are drawn at the five percent significance level. However, the results do not change when significance level is changed from 5 to 10 percent level. 3.4.3. Results on Selected Parameter Estimates Further to Kiani and Bidarkota (2004) the primary objective of this study is to find possible existence of business cycle asymmetries in Asian economies. However, data limitations hampered out analysis to a limited number of countries because most countries included in the initial sample do not have sufficient number of observation in quarterly series. Therefore, many countries8 including India, and China that would be good examples of fastest growing economies, were dropped from this empirical exercise, however, the sample used encompassed data from highly developed countries, fastest growing countries, transition economies,

8

Afghanistan, Azerbaijan, Bahrain, Bangladesh, Bhutan, Brunei, Burma, Cambodia, China, Georgia, Hong Kong, India, Iraq, Israel, Jordan, Korea, North, Korea, South, Kuwait, Kyrgyzstan, Laos, Lebanon, Maldives, Mongolia, Myanmar, Nepal, Oman, Pakistan, Qatar, Saudi Arabia, Sri Lanka, Syria, Taiwan, Tajikistan, Turkmenistan, United Arab Emirates, Uzbekistan, Vietnam, and Yemen.


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and other developing countries, which helped finding possible existence of business cycle asymmetries in the selected Asian economies9. Results for selected parameters from the most general switching models estimated for each country are shown in Table 6. In this Table, columns 2 to 3 show parameter estimates for the switching parameter γ , characteristic exponent

α from SETAR-Switching models and parameter estimate for nonlinearity term ω1 from CDR-Augmented for all countries starting Armenia in alphabetical order. Standard errors for these parameter estimates are placed below each parameter estimate in parenthesis. Table 6. Selected Parameter Estimates for Most General Models Models

CDR-Switching

Parameters of interest Armenia

γ

α

ω1

3.0193 (1.604) 1.669 (0.311) 1.279 (0.283) 0.964 (0.154) 1.244 (0.385) 1.718 (0.143) 1.246 (0.197) 0.7301 (0.150) 1.074 (0.166) 2.383 (1.074)

1.560 (0.277) 1.999 (0.000) 1.999 (0.005) 1.999 (0.000) 1.999 (0.000) 1.718 (0.093) 1.400 (0.000) 1.999 (0.000) 1.718 (0.092) 1.652 (0.203)

-2.097e-5 (0.011) -0.035 (0.100) 0.810 (0.000) -0.295 (0.334) 0.294 (0.023) 0.843 (0.954) -0.004 (0.013) 0.001 (0.012) 1.090 (0.374) 0.028 (0.208)

Indonesia Iran Japan Kazakhstan Malaysia Philippines Russia Singapore Thailand

9

SETARSwitching

Armenia, Indonesia, Iran, Malaysia, Philippines, Japan, Kazakhstan, Russia, Singapore, Thailand, and Turkey,

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Table 6. (Continued) Models

CDR-Switching

SETARSwitching

Parameters of interest Turkey

γ

α

ω1

1.531 (0.086)

1.882 (0.099)

0.082 (0.090)

Notes: 1. Table presents selected parameter estimates in rows 1 to 11 respectively for Armenia, Indonesia, Iran, Japan, Kazakhstan, Malaysia, Philippines, Russia, Singapore, Thailand, and Turkey. Standard error for each of the parameter estimates are shown in parenthesis beneath each parameter estimates. 2. In this Table column 2 to 4 show parameter estimates for switching parameter gamma, and characteristic exponent alpha ( α ) for CDR-switching models and parameter estimated for omega ( ω1 ) from SETAR-switching models for all the countries starting Armenia are shown in alphabetical order.

3.5. Nature of Asymmetries The results reveal that parameter estimates for characteristic exponent ( α ) is close to normal behavior in Indonesia, Iran, Japan, Kazakhstan, and Russia. However, the values for the characteristic exponent ( α ) for Armenia, Malaysia, Philippines, Singapore, Thailand, and Turkey show fat tails. The values of the switching parameter ( γ ) are different across different countries. For example, high values of this parameter for Armenia, Indonesia, Iran, Kazakhstan, Malaysia, Philippines, Singapore, Thailand, and Turkey shows that volatility in high regimes in these countries is lower than in the low regimes. The results for Japan, and Russia though in sharp contrast reveal that compared to developing countries developed countries have different volatility patterns during different phases of business cycles. Developing countries’ volatility has similar patterns irrespective of their geographical locations. The values of ω1 for Armenia, Indonesia, Japan, and Philippines are positive implying that negative shocks are less persistence in these economies. On the other hands the value of ω1 for Iran, Kazakhstan, Malaysia, Singapore, Thailand, and Turkey being negative reveals that positive shocks are less persistent in these economies.


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The results on nonlinearity tests based on time series models provide statistically significant evidence of business cycle asymmetries in Indonesia, Iran, Japan, Malaysia, Philippines, Singapore, and Turkey only. Neural network nonlinearity tests based on in-sample as well jackknife out-of-sample forecasts approximated from artificial neural networks, in conjunction with linear models show that nonlinearities do prevail in all the series, including Iran, Kazakhstan, and Russia. The results based on neural network linearity test show statistically significant evidence of nonlinearities in Armenia, Indonesia, Iran, Japan, Kazakhstan, Malaysia, Philippines, Russia, Singapore, Thailand, and Turkey real GDP growth rates which is an improvement over the linearity tests that are obtained from nonlinear time series and switching models. This shows that neural networks linearity tests outperformed the linearity tests that are constructed from forecasts from the nonlinear time series models employed for detecting possible existence of nonlinearities in all the series. These results are in line with those obtained from previous studies including Lee, et al. (1993) Terasvirta, et al. (1993), and Kiani et al. (2005). Additionally, volatility patterns in developed countries are in sharp contrast with those from the developing countries when switching from lower regime of the business cycle to the high regime and vice versa. The characteristic exponent α , that governs the tail behavior in these countries, did not show any pattern like the other parameters did. Similarly, the results also show that negative shocks are less persistent in Iran, Kazakhstan, Malaysia, Singapore, Thailand, and Turkey, whereas positive shocks are less persistent for Armenia, Indonesia, Japan, and Philippines The results on nonlinearity in the conditional mean for Japan are in line with Bidarkota (2000) and Kiani and Bidarkota (2004). This shows that the evidence against linearity in conditional mean for the Japan is robust to changes in the sample period. Koop and Potter (2001) investigated whether nonlinearities could arise from structural instability. Blanchard and Simon (2001) show a possible structural change in the early 1980s; however, this is not accounted for in the present work. Similarly, Diebold and Inoue (2001) show that the series that undergo occasional structural change may show spurious evidence of long memory or spurious evidence of unit roots (Perron, 1989). These limitations also apply to the present work.

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4. CONCLUSION Nonlinear time series models are employed to test possible existence of business cycle asymmetries (if any) in Armenia, Indonesia, Iran, Japan, Kazakhstan, Malaysia, Philippines, Russia, Singapore, Thailand and Turkey real GDP growth rates. Time series models employed are fully parametric, and are capable to account for time varying volatility, long memory in the process, and outliers that might be present in the series. In addition, artificial neural networks are also employed in order to find possible existence of nonlinearities in all the series. The findings from nonlinear augmented and switching time series models reveal strong evidence of business cycle asymmetries in Indonesia, Japan, Malaysia, Philippines, real GDP growth rates. The results show weak evidence of business cycle asymmetries in Armenia, Singapore, Thailand, and Turkey. However, the results do not show statistically significant evidence of business cycle asymmetries in real GDP growth rates of Iran, Kazakhstan, and Russia. The results from neural network linearity tests that are based on in-sample approximations from neural networks show statistically significant evidence of business cycle asymmetries in real GDP growth rates in Armenia, Indonesia, Iran, Japan, Kazakhstan, Malaysia, Philippines, Russia, Singapore, Thailand, and Turkey. Similarly, results from neural network linearity tests that are constructed from jackknife out-of-sample approximations also show statistically significant evidence of business cycle asymmetries in real GDP growth rates in Armenia, Indonesia, Iran, Japan, Kazakhstan, Malaysia, Philippines, Russia, Singapore, Thailand, and Turkey. Thus, these results reveal that business cycle fluctuations in all the Asian economies studied are asymmetric, thus alike. The study results suggest that forecasts from linear models as well as those derived from linear vector autoregression cannot be employed to forecast the impact of monetary policy or any other shock on output in Armenia, Indonesia, Iran, Japan, Kazakhstan, Malaysia, Philippines, Russia, Singapore, Thailand, and Turkey. Therefore, the policymakers in these countries will have to employ appropriate nonlinear time series forecasting models to anticipate the impact of monetary policy or any other shock on output in these countries.


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INDEX A accounting, 3, 76 accuracy, 5, 24 ad hoc, 3, 88, 93 Afghanistan, 86, 100 agents, 24, 59, 106 aggregate supply, 2 agricultural, 24, 42 algorithm, 9, 31, 47, 51, 70, 93 alpha, 102 alternative, 21, 30, 38, 39, 47, 50, 52, 58, 67, 75, 92, 95, 100, 106 alternative hypothesis, 21, 30, 38, 39, 50, 52, 67, 75, 92, 95, 100 amplitude, 2, 49 ANN, 1, 4, 7, 28, 29, 30, 31, 34, 36, 39, 40, 41, 47, 49, 50, 56, 63, 65, 66, 67, 68, 69, 70, 75, 77, 78, 79, 91, 93 Annealing, 23, 57, 80, 105 application, 9, 32, 51, 71 Arabia, 86, 100 Armenia, 85, 86, 87, 88, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104 artificial, vii, viii, 1, 4, 24, 25, 28, 29, 34, 35, 36, 37, 40, 41, 44, 45, 47, 49, 56, 59, 60, 61, 63, 65, 66, 78, 79, 81, 87, 88, 91, 92, 93, 100, 103, 104, 106, 107 artificial intelligence, 4, 49, 66 Asian, v, vii, viii, 85, 86, 94, 100, 104

Asian countries, 86 assumptions, 15, 32, 44 asymmetry, 2, 25, 44, 60, 107 asymptotic, 44, 60, 61, 95, 108 asymptotically, 93 autoregressive model, 3, 23, 24, 25 Azerbaijan, 86, 100

B Bahrain, 86, 100 Bangladesh, 86, 100 basic research, 28 Bayesian, 6, 88 behavior, 23, 24, 43, 51, 58, 66, 86, 89, 102, 103, 106 Bhutan, 86, 100 bias, 32, 44, 60, 66 biology, 9, 29, 51, 71, 93 bivariate analysis, 79 brain, 4, 29, 49, 66 Burma, 86, 100 business, vii, viii, 1, 2, 3, 6, 17, 18, 20, 22, 23, 24, 25, 27, 28, 29, 30, 41, 42, 43, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59, 60, 63, 64, 65, 66, 69, 76, 77, 78, 79, 81, 85, 86, 87, 88, 94, 99, 100, 102, 103, 104, 106, 107, 108 business cycle, vii, viii, 1, 2, 3, 6, 17, 18, 20, 22, 23, 24, 25, 27, 28, 29, 30, 41, 42, 43, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59,

Index

110 60, 63, 64, 65, 66, 69, 76, 77, 78, 79, 81, 85, 86, 87, 88, 94, 99, 100, 102, 103, 104, 106, 107, 108

C Cambodia, 86, 100 Canada, 1, 3, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27, 28, 29, 30, 34, 35, 36, 37, 38, 40, 41, 42, 45, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79 capacity, 4 capitalist, 25, 60 causality, 42 CDR, 88, 94, 95, 96, 101, 102 Chicago, 23, 43, 58, 59, 80, 81, 105 China, 86, 100 classes, 88, 94 classical, 25, 48 classified, 2, 48 codes, 27, 46, 64 commerce, 43, 59 Commerce Department, 81 components, 2, 6, 7, 13, 18, 19, 23, 57, 105 computers, 44, 60, 83 computing, 24 conditional mean, 22, 23, 42, 58, 90, 94, 95, 99, 103, 105 confidence, 44, 59, 60, 107 consensus, 87 construction, 47, 91 consumption, 42 continuity, 43, 59 convergence, 9, 93 convex, 31 critical value, 12, 39, 53, 55 criticism, 47 CSS, 93 cumulative distribution function, 5, 91 cycles, viii, 2, 3, 23, 24, 25, 27, 28, 43, 44, 45, 46, 48, 49, 58, 59, 60, 63, 64, 65, 77, 79, 86, 88, 94, 102

D data analysis, 29, 65, 71 dating, 48 decay, 89 decision-making process, 29, 49, 66 degrees of freedom, 16, 17, 37, 92 demand, 2, 43 dependent variable, 7, 20, 67, 95 depression, 2, 48 derivatives, 24 developed countries, 86, 100, 102, 103 developing countries, 101, 102, 103 deviation, 34, 35 dispersion, 93 disseminate, vii distribution, 4, 5, 32, 37, 39, 47, 49, 51, 53, 55, 65, 67, 89, 90, 91, 95 distribution function, 5, 91 duration, 2, 23, 43, 48, 58

E econometrics, 60, 82, 107 economic, 2, 9, 23, 24, 43, 49, 51, 58, 59, 60, 64, 71, 79, 86, 88, 105, 106, 107 economic activity, 24, 49 economics, vii, 28, 47, 65, 93 economies, viii, 77, 79, 85, 86, 94, 100, 102, 104 economy, viii, 24, 48, 59, 77, 79, 86, 87, 89, 106 employment, 48 endogenous, 50, 70 engineering, 9, 28, 29, 51, 71, 93 estimating, 9, 31, 32, 33, 36, 66, 70, 93 estimation problems, 93 estimation process, 93 estimator, 93 estimators, 32, 66, 93 European, 46 European Union, 46 evidence, vii, 1, 3, 6, 22, 23, 24, 27, 28, 39, 42, 43, 45, 46, 55, 56, 57, 58, 63, 64, 65,

Index 69, 76, 77, 79, 85, 86, 87, 99, 100, 103, 104, 105 evolution, 23, 57, 105 exchange rate, 24, 42 exchange rates, 24, 42 exercise, 86, 94, 100 exogenous, 50, 66, 70 expansions, 48, 89

F false, 12, 38, 52, 75 family, 90 fat, 10, 71, 86, 102 Federal Reserve, 48 finance, 29, 47, 65 financial crises, 87 fluctuations, vii, 2, 3, 23, 24, 28, 45, 46, 48, 49, 57, 59, 60, 63, 64, 65, 76, 77, 79, 85, 86, 87, 93, 94, 100, 104, 105 food, 43 forecasting, viii, 5, 14, 25, 27, 41, 43, 48, 49, 60, 64, 87, 104, 107 foreign exchange, 24, 58 France, 1, 3, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27, 28, 29, 30, 34, 35, 36, 37, 38, 40, 41, 42, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79 freedom, 16, 17, 37, 92

G G7 countries, vii, 9, 24, 28, 35, 42, 46, 57, 59, 65, 66, 76, 79, 86, 87, 106 Gaussian, 89 GDP, 1, 2, 3, 9, 10, 11, 15, 17, 18, 19, 20, 21, 22, 23, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 41, 42, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 75, 76, 77, 78, 80, 85, 86, 87, 88, 89, 91, 94, 95, 96, 100, 103, 104, 105, 106 generalization, 5

111

genetic, 9, 31, 47, 51, 70, 93 Georgia, 86, 100 Germany, 3, 9, 28, 35, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79 GNP, 3, 23, 25, 42, 43, 44, 46, 58, 60, 80, 82, 87, 105, 106, 107 graduate students, viii Great Depression, 48, 58 gross domestic product, 63 gross national product, 46, 64 growth, 1, 3, 9, 10, 11, 12, 15, 17, 18, 19, 20, 21, 22, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 41, 42, 45, 46, 47, 50, 52, 53, 55, 56, 57, 63, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 78, 85, 86, 87, 88, 89, 94, 95, 100, 103, 104 growth rate, 1, 3, 9, 10, 11, 12, 15, 17, 18, 19, 20, 21, 22, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 41, 42, 45, 46, 47, 50, 52, 53, 55, 56, 57, 63, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 78, 85, 86, 87, 88, 89, 94, 95, 100, 103, 104

H hands, 21, 102 hedging, 58, 59 heterogeneous, viii heteroskedasticity, 3, 11, 71, 88, 90 histogram, 44, 61, 108 homogeneity, 58 Hong Kong, 86, 100 human, 4, 29, 49, 66 human brain, 4, 49, 66 hyperbolic, 5, 91 hypothesis, 2, 9, 12, 14, 18, 20, 21, 22, 30, 31, 36, 37, 38, 39, 50, 52, 54, 57, 67, 72, 75, 77, 85, 92, 95, 100, 105

I identification, 48 Illinois, 81

Index

112 imports, 77 inclusion, 77 income, 48 independent variable, 7 India, 86, 100 indicators, 42, 44, 57, 60, 107 indices, 57, 105 Indonesia, 85, 86, 87, 88, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104 industrial, 57, 105 industrial production, 57, 105 industrialized countries, 3, 63, 79 inferences, 21, 100 infinite, 89 information processing, 4, 29, 49, 66 instability, 103 intelligence, 4, 49, 66 international, 46, 65 Iran, 86, 87, 88, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104 Iraq, 86, 100 Israel, 86, 100 Italy, 3, 9, 28, 35, 45, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79 iteration, 94

J jackknife, 27, 28, 29, 32, 35, 36, 37, 38, 39, 41, 42, 44, 45, 47, 49, 51, 52, 55, 56, 57, 61, 63, 65, 66, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 92, 99, 100, 103, 104, 108 Japan, 1, 3, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27, 28, 29, 30, 34, 35, 36, 37, 38, 40, 41, 42, 45, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 85, 86, 87, 88, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104 Jordan, 86, 100 judge, 39 justification, 47

K Kazakhstan, 85, 86, 87, 88, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104 Keynesian, 2, 48 Keynesians, 48 Korea, 86, 100 Kuwait, 86, 100 Kyrgyzstan, 86, 100

L Laos, 86, 100 laws, 30, 47 learning, 4, 29, 49, 59, 60, 66, 108 Lebanon, 86, 100 likelihood, 9, 51, 70, 89, 93, 96, 97, 99 limitations, 86, 94, 100, 103 linear, vii, viii, 2, 6, 7, 8, 9, 13, 14, 16, 18, 21, 25, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 60, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 85, 86, 87, 90, 92, 93, 95, 99, 100, 103, 104, 107 linear model, vii, viii, 2, 6, 7, 8, 13, 16, 18, 21, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 45, 47, 49, 50, 51, 52, 53, 55, 56, 57, 60, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 85, 86, 87, 92, 93, 95, 99, 100, 103, 104, 107 linear regression, 14 literature, 5, 48 livestock, 42 location, 89

M M1, 35, 36 macroeconomic, vii, viii, 2, 3, 23, 28, 46, 47, 48, 49, 60, 64, 86, 87, 105, 107 macroeconomics, 2, 48, 60

Index Malaysia, 85, 86, 87, 88, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104 management, 2, 48 mapping, 24 market, 60 Markov, 46, 47, 65 mathematical, 32, 44, 51, 61, 93, 108 mathematical programming, 44, 61, 108 matrix, 6, 7, 13, 30, 91 Maximum Likelihood, 107 measurement, 42, 57 measures, 2, 10, 47, 64, 86, 87, 89 mechanical, iv medium of exchange, 24 memory, 3, 24, 46, 65, 85, 86, 87, 88, 89, 90, 103, 104 Mexico, 79 MLE, 93 modeling, vii, 2, 4, 28, 30, 47, 65 models, vii, viii, 1, 2, 4, 6, 9, 13, 16, 18, 20, 21, 22, 23, 24, 25, 27, 28, 29, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 85, 86, 87, 88, 89, 90, 91, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 106, 107, 108 monetarists, 48 monetary policy, viii, 1, 2, 42, 45, 47, 49, 57, 63, 64, 79, 85, 86, 87, 104 money, 2 Mongolia, 86, 100 Myanmar, 86, 100

N national product, 46, 64 Nepal, 86, 100 network, vii, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 20, 21, 22, 24, 25, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 47, 49, 50, 51, 52, 53, 54, 55, 56, 59, 60, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 85, 91, 92, 93, 95, 99, 100, 103, 104, 106, 107, 108

113

neural network, vii, viii, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 20, 21, 22, 24, 25, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 85, 86, 87, 88, 91, 92, 93, 95, 99, 100, 103, 104, 106, 107, 108 Neural Network Model, 56, 82 neural networks, vii, viii, 1, 5, 6, 8, 20, 22, 24, 25, 27, 28, 29, 34, 36, 37, 39, 40, 41, 43, 44, 45, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 69, 71, 73, 74, 76, 77, 78, 79, 86, 87, 88, 92, 93, 95, 100, 103, 104, 106, 107 neurons, 91 New York, 25, 44, 58, 59, 60, 82, 105 nodes, 4, 5, 30, 34, 35, 36, 91 noise, 33 nonlinear, vii, viii, 1, 3, 4, 5, 9, 14, 16, 17, 18, 21, 22, 24, 25, 28, 29, 30, 31, 38, 44, 45, 46, 47, 49, 51, 57, 58, 60, 64, 65, 66, 67, 70, 75, 85, 87, 88, 89, 90, 91, 93, 94, 95, 96, 98, 103, 104, 105, 107 nonlinearities, 1, 2, 3, 6, 8, 9, 17, 21, 22, 24, 25, 38, 41, 42, 45, 49, 52, 55, 56, 59, 63, 64, 66, 75, 76, 77, 79, 85, 86, 90, 92, 95, 100, 103, 104 nonparametric, 9, 23, 30, 43, 49, 58, 59, 66, 91, 106 normal, 10, 89, 90, 93, 102 normal distribution, 89 North America, 79 North American Free Trade Agreement (NAFTA), 79 null hypothesis, 18, 20, 21, 22, 30, 38, 39, 52, 75, 95, 100

O observations, 8, 14, 16, 20, 31, 32, 33, 38, 51, 55, 66, 67, 70, 72, 75, 92 oil, 86 Oman, 86, 100 operator, 89

Index

114 optimization, 9, 51, 70, 93 organ, 57 oscillator, 105 outliers, 3, 23, 87, 104, 105

P Pakistan, 86, 100 paper, 29, 48 parameter, 9, 16, 32, 40, 50, 51, 57, 58, 67, 70, 89, 91, 92, 94, 101, 102, 105, 106 parameter estimates, 40, 89, 91, 101, 102 perception, 3, 87 performance, vii, 27, 28, 29, 39, 40, 41, 47, 49, 55, 56, 59, 66, 78, 79, 106 permit, 89 Philippines, 85, 86, 87, 88, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104 policymakers, viii, 1, 42, 86, 87, 104 polynomial, 18, 89 polynomials, 18, 19, 94 population, 25, 108 power, 5, 14 prediction, 58, 91 preparation, iv prices, 42 principal component analysis, 7, 18 probability, 32, 39, 94 production, 57, 105 programming, 44, 61, 108 propagation, 25, 48, 90

Q Qatar, 86, 100

R random, 9, 51, 70, 89, 90, 94 random walk, 90 range, 9, 10, 34 real time, 60, 107 reality, 25

recession, 2, 48, 89 recessions, 23, 42, 44, 48, 57, 60, 89, 105, 107 recovery, 2 regression, 7, 8, 14, 16, 19, 25, 33, 38, 40, 75 regression analysis, 25 regression equation, 7, 8, 14 regressions, viii, 33 rejection, 95 relationship, 4, 29, 45 relationships, 24, 29 research, vii, 2, 6, 9, 28, 46, 48, 51, 64, 65, 66, 71, 79, 93 researchers, viii, 4, 5, 21, 30, 87 residuals, 7, 8, 14, 17, 18, 19, 21, 31, 33, 35, 36, 50, 69, 70, 92 reunification, 9, 35 Russia, 85, 86, 87, 88, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104

S sales, 42 sample, vii, 6, 22, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 47, 49, 51, 52, 53, 55, 56, 57, 59, 63, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 86, 87, 92, 95, 99, 100, 103, 104, 107 sample mean, 33 sampling, 29, 32, 36, 51, 60, 61, 66, 107, 108 Saudi Arabia, 86, 100 science, 28 search, 94 securities, 58, 59 selecting, 7 series, vii, viii, 1, 2, 3, 4, 6, 9, 11, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 55, 56, 58, 59, 60, 63, 64, 65, 66, 67, 68, 69, 70, 71, 75, 76, 77, 78, 79, 81, 85, 86, 87, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100, 103, 104, 105, 106, 107, 108 shock, viii, 1, 2, 42, 45, 48, 57, 63, 64, 79, 85, 87, 104

Index shocks, viii, 2, 23, 47, 49, 64, 86, 89, 90, 93, 102, 103, 105 sigmoid, 5, 91 sign, 33, 39 signals, 5 significance level, 12, 21, 39, 55, 76, 100 similarity, vii Singapore, 85, 86, 87, 88, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104 skewness, 11, 12, 71 speed, 93 spillover effects, 65 spillovers, 76 Sri Lanka, 86, 100 stabilization, viii stabilize, viii standard deviation, 10, 34, 35 standard error, 101, 102 statistical analysis, 25, 60 statistical inference, 100 statistics, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 33, 36, 37, 38, 39, 40, 52, 53, 54, 55, 67, 71, 72, 73, 74, 75, 76, 98, 99 STD, 34, 35 Stochastic, 106 stock, 60 strategies, 23, 57, 105 students, viii superiority, 29 supply, 2 switching, vii, viii, 46, 47, 65, 85, 90, 94, 101, 102, 103, 104 Syria, 86, 100 systems, 25, 44, 57, 60, 105, 107

T Taiwan, 86, 100 Tajikistan, 86, 100 technology, 4, 29, 49, 66 test statistic, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 30, 31, 33, 37, 38, 39, 50, 52, 53, 54, 55, 67, 68, 70, 71, 72, 73, 75, 76, 90, 92, 93, 98, 99

115

Thailand, 85, 86, 87, 88, 94, 96, 97, 98, 99, 100, 101, 102, 103, 104 theoretical, 25, 47, 60 theory, 42, 43, 57 thinking, 48 threshold, 90, 91 time series, vii, viii, 1, 2, 3, 4, 6, 17, 20, 21, 22, 23, 24, 25, 28, 43, 44, 46, 47, 49, 58, 59, 60, 64, 65, 69, 81, 85, 86, 87, 88, 94, 96, 98, 103, 104, 105, 106, 107, 108 trade, 48, 79 trading, 58 training, 5 transfer, 5, 13, 91, 92 transition, vii, 3, 23, 24, 25, 86, 100 transition economies, 86, 100 transmission, 90 trend, 10, 11, 12, 71 trial, 5, 29 trial and error, 5 Turkey, 85, 86, 87, 88, 94, 96, 97, 98, 99, 100, 101, 102, 103, 104 Turkmenistan, 86, 100

U Uunemployment, 46, 64 unemployment rate, 46, 64 United Arab Emirates, 86, 100 United Kingdom (UK), K, 1, 3, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27, 28, 29, 30, 34, 35, 36, 37, 38, 40, 41, 42, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 57, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79 United States, 48, 52 univariate, 28, 55, 56, 69, 71, 72, 75, 76, 77, 79 Uzbekistan, 86, 100

Index

116

V values, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 38, 40, 53, 55, 71, 72, 73, 75, 76, 89, 98, 99, 102 VAR models, 70 variable, 7, 17, 18, 19, 89, 95 variables, 2, 7, 20, 24, 30, 33, 48, 50, 66, 70, 86, 91 variance, 24, 89 vector, viii, 2, 9, 30, 32, 45, 47, 51, 57, 63, 64, 69, 70, 79, 91, 94, 104 Vietnam, 86, 100

volatility, 3, 85, 87, 102, 103, 104

W warrants, vii

Y Yemen, 86, 100

Business Cycle Fluctuations and Economic Policy

Business Fluctuations and Cycles

Inflation, Employment and Business Fluctuations

The Role of Policymakers in Business Cycle Fluctuations

International policy coordination and exchange rate fluctuations

Currency Interventions, Fluctuations and Economic Issues

Economic Forecasting and Policy

International Migration: Trends, Policy and Economic Impact (Contemporary Economic Policy)

Trade policy and economic welfare

Trade Policy and Economic Welfare

Globalization and progressive economic policy

Economic Policy and Technological Performance

International Policy Coordination and Exchange Rate Fluctuations (National Bureau of Economic Research Conference Report)

Power tails and fluctuations

Economic Policy 56

Frontiers of Business Cycle Research

Economic Liberalization and Integration Policy: Options for Eastern Europe and Russia (Economic Liberalization and Integration Policy)

Universal Fluctuations

Structural Reform and Economic Policy (International Economic Association)

Tax Policy Reform and Economic Growth, OECD Tax Policy Studies

Enterprise restructuring and economic policy in Russia

Tax Policy Reform and Economic Growth (OECD Tax Policy Studies)

Economic Sanctions: Law and Public Policy

Economic Analysis, Moral Philosophy and Public Policy

Quantum fluctuations

Quantum fluctuations

Cities, Economic Competition and Urban Policy

Canadian foreign policy and international economic regimes

Reforming Britain's Economic and Financial Policy: Towards Greater Economic Stability

Fat Economics: Nutrition, Health, and Economic Policy

Economic Policy Proposals for Germany and Europe

Business Cycle Fluctuations and Economic Policy

Business Fluctuations and Cycles

Inflation, Employment and Business Fluctuations

The Role of Policymakers in Business Cycle Fluctuations

International policy coordination and exchange rate fluctuations

Currency Interventions, Fluctuations and Economic Issues

Economic Forecasting and Policy

International Migration: Trends, Policy and Economic Impact (Contemporary Economic Policy)

Trade policy and economic welfare

Trade Policy and Economic Welfare

Globalization and progressive economic policy

Economic Policy and Technological Performance

International Policy Coordination and Exchange Rate Fluctuations (National Bureau of Economic Research Conference Report)

Power tails and fluctuations

Economic Policy 56

Frontiers of Business Cycle Research

Economic Liberalization and Integration Policy: Options for Eastern Europe and Russia (Economic Liberalization and Integration Policy)

Universal Fluctuations

Structural Reform and Economic Policy (International Economic Association)

Tax Policy Reform and Economic Growth, OECD Tax Policy Studies

Enterprise restructuring and economic policy in Russia

Tax Policy Reform and Economic Growth (OECD Tax Policy Studies)

Economic Sanctions: Law and Public Policy

Economic Analysis, Moral Philosophy and Public Policy

Quantum fluctuations

Quantum fluctuations

Cities, Economic Competition and Urban Policy

Canadian foreign policy and international economic regimes

Reforming Britain's Economic and Financial Policy: Towards Greater Economic Stability

Fat Economics: Nutrition, Health, and Economic Policy

Economic Policy Proposals for Germany and Europe

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