krolzig markov-switching vector autoregressions_ modelling, statistical inference, and application...

8/18/2019 Krolzig Markov-Switching Vector Autoregressions_ Modelling, Statistical Inference, And Application to Business Cycle A

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Hans–Martin Krolzig

Markov–Switching

Vector AutoregressionsModelling, Statistical Inference, and Applicationto Business Cycle Analysis


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To my parents, Grete and Walter


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Preface

This book contributes to recent developments on the statistical analysis of multiple

time series in the presence of regime shifts. Markov-switching models have be-

come popular for modelling non-linearities and regime shifts, mainly, in univariateeconomic time series. This study is intended to provide a systematic and opera-

tional approach to the econometric modelling of dynamic systems subject to shifts

in regime, based on the Markov-switching vector autoregressive model. The study

presents a comprehensive analysis of the theoretical properties of Markov-switching

vector autoregressive processes and the related statistical methods. The statistical

concepts are illustrated with applications to empirical business cycle research.

This monograph is a revised version of my dissertation which has been accepted by

the Economics Department of the Humboldt-University of Berlin in 1996. It con-

sists mainly of unpublished material which has been presented during the last years

at conferences and in seminars. The major parts of this study were written while I

was supported by the Deutsche Forschungsgemeinschaft (DFG), Berliner Graduier-

tenkolleg Angewandte Mikro ¨ okonomik and Sonderforschungsbereich 373 at the Free

University and Humboldt-University of Berlin. Work was finally completed in the

project The Econometrics of Macroeconomic Forecasting founded by the Economic

and Social Research Council (ESRC) at the Institute of Economics and Statistics,

University of Oxford. It is a pleasure to record my thanks to these institutions for

their support of my research embodied in this study.

The author is indebted to numerous individuals for help in the preparation of this

study. Primarily, I owe a great debt to Helmut Lütkepohl, who inspired me for mul-

tiple time series econometrics, suggested the subject, advised and encouraged my

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viii Preface

research. The many hours Helmut Lütkepohl and Jürgen Wolters spent in discussing

the issues of this study have been an immeasurable help.

The results obtained and their presentation have been profoundly affected by the in-

spiration of and interaction with numerous colleagues in Berlin and Oxford. Of the

many researchers from whom I have benefited by discussing with them various as-

pects of the work presented here, I would like especially to thank Ralph Friedmann,

David Hendry and D.S. Poskitt.

I wish to express my sincere appreciation of the helpful discussions, suggestions

and comments of the audiences at the 7th World Congress of the Econometric So-

ciety, the SEDC 1996 Annual Meeting, the ESEM96 , the American Wintermeeting

of the Econometric Society 1997 , the 11th Annual Congress of the European Eco-

nomic Association, the Workshop Zeitreihenanalyse und stochastische Prozesse and

the Pfingsttreffen 1996 of the Deutsche Statistische Gesellschaft , the Jahrestagungen

1995 and 1996 of the Verein f ¨ ur Socialpolitik , and in seminars at the Free-University

Berlin, the Humboldt-University of Berlin, the University College London and Nuf-

field College, Oxford.

Many people have helped with the reading of the manuscript. Special thanks go

to Paul Houseman, Marianne Sensier, Dirk Soyka and Don Indra Asoka Wijewick-

rama; they pointed out numerous errors and provided helpful suggestions.

I am very grateful to all of them, but they are of course, absolved from any respons-

ibility for the views expressed in the book. Any errors that may remain are my own.

Finally, I am greatly indebted to my parents and friends for their support and en-

couragement while I was struggling with the writing of the thesis.

Hans-Martin KrolzigOxford, March 1997


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Contents

Prologue 1

1 The Markov–Switching Vector Autoregressive Model 6

1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Markov-Switching Vector Autoregressions . . . . . . . . . . . . . . . . . . . . 10

1.2.1 The Vector Autoregression . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.2 Particular MS–VAR Processes . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.3 The Regime Shift Function . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.4 The Hidden Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3 The Data Generating Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Features of MS-VAR Processes and Their Relation to Other Non-

linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.1 Non-Normality of the Distribution of the Observed Time

Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4.2 Regime-dependent Variances and Conditional Heteroske-

dasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.4.3 Regime-dependent Autoregressive Parameters: ARCH and

Stochastic Unit Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.A Appendix: A Note on the Relation of SETAR to MS-AR Processes 27

2 The State-Space Representation 29

2.1 A Dynamic Linear State-Space Representation for MS-VAR Pro-

cesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.1.1 The Gaussian Measurement Equation . . . . . . . . . . . . . . . . . 33

2.1.2 The Non–Normal VAR(1)–Representation of the Hidden

Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1.3 Linearity of the State-Space Representation . . . . . . . . . . . . 34

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2.1.4 Markov Property of the State-Space Representation . . . . . . 35

2.2 Specification of the State–Space Representation . . . . . . . . . . . . . . . . 38

2.3 An Unrestricted State-Space Representation . . . . . . . . . . . . . . . . . . . 41

2.4 Prediction-Error Decomposition and the Innovation State-Space

Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.5 The MS-VAR Model and Time–Varying Coefficient Models . . . . . 45

3 VARMA-Representation of MSI-VAR and MSM-VAR Processes 49

3.1 Linearly Transformed Finite Order VAR Representations . . . . . . . . 50

3.2 VARMA Representation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.1 VARMA Representation of Linearly Transformed Finite

Order VAR Representations . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.2 ARMA Representation of a Hidden Markov Chain . . . . . . 56

3.2.3 VARMA Representations of MSI(M )–VAR(0) Processes . 56

3.2.4 VARMA Representations of MSI(M )–VAR( p) Processes . 57

3.2.5 VARMA Representations of MSM(M )–VAR( p) Processes 58

3.3 The Autocovariance Function of MSI–VAR and MSM-VAR Pro-

cesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3.1 The ACF of the Regime Generating Process . . . . . . . . . . . . 60

3.3.2 The ACF of a Hidden Markov Chain Process . . . . . . . . . . . 61

3.3.3 The ACF of MSM–VAR Processes . . . . . . . . . . . . . . . . . . . 62

3.3.4 The ACF of MSI-VAR Processes . . . . . . . . . . . . . . . . . . . . . 64

3.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 Forecasting MS–VAR Processes 67

4.1 MSPE-Optimal Predictors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2 Forecasting MSM–VAR Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 Forecasting MSI–VAR Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4 Forecasting MSA–VAR Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 The BLHK Filter 79

5.1 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.A Supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.A.1 Conditional Moments of Regime . . . . . . . . . . . . . . . . . . . . . 89

5.A.2 A Technical Remark on Hidden Markov-Chains: The

MSI/MSIH(M )-VAR(0) Model . . . . . . . . . . . . . . . . . . . . . . 90


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6 Maximum Likelihood Estimation 91

6.1 The Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2 The Identification Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.3 Normal Equations of the ML Estimator . . . . . . . . . . . . . . . . . . . . . . . 97

6.3.1 Derivatives with Respect to the VAR Parameters . . . . . . . . 986.3.2 Derivatives with Respect to the Hidden Markov-Chain

Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.3.3 Initial State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.4 The EM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.4.1 Estimation of γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.4.2 Estimation of σ under Homoskedasticity . . . . . . . . . . . . . . . 109

6.4.3 Estimation of σ under Heteroskedasticity . . . . . . . . . . . . . . 110

6.4.4 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.5 Extensions and Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.5.1 The Scoring Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.5.2 An Adaptive EM Algorithm (Recursive Maximum Likeli-

hood Estimation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.5.3 Incorporating Bayesian Priors . . . . . . . . . . . . . . . . . . . . . . . . 117

6.5.4 Extension to General State-Space Models with Markovian

Regime Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.6 Asymptotic Properties of the Maximum Likelihood Estimator . . . . 120

6.6.1 Asymptotic Normal Distribution of the ML Estimator . . . . 120

6.6.2 Estimation of the Asymptotic Variance–Covariance Matrix 122

6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7 Model Selection and Model Checking 125

7.1 A Bottom-up Strategy for the Specification of MS–VAR Models . . 126

7.2 ARMA Representation Based Model Selection . . . . . . . . . . . . . . . . 132

7.3 Model Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.3.1 Residual Based Model Checking . . . . . . . . . . . . . . . . . . . . . 135

7.3.2 The Coefficient of Determination . . . . . . . . . . . . . . . . . . . . . 137

7.4 Specification Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.4.1 Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.4.2 Lagrange Multiplier Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.4.3 Wald Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.4.4 Newey-Tauchen-White Test for Dynamic Misspecification 142

7.5 Determination of the Number of Regimes . . . . . . . . . . . . . . . . . . . . . 144


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7.6 Some Critical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

8 Multi-Move Gibbs Sampling 148

8.1 Bayesian Analysis via the Gibbs Sampler . . . . . . . . . . . . . . . . . . . . . 150

8.2 Bayesian Analysis of Linear Markov-Switching Regression Models 152

8.3 Multi–Move Gibbs Sampling of Regimes . . . . . . . . . . . . . . . . . . . . . 155

8.3.1 Filtering and Smoothing Step . . . . . . . . . . . . . . . . . . . . . . . . 156

8.3.2 Stationary Probability Distribution and Initial Regimes . . . 157

8.4 Parameter Estimation via Gibbs Sampling . . . . . . . . . . . . . . . . . . . . 158

8.4.1 Hidden Markov Chain Step . . . . . . . . . . . . . . . . . . . . . . . . . 158

8.4.2 Inverted Wishart Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

8.4.3 Regression Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8.5 Forecasting via Gibbs Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

9 Comparative Analysis of Parameter Estimation in Particular MS-VARModels 170

9.1 Analysis of Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

9.2 Comparison of the Gibbs Sampler with the EM Algorithm . . . . . . . 174

9.3 Estimation of VAR Parameters for Given Regimes. . . . . . . . . . . . . . 175

9.3.1 The Set of Regression Equations . . . . . . . . . . . . . . . . . . . . . 175

9.3.2 Maximization Step of the EM Algorithm . . . . . . . . . . . . . . . 177

9.3.3 Regression Step of the Gibbs Sampler . . . . . . . . . . . . . . . . . 180

9.3.4 MSI Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9.3.5 MSM Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

9.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1869.A Appendix: Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

10 Extensions of the Basic MS-VAR Model 202

10.1 Systems with Exogenous Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 202

10.2 Distributed Lags in the Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

10.2.1 The MSI(M, q )-VAR( p) Model . . . . . . . . . . . . . . . . . . . . . . 205

10.2.2 VARMA Representations of MSI(M, q )–VAR( p) Processes 206

10.2.3 Filtering and Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

10.3 The Endogenous Markov-Switching Vector Autoregressive Model 208

10.3.1 Models with Time-Varying Transition Probabilities . . . . . . 208

10.3.2 Endogenous Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

10.3.3 Filtering and Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212


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10.3.4 A Modified EM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 213

10.4 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

11 Markov-Switching Models of the German Business Cycle 215

11.1 MS-AR Processes as Stochastic Business Cycle Models . . . . . . . . . 218

11.2 Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

11.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

11.2.2 Traditional Turning Point Dating . . . . . . . . . . . . . . . . . . . . . 221

11.2.3 ARMA Representation Based Model Pre-Selection . . . . . . 222

11.3 The Hamilton Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

11.3.1 Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

11.3.2 Contribution to the Business Cycle Characterization . . . . . 226

11.3.3 Impulse Response Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . 229

11.3.4 Asymmetries of the Business Cycle . . . . . . . . . . . . . . . . . . . 230

11.3.5 Kernel Density Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 231

11.4 Models with Markov-Switching Intercepts . . . . . . . . . . . . . . . . . . . . 233

11.5 Regime-Dependent and Conditional Heteroskedasticity . . . . . . . . . 237

11.6 Markov-Switching Models with Multiple Regimes . . . . . . . . . . . . . 243

11.6.1 Outliers in a Three-Regime Model . . . . . . . . . . . . . . . . . . . . 243

11.6.2 Outliers and the Business Cycle . . . . . . . . . . . . . . . . . . . . . . 245

11.6.3 A Hidden Markov-Chain Model of the Business Cycle . . . 246

11.6.4 A Highly Parameterized Model . . . . . . . . . . . . . . . . . . . . . . 248

11.6.5 Some Remarks on Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

11.7 MS-AR Models with Regime-Dependent Autoregressive Parameters 250

11.8 An MSMH(3)-AR(4) Business Cycle Model . . . . . . . . . . . . . . . . . . 253

11.9 Forecasting Performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

11.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

11.A Appendix: Business Cycle Analysis with the Hodrick-Prescott Filter 260

12 Markov–Switching Models of Global and International Business

Cycles 262

12.1 Univariate Markov-Switching Models . . . . . . . . . . . . . . . . . . . . . . . . 263

12.1.1 USA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

12.1.2 Canada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

12.1.3 United Kingdom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

12.1.4 Germany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

12.1.5 Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

12.1.6 Australia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276


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12.1.7 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

12.2 Multi-Country Growth Models with Markov-Switching Regimes . . 282

12.2.1 Common Regime Shifts in the Joint Stochastic Process of

Economic Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

12.2.2 Structural Breaks and the End of the Golden Age . . . . . . . . 28312.2.3 Global Business Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

12.2.4 Rapid Growth Episodes and Recessions . . . . . . . . . . . . . . . 289

12.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

12.A Appendix: Estimated MS-DVAR Models . . . . . . . . . . . . . . . . . . . . . 295

13 Cointegration Analysis of VAR Models with Markovian Shifts in Re-

gime 302

13.1 Cointegrated VAR Processes with Markov-Switching Regimes . . . 303

13.1.1 Cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

13.1.2 The MSCI-VAR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

13.1.3 A State-Space Representation for MSCI-VAR Processes . . 307

13.2 A Cointegrated VARMA Representation for MSCI-VAR Processes 311

13.3 A Two-Stage Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

13.3.1 Cointegration Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

13.3.2 EM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

13.4 Global and International Business Cycles . . . . . . . . . . . . . . . . . . . . . 317

13.4.1 VAR Order Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

13.4.2 Cointegration Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

13.4.3 Granger Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

13.4.4 Forecast Error Decomposition . . . . . . . . . . . . . . . . . . . . . . . 324

13.5 Global Business Cycles in a Cointegrated System . . . . . . . . . . . . . . 32513.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

13.A Appendix: Estimated CI-VAR and MSCI-VAR Models . . . . . . . . . . 331

Epilogue 335

References 337

Tables 353

Figures 357

List of Notation 359


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Prologue

Objective of the Study

In the last decade time series econometrics has changed dramatically. One increas-

ingly prominent field has become the treatment of regime shifts and non-linear mod-

elling strategies. While the importance of regime shifts, particularly in macroecono-

metric systems, seems to be generally accepted, there is no established theory sug-

gesting a unique approach for specifying econometric models that embed changes

in regime.

Structural changes such as the oil price shocks, the introduction of European Mon-

etary System, the German reunification, the European Monetary Union and Eastern

European economies in transition, are often incorporated into a dynamic system in

a deterministic fashion. A time-varying process poses problems for estimation and

forecasting when a shift in parameters occurs. The degradation of performance of structural macroeconomic models seems at least partly due to regime shifts. In-

creasingly, regime shifts are not considered as singular deterministic events, but the

unobservable regime is assumed to be governed by an exogenous stochastic process.

Thus regime shifts of the past are expected to occur in the future in a similar fashion.

The main aim of this study is to construct a general econometric framework for

the statistical analysis of multiple time series when the mechanism which generated

the data is subject to regime shifts. We build-up a stationary model where a stable

vector autoregression is defined conditional on the regime and where the regime

generating process is given by an irreducible ergodic Markov chain.

The primary advantage of the Markov-switching vector autoregressive model is to

provide a systematic approach to deliver statistical methods for: (i.) extracting the

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2 Prologue

information in the data about regime shifts in the past, (ii.) estimating consistently

and efficiently the parameters of the model, (iii.) detecting recent regime shifts,

(iv.) correcting the vector autoregressive model at times when the regime alters,

and finally (v.) incorporating the probability of future regime shifts into forecasts.

This Markov-switching vector autoregressive model represents a very general class

which encompasses some alternative non-linear and time-varying models. In gen-

eral, the model generates conditional heteroskedasticity and non-normality; predic-

tion intervals are asymmetric and reflect the prevailing uncertainty about the regime.

We will investigate the issues of detecting multiple breaks in multiple time series,

modelling, specification, estimation, testing and forecasting. En route, we discuss

the relation to alternative non-linear models and models with time-varying para-

meters. In course of this study we will also propose new directions to generalize the

MS-VAR model. Although some methodological and technical ideas are discussed

in detail, the focus is on modelling, specification and estimation of suitable models.

The previous literature on this topic is often characterized by imprecise generalities

or the restriction of empirical analysis to a very specific model whose specifica-

tion is motivated neither statistically nor theoretically. These limitations have to be

overcome. Therefore, the strategy of this study has to be twofold: (i.) to provide a

general approach to model building and (ii.) to offer concrete solutions for special

problems. This strategy implies an increase in the number of models as well as in

the complexity of the analysis. We believe, however, that this price will be proven

in practice to be offset by the increased flexibility for empirical research.

Survey of the Study

The first part of the book gives a comprehensive mathematical and statistical ana-

lysis of the Markov-switching vector autoregressive model. In the first chapters,

Markov-switching vector autoregressive (MS-VAR) processes are introduced and

their basic properties are investigated. We discuss the relation of the MS-VAR

model to the time invariant vector autoregressive model and against alternative non-

linear time series models. The preliminary considerations of Chapter 1 are formal-

ized in the state-space representation given in Chapter 2 , which will be the frame-

work for analyzing the stochastic properties of MS-VAR processes and for devel-

oping statistical techniques for the specification and estimation of MS-VAR models


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Survey of the Study 3

to fit data which exhibits regime shifts in a stationary manner. In Chapter 3, vector

autoregressive moving average (VARMA) representation theorems for VAR models

with Markov-switching means or intercepts are given.

In Chapter 4 and Chapter 5 , the statistical analysis of MS-VAR models is considered

for known parameters. In Chapter 4 , optimal predictors for MS-VAR processes are

derived. Chapter 5 is devoted to an intensive discussion of the filtering and smooth-

ing techniques for MS-VAR processes which the following statistical analysis is

based on. These statistical tools produce an inference for the time paths of unob-

served regimes under alternative information sets and given parameters. It is shown

that a modification of the model by introducing time-varying transition probabilities

can be analyzed with only slight modifications within our framework.

The main part of this study (Chapters 6 – 10 ) is devoted to the discussion of para-

meter estimation for this class of models. The classical method of maximum like-

lihood estimation is considered in Chapter 6 , where due to the nonlinearity of the

model, iterative procedures have to be introduced. While various approaches are

discussed, major attention is given to the EM algorithm, at which the limitation in

the previous literature of using special MS-VAR models is overcome. The issues

of identifiability and consistency of the maximum likelihood (ML) estimation are

investigated. Techniques for the calculation of the asymptotic variance-covariance

matrix of ML estimates are presented.

In Chapter 7 the issue of model selection and model checking is investigated. The

focus is maintained on the specification of MS-VAR models. A strategy for sim-

ultaneously selecting the number of regimes and the order of the autoregression inMarkov-switching time series models based on ARMA representations is proposed

and combined with classical specification testing procedures.

Chapter 8 introduces a multi-move Gibbs-Sampler for multiple time series subject

to regime shifts. Even for univariate time series analysis, an improvement over the

approaches described in the literature is achieved by an increased convergence due

to the simultaneous sampling of the regimes from their joint posterior distribution

using the methods introduced in Chapter 5 . Here again, a thorough analysis of vari-

ous MS-VAR specifications allows for a greater flexibility in empirical research.

The main advantage of the Gibbs sampler is that (by invoking Bayesian theory) this

simulation technique enables us to gain new insights into the unknown parameters.

Without informative priors, the Gibbs sampler reproduces the ML estimator as mode


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Survey of the Study 5

a multiple time series framework. Chapter 12 contributes to the research of inter-

national and global business cycles by analyzing a six-dimensional system for the

USA, Japan, West Germany, the UK, Canada, and Australia. The considerations

formulated in Chapter 13 suggest a new methodological approach to the analysis of

cointegrated linear systems with shifts in regime. This methodology is then illus-

trated with a reconsideration of international and global business cycles. The study

concludes with a brief discussion of our major findings and remaining problems.

The study has a modular structure. Given the notation and basic structures intro-

duced in the first two chapters, most of the following chapters can stand alone.

Hence, the reader, who is primarily interested in empirical applications and less in

statistical techniques, can decide to read first the fundamental Chapters 1 and 2 ,

then Chapter 5 and Chapter 6 followed by the empirical analyses in Chapters 11 and

12 alongside the more technically demanding Chapter 13 and to decide afterwards

which of the remaining chapters will be of interest to him or her.

Although it is not necessary for the reader to be familiar with all fundamental meth-

ods of multiple time series analysis, the subject of interest requires the application

of some formal techniques. A number of references to standard results are given

throughout the study, while to simplify things for the reader we have remained as

close as possible to the notation used in L ÜTKEPOHL [1991]. In order to achieve

compactness in our presentation, we have dispensed with a more general introduc-

tion of the topic since these are already available in H AMILTON [1993], [1994b,

ch. 22] and KROLZIG AND L ÜTKEPOHL [1995].


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Chapter 1

The Markov–Switching

Vector Autoregressive Model

This first chapter is devoted to a general introduction into the Markov–switching

vector autoregressive (MS-VAR) time series model. In Section 1.2 we present the

fundamental assumptions constituting this class of models. The discussion of the

two components of MS-VAR processes will clarify their on time invariant vector

autoregressive and Markov-chain models. Some basic stochastic properties of MS-

VAR processes are presented in Section 1.3. Finally, MS-VAR models are compared

to alternative non-normal and non-linear time series models proposed in the literat-

ure. As most non-linear models have been developed for univariate time series, this

discussion is restricted to this case. However, generalizations to the vector case are

also considered.

1.1 General Introduction

Reduced form vector autoregressive (VAR) models have been become a dominant

research strategy in empirical macroeconomics since S IM S [1980]. In this study we

will consider VAR models with changes in regime, most results will carry over to

structural dynamic econometric models by treating them as restricted VAR models.

When the system is subject to regime shifts, the parameters θ of the VAR process

will be time-varying. But the process might be time-invariant conditional on an

unobservable regime variable s t which indicates the regime prevailing at time t.

Let M denote the number of feasible regimes, so that s t ∈ {1, . . . , M }. Then the

6


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1.1. General Introduction 7

conditional probability density of the observed time series vector y t is given by

p(yt|Y t−1, st) =

f (yt|Y t−1, θ1) if st = 1...

f (yt|Y t−1, θM ) if st = M,

(1.1)

where θm is the VAR parameter vector in regime m = 1, . . . , M and Y t−1 are the

observations {yt−j}∞j=1.Thus, for a given regime s t, the time series vector yt is generated by a vector auto-

regressive process of order p (VAR( p) model) such that

E[yt|Y t−1, st] = ν (st) + p

j=1

Aj (st)yt−j ,

where ut is an innovation term,

ut = yt − E[yt|Y t−1, st].

The innovation process ut is a zero-mean white noise process with a variance-

covariance matrix Σ(st), which is assumed to be Gaussian:

ut ∼ NID(0, Σ(st)).

If the VAR process is defined conditionally upon an unobservable regime as in equa-

tion (1.1), the description of the data generating mechanism has to be completed by

assumptions regarding the regime generating process. In Markov-switching vector

autoregressive (MS-VAR) models – the subject of this study – it is assumed that theregime st is generated by a discrete-state homogeneous Markov chain: 1

Pr(st|{st−j}∞j=1, {yt−j}∞j=1) = Pr(st|st−1; ρ),

where ρ denotes the vector of parameters of the regime generating process.

The vector autoregressive model with Markov-switching regimes is founded

on at least three traditions. The first is the linear time-invariant vector auto-

regressive model, which is the framework for the analysis of the relation of

the variables of the system, the dynamic propagation of innovations to the

1The notation Pr(·) refers to a discrete probability measure, while p(·) denotes a probability density

function.


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8 The Markov–Switching Vector Autoregressive Model

system, and the effects of changes in regime. Secondly, the basic statistical

techniques have been introduced by BAUM AND PETRIE [1966] and BAUM

et al. [1970] for probabilistic functions of Markov chains, while the MS-VAR

model also encompasses older concepts as the mixture of normal distributions

model attributed to PEARSON [1894] and the hidden Markov-chain model

traced back to BLACKWELL AND KOOPMANS [1975] and HELLER [1965].

Thirdly, in econometrics, the first attempt to create Markov-switching regression

models were undertaken by GOLDFELD AND QUANDT [1973], which remained,

however, rather rudimentary. The first comprehensive approach to the statistical

analysis of Markov-switching regression models has been proposed by L INDGREN

[1978] which is based on the ideas of BAUM et al. [1970]. In time series analysis,

the introduction of the Markov-switching model is due to HAMILTON [1988],

[1989] on which most recent contributions (as well as this study) are founded.

Finally, our consideration of MS-VAR models as a Gaussian vector autoregressive

process conditioned on an exogenous regime generating process is closely related to

state space models as well as the concept of doubly stochastic processes introduced

by TJØSTHEIM [1986b].

The MS-VAR model belongs to a more general class of models that characterize a

non-linear data generating process as piecewise linear by restricting the process to

be linear in each regime, where the regime is conditioned is unobservable, and only

a discrete number of regimes are feasible.2 These models differ in their assumptions

concerning the stochastic process generating the regime:

(i.) The mixture of normal distributions model is characterized by serially inde-pendently distributed regimes:

Pr(st|{st−j}∞j=1, {yt−j}∞j=1) = Pr(st; ρ).

In contrast to MS-VAR models, the transition probabilities are independent of

the history of the regime. Thus the conditional probability distribution of y t

is independent of st−1,

Pr(yt|Y t−1, st−1) = Pr(yt|Y t−1),

2In the case of two regimes, P OTTER [1990],[1993] proposed to call this class of non-linear, non-

normal models the single index generalized multivariate autoregressive (SIGMA) model.


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1.1. General Introduction 9

and the conditional mean E[yt|Y t−1, st−1] is given by E[yt|Y t−1].3 Even so,this model can be considered as a restricted MS-VAR model where the trans-

ition matrix has rank one. Moreover, if only the intercept term will be regime-

dependent, MS(M )-VAR( p) processes with Gaussian errors and i.i.d. switch-

ing regimes are observationally equivalent to time-invariant VAR( p) processes

with non-normal errors. Hence, the modelling with this kind of model is very

limited.

(ii.) In the self-exciting threshold autoregressive SETAR( p, d, r) model, the

regime-generating process is not assumed to be exogenous but directly linked

to the lagged endogenous variable y t−d.4 For a given but unknown threshold

r, the ‘probability’ of the unobservable regime s t = 1 is given by

Pr(st = 1|{st−j}∞j=1, {yt−j}∞j=1) = I (yt−d ≤ r) = 1 if yt−d ≤ r0 if yt−d > r,

While the presumptions of the SETAR and the MS-AR model seem to be

quite different, the relation between both model alternatives is rather close.

This is also illustrated in the appendix which gives an example showing that

SETAR and MS-VAR models can be observationally equivalent.

(iii.) In the smooth transition autoregressive (STAR) model popularized by GRA N-

GER AND T ER ÄSVIRTA [1993], exogenous variables are mostly employed to

model the weights of the regimes, but the regime switching rule can also be

dependent on the history of the observed variables, i.e. y t−d:

Pr(st = 1|{st−j}∞j=1, {yt−j}∞j=1, ) = F (yt−dδ − r),

where F (yt−dδ − r) is a continuous function determining the weight of re-

3The likelihood function is given by

p(Y T |Y 0; θ, ξ̄) =

T t=1

M m=1

ξ̄m p(yt|Y t−1, θm),

where θ = (θ1, . . . , θM )

collects the VAR parameters and ξ̄m is the ergodic probability of regime

m.4

In threshold autoregressive (TAR) processes, the indicator function is defined in a switching variablezt−d, d ≥ 0. In addition, indicator variables can be introduced and treated with error-in-variables

techniques. Refer for example to COSSLETT AND L EE [1985] and KAMINSKY [1993].


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gime 1. For example, TER ÄSVIRTA AND A NDERSON [1992] use the logistic

distribution function in their analysis of the U.S. business cycle. 5

(iv.) All the previously mentioned models are special cases of an endogenous se-

lection Markov-switching vector autoregressive model. In an EMS(M, d)-

VAR( p) model the transition probabilities pij (·) are functions of the observedtime series vector yt−d:

Pr(st = m|st−1 = i, yt−d) = pim(yt−dδ ).

Thus the observed variables contain additional information on the conditional

probability distribution of the states:

Pr(st|{st−j}∞j=1)a.e.

= Pr(st|{st−j}∞j=1, {yt−j}∞j=1).

Thus the regime generating process is no longer Markovian. In contrast to the

SETAR and the STAR model, EMS-VAR models include the possibility thatthe threshold depends on the last regime, e.g. that the threshold for staying

in regime 2 is different from the threshold for switching from regime 1 to

regime 2 . The EMS(M, d)-VAR( p) model will be presented in Section 10.3.

It is shown that the methods developed in this study for MS-VAR processes

can easily be extended to capture EMS-VAR processes.

In this study, it will be shown that the MS-VAR model can encompass a wide spec-

trum of non-linear modifications of the VAR model proposed in the literature.

1.2 Markov-Switching Vector Autoregressions

1.2.1 The Vector Autoregression

Markov-switching vector autoregressions can be considered as generalizations of

the basic finite order VAR model of order p. Consider the p-th order autoregression

for the K -dimensional time series vector y t = (y1t, . . . , yKt ), t = 1, . . . , T ,

yt = ν + A1yt−1 + . . . + A pyt− p + ut, (1.2)

5If F (·) is even, e.g. F (yt−d − r) = 1 − exp−(yt−d − r)2, a generalized exponential auto-

regressive model as proposed by O ZAKI [1980] and HAGGAN AND O ZAKI [1981] ensues.


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1.2. Markov-Switching Vector Autoregressions 11

where ut ∼ IID(0, Σ) and y0, . . . , y1− p are fixed. Denoting A(L) =IK − A1L − . . . − A pL p as the (K × K ) dimensional lag polynomial, we as-sume that there are no roots on or inside the unit circle |A(z)| = 0 for |z| ≤ 1where L is the lag operator, so that y t−j = L

jyt . If a normal distribution of the

error is assumed, ut ∼ NID(0, Σ), equation (1.2) is known as the intercept form of a stable Gaussian VAR( p) model. This can be reparametrized as the mean adjusted

form of a VAR model:

yt − µ = A1(yt−1 − µ) + . . . + A p(yt− p − µ) + ut, (1.3)where µ = (IK −

pj=1 Aj)

−1ν is the (K × 1) dimensional mean of yt.If the time series are subject to shifts in regime, the stable VAR model with its time

invariant parameters might be inappropriate. Then, the MS–VAR model might be

considered as a general regime-switching framework. The general idea behind this

class of models is that the parameters of the underlying data generating process 6 of

the observed time series vector y t depend upon the unobservable regime variablest, which represents the probability of being in a different state of the world.

The main characteristic of the Markov-switching model is the assumption that the

unobservable realization of the regime s t ∈ {1, . . . , M } is governed by a discretetime, discrete state Markov stochastic process, which is defined by the transition

probabilities

pij = Pr(st+1 = j |st = i),M

j=1

pij = 1 ∀i, j ∈ {1, . . . , M }. (1.4)

More precisely, it is assumed that st follows an irreducible ergodic M state Markov

process with the transition matrix P. This will be discussed in Section 1.2.4 in moredetail.

In generalization of the mean-adjusted VAR( p) model in equation (1.3) we would

like to consider Markov-switching vector autoregressions of order p and M regimes:

yt−µ(st) = A1(st) (yt−1 − µ(st−1))+ . . .+A p(st) (yt− p − µ(st− p))+ ut, (1.5)where ut ∼ NID(0, Σ(st)) and µ(st), A1(st), . . . , A p(st), Σ(st) are parametershift functions describing the dependence of the parameters 7 µ, A1, . . . , A p, Σ on

6For reasons of simplicity in notation, we do not introduce a separate notation for the theoretical

representation of the stochastic process and its actual realizations.7In the notation of state-space models, the varying parameters µ,ν, A1, . . . , Ap, Σ become functions

of the model’s hyper-parameters.


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the realized regime st, e.g.

µ(st) =

µ1 if st = 1,...

µM if st = M.

(1.6)

In the model (1.5) there is after a change in the regime an immediate one–time jump

in the process mean. Occasionally, it may be more plausible to assume that the mean

smoothly approaches a new level after the transition from one state to another. In

such a situation the following model with a regime-dependent intercept term ν (s t)

may be used:

yt = ν (st) + A1(st)yt−1 + . . . + A p(st)yt− p + ut. (1.7)

In contrast to the linear VAR model, the mean adjusted form (1.5) and the intercept

form (1.7) of an MS(M )–VAR( p) model are not equivalent. In Chapter 3 it will be

seen that these forms imply different dynamic adjustments of the observed variables

after a change in regime. While a permanent regime shift in the mean µ(s t) causes

an immediate jump of the observed time series vector onto its new level, the dynamic

response to a once-and-for-all regime shift in the intercept term ν (s t) is identical to

an equivalent shock in the white noise series u t.

In the most general specification of an MS-VAR model, all parameters of the autore-

gression are conditioned on the state s t of the Markov chain. We have assumed that

each regime m possesses its VAR( p) representation with parameters ν (m) (or µ m),Σm, A1m, . . . , Ajm , m = 1, . . . , M , such that

yt =

ν 1 + A11yt−1 + . . . + A p1yt− p + Σ

1/21 ut, if st = 1

...

ν M + A1M yt−1 + . . . + A pM yt− p + Σ1/2M ut, if st = M

where ut ∼ NID(0, IK ).8

However for empirical applications, it might be more helpful to use a model where

only some parameters are conditioned on the state of the Markov chain, while the

8Even at this early stage a complication arises if the mean adjusted form is considered. The conditionaldensity of yt depends not only on st but also on st−1, . . . , st−p, i.e. M p+1 different conditional


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other parameters are regime invariant. In Section 1.2.2 some particular MS-VAR

models will be introduced where the autoregressive parameters, the mean or the in-

tercepts, are regime-dependent and where the error term is hetero- or homoskedastic.

Estimating these particular MS-VAR models is discussed separately in Chapter 9 .

1.2.2 Particular MS–VAR Processes

The MS-VAR model allows for a great variety of specifications. In principle, it

would be possible to (i.) make all parameters regime-dependent and (ii.) to intro-

duce separate regimes for each shifting parameter. But, this would be no practicable

solution as the number of parameters of the Markov chain grows quadratic in the

number of regimes and coincidently shrinks the number of observations usable for

the estimation of the regime-dependent parameter. For these reasons a specific-to-

general approach may be preferred for the determination of the regime generating

process by restricting the shifting parameters (i.) to a part of the parameter vector

and (ii.) to have identical break-points.

In empirical research, only some parameters will be conditioned on the state of

the Markov chain while the other parameters will be regime invariant. In order to

establish a unique notation for each model, we specify with the general MS(M )

term the regime-dependent parameters:

M Markov-switching mean ,

I Markov-switching intercept term ,

A Markov-switching autoregressive parameters ,

H Markov-switching heteroskedasticity .To achieve a distinction of VAR models with time-invariant mean and intercept

term, we denote the mean adjusted form of a vector autoregression as MVAR( p).

means of yt are to be distinguished:

yt=

µ1 +A11 (yt−1−µ1 )+ . . . +Ap1(yt−p−µ1 )+Σ1/21 ut, if st=1, . . . , st−p=1

µ1 +A11 (yt−1−µ1 )+ . . . +Ap1(yt−p−µ2 )+Σ1/21 ut, if st=1, . . . , st−p+1=1, st−p=2

.

.

.µ1 +A11(yt−1−µM )+ . . . +Ap1(yt−p−µM )+Σ

1/21 ut, if st=1, st−1 =M,. . . , st−p=M

.

.

.

µM +A1M (yt−1−µ1 )+ . . . +ApM (yt−p−µ1)+Σ1/2M ut, if st=M, st−1=1, . . . , st−p=1

.

.

.

µM +A1M (yt−1−µ1 )+ . . . +ApM (yt−p−µM −1)+Σ1/2M ut, if st=M...st−p+1 =M, st−p=M −1

µM +A1M (yt−1−µM )+ . . . +ApM (yt−p−µM )+Σ1/2M ut, if st=M,. . . , st−p=M


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Table 1.1: Special Markov Switching Vector Autoregressive Models

MSM MSI Specification

µ varying µ invariant ν varying ν invariant

Aj Σ invariant MSM–VAR linear MVAR MSI–VAR linear VAR

invariant Σ var yi ng MSMH– VAR MS H–MVAR MSI H–VAR MS H–VAR

Aj Σ invariant MSMA–VAR MSA–MVAR MSIA–VAR MSA–VAR

varying Σ varying MSMAH–VAR MSAH–MVAR MSIAH–VAR MSAH–VAR

An overview is given in Table 1.1. Obviously the MSI and the MSM specifications

are equivalent if the order of the autoregression is zero. For this so-called hidden

Markov-chain model, we prefer the notation MSI(M )-VAR(0). As it will be seen

later on, the MSI(M )-VAR(0) model and MSI(M )-VAR( p) models with p > 0 are

isomorphic concerning their statistical analysis. In Section 10.3 we will further

extend the class of models under consideration.

The MS-VAR model provides a very flexible framework which allows for hetero-

skedasticity, occasional shifts, reversing trends, and forecasts performed in a non-

linear manner. In the following sections the focus is on models where the mean

(MSM(M )–VAR( p) models) or the intercept term (MSI(M )–VAR( p) models) aresubject to occasional discrete shifts; regime-dependent covariance structures of the

process are considered as additional features.

1.2.3 The Regime Shift Function

At this stage it is useful to define the parameter shifts more clearly by formulating

the system as a single equation by introducing “dummy” (or more precisely) indic-

ator variables:

I (st = m) = 1 if st = m

0 otherwise,


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where m = 1, . . . , M . In the course of the following chapters it will prove helpful

to collect all the information about the realization of the Markov chain in the vector

ξ t as

ξ t = I (st = 1)

...

I (st = M )

.Thus, ξ t denotes the unobserved state of the system. Since ξ t consists of binary

variables, it has some particular properties:

E[ξ t] =

Pr(st = 1)

...

Pr(st = M )

=

Pr(ξ t = ι1)...

Pr(ξ t = ιM )

,where ιm is the m-th column of the identity matrix. Thus E[ξ t], or a well defined

conditional expectation, represents the probability distribution of s t. It is easily

verified that 1 M ξ t = 1 as well as ξ tξ t = 1 and ξ tξ

t = diag(ξ t), where 1 M =

(1, . . . , 1) is an (M × 1) vector.For example, we can now rewrite the mean shift function (1.6) as

µ(st) =

M m=1

µmI (st = m).

In addition, we can use matrix notation to derive

µ(st) = Mξ t,

where M is a (K × M ) matrix containing the means,M = µ1 . . . µM , µ = vec(M).

We will occasionally use the following notation for the variance parameters:

Σ

(K ×MK )

=

Σ1 . . . ΣM

σmK(K+1)

2 ×1 = vech (Σm), σ = (σ1, . . . , σM )

such that

Σt = Σ(st) = Σ(ξ t ⊗ IK )is a (K × K ) matrix.


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1.2.4 The Hidden Markov Chain

The description of the data-generatingprocess is not completed by the observational

equations (1.5) or (1.7). A model for the parameter generating process has to be

formulated. if the parameters depend on a regime which is assumed to be stochasticand unobservable, a generating process for the states s t must be postulated. Using

this law, the evolution of regimes then might be inferred from the data. In the MS-

VAR model the state process is an ergodic Markov chain with a finite number of

states st = 1, . . . , M and transition probabilities pij .

It is convenient to collect the transition probabilities in the transition matrix P,

P =

p11 p12 · · · p1M p21 p22 · · · p2M

......

. . . ...

p11 p12 · · · p1M

, (1.8)

where piM = 1 − pi1 − . . . − pi,M −1 for i = 1, . . . , M . To be more precise, allrelevant information about the future of the Markovian process is included in the

present state ξ t

Pr(ξ t+1|ξ t, ξ t−1, . . . ; yt, yt−1, . . .) = Pr(ξ t+1|ξ t)

where the past and additional variables such as y t reveal no relevant information

beyond that of the actual state. The assumption of a first-order Markov process is

not especially restrictive, since each Markov chain of an order greater than one canbe reparametrized as a higher dimensional first-order Markov process (cf. FRIED-

MANN [1994]). A comprehensive discussion of the theory of Markov chains with

application to Markov-switching models is given by HAMILTON [1994b, ch. 22.2].

We will just give a brief introduction to some basic concepts related to MS-VAR

models, in particular to the state-space form and the filter.

It is usually assumed that the Markov process is ergodic. A Markov chain is said

to be ergodic if exactly one of the eigenvalues of the transition matrix P is unity

and all other eigenvalues are inside the unit circle. Under this condition there exists

a stationary or unconditional probability distribution of the regimes. The ergodic

probabilities are denoted by ξ̄ = E[ξ t]. They are determined by the stationarity

restriction Pξ̄ = ξ̄ and the adding up restriction 1 M ξ̄ = 1, from which it follows


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1.3. The Data Generating Process 17

that

ξ̄ =

IM −1 − P1.M −1,1.M −1 P1.M −1,M

1M −1 1

−1 0M −1

1

. (1.9)

if ξ̄ is strictly positive, such that all regimes have a positive unconditional probab-

ility ξ̄ i > 0, i = 1, . . . , M , the process is called irreducible. The assumptions of

ergodicity and irreducibility are essential for the theoretical properties of MS-VAR

models, e.g. its property of being stationary. The estimation procedures, which will

be introduced in Chapter 6 and Chapter 8 are flexible enough to capture even these

degenerated cases, e.g. when there is a single jump (“structural break”) into the

absorbing state that prevails until the end of the observation period.

1.3 The Data Generating Process

After this introduction of the two components of MS-VAR models, (i.) the Gaussian

VAR model as the conditional data generating process and (ii.) the Markov chain

as the regime generating process, we will briefly discuss their main implications for

the data generating process.

For given states ξ t and lagged endogenous variables Y t−1 = (yt−1, y

t−2, . . . , y

1,

y0, . . . , y1− p)

the conditional probability density function of y t is denoted by

p(yt|ξ t, Y t−1). It is convenient to assume in (1.5) and (1.7) a normal distributionof the error term ut, so that

p(yt|ξ t = ιm, Y t−1)= ln(2π)−1/2 ln |Σ|−1/2 exp{(yt − ȳmt)Σ−1m (yt − ȳmt)}, (1.10)

where ȳmt = E[yt|ξ t, Y t−1] is the conditional expectation of y t in regime m. Thusthe conditional density of y t for a given regime ξ t is normal as in the VAR model

defined in equation (1.2). Thus:

yt|ξ t = ιm, Y t−1

∼ NID (ȳmt, Σm) ,

∼ NID (ȳtξ t, Σ(ξ t ⊗ IK )) , (1.11)


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where the conditional means ȳmt are summarized in the vector ȳt which is e.g. in

MSI specifications of the form

ȳt =

ȳ1t..

.ȳMt

=

ν 1 +

pj=1 A1j yt−j

..

.ν M +

pj=1 AMj yt−j

.

Assuming that the information set available at time t − 1 consists only of thesample observations and the pre-sample values collected in Y t−1 and the states of

the Markov chain up to ξ t−1, the conditional density of yt is a mixture of normals9:

p(yt|ξ t−1 = ιi, Y t−1)

=M

m=1

p(yt|ξ t−1 = ιm, Y t−1)Pr(ξ t|ξ t−1 = ιi)

=

M m=1

pim ln(2π)− 12 ln |Σm|− 12 exp{(yt − ȳmt)Σ−1m (yt − ȳmt)} .(1.12)if the densities of yt conditional on ξ t and Y t−1 are collected in the vector ηt as

ηt =

p(yt|ξ t = ι1, Y t−1)

...

p(yt|ξ t = ιM , Y t−1)

, (1.13)equation (1.12) can be written as

p(yt|ξ t−1, Y t−1) = η t Pξ t−1. (1.14)

Since the regime is assumed to be unobservable, the relevant information set avail-able at time t − 1 consists only of the observed time series until time t and theunobserved regime vector ξ t has to be replaced by the inference Pr(ξ t|Y τ ). Theseprobabilities of being in regime m given an information set Y τ are denoted ξ mt|τ

and collected in the vector ξ̂ t|τ as

ξ̂ t|τ =

Pr(ξ t = ι1|Y τ )

...

Pr(ξ t = ιM |Y τ ),

9

The reader is referred to HAMILTON [1994a] for an excellent introduction into the major conceptsof Markov chains and to T ITTERINGTON , S MITH & MAKOV [1985] for the statistical properties of

mixtures of normals.


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1.3. The Data Generating Process 19

which allows two different interpretations. First, ξ̂ t|τ denotes the discrete condi-

tional probability distribution of ξ t given Y τ . Secondly, ξ̂ t|τ is equivalent to the

conditional mean of ξ t given Y τ . This is due to the binarity of the elements of ξ t,

which implies that E[ξ mt] = Pr(ξ mt = 1) = Pr(st = m). Thus, the conditional

probability density of yt based upon Y t−1 is given by

p(yt|Y t−1) =M

m=1

p(yt, ξ t−1 = ιm|Y t−1)

=

M m=1

p(yt|ξ t−1 = ιm, Y t−1)Pr(ξ t−1 = ιm|Y t−1) (1.15)

= ηt Pξ̂ t−1|t−1.

As with the conditional probability density of a single observation y t in (1.15) the

conditional probability density of the sample can be derived analogously. The tech-

niques of setting-up the likelihood function in practice are introduced in Section 6.1.

Here we only sketch the basic approach.

Assuming presample values Y 0 are given, the density of the sample Y ≡ Y T forgiven states ξ is determined by

p(Y |ξ ) =T

t=1

p(yt|ξ t, Y t−1). (1.16)

Hence, the joint probability distribution of observations and states can be calculated

as

p(Y, ξ ) = p(Y |ξ ) Pr(ξ )

=T

t=1

p(yt|ξ t, Y t−1)T

t=2

Pr(ξ t|ξ t−1) Pr(ξ 1). (1.17)

Thus, the unconditional density of Y is given by the marginal density

p(Y ) =

p(Y, ξ ) dξ, (1.18)

where f (x, ξ )dξ := M i1=1 . . .M iT =1 f (x, ξ T = ιiT , . . . , ξ 1 = ιi1 ) denotessummation over all possible values of ξ = ξ T ⊗ ξ T −1 ⊗ . . . ⊗ ξ 1 in equation (1.18).


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Finally, it follows by the definition of the conditional density that the conditional

distribution of the total regime vector ξ is given by

Pr(ξ |Y ) = p(Y, ξ ) p(Y )

.

Thus, the desired conditional regime probabilities Pr(ξ t|Y ) can be derived by mar-ginalization of Pr(ξ |Y ). In practice these cumbrous calculations can be simplifiedby a recursive algorithm, a matter which is discussed in Chapter 5 .

The regime probabilities for future periods follow from the exogenous stochastic

process of ξ t, more precisely the Markov property of regimes, Pr(ξ T +h|ξ T , Y ) =Pr(ξ T +h|ξ T ),

Pr(ξ T +h|Y ) =

ξtPr(ξ T +h|ξ T , Y )Pr(ξ T |Y )

= ξt

Pr(ξ T +h|ξ T )Pr(ξ T |Y ).

These calculations can be summarized in the simple forecasting rule:Pr(sT +h = 1|Y )

...

Pr(sT +h = M |Y )

= [P]h

Pr(sT = 1|Y )...

Pr(sT = M |Y )

,where P is the transition matrix as in (1.8). Forecasting MS-VAR processes is

discussed in full length in Chapter 4 .

In this section we have given just a short introduction to some basic concepts related

to MS-VAR models; the following chapters will provide broader analyses of the

various topics.

1.4 Features of MS-VAR Processes and Their Rela-

tion to Other Non-linear Models

The Markov switching vector autoregressive model is a very general approach for

modelling time series with changes in regime. In Chapter 3 it will be shown that MS-

VAR processes with shifting means or intercepts but regime-invariant variances and


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1.4. Features of MS-VAR Processes and Their Relation to Other Non-linear Models 21

autoregressive parameters can be represented as non-normal linear state space mod-

els. Furthermore, MSM-VAR and MSI-VAR models possess linear representations.

These processes may be better characterized as non-normal than as non-linear time

series models as the associated Wold representations coincide with those of linear

models. While our primary research interest concerns the modelling of the condi-

tional mean, we will exemplify the effects of Markovian switching regimes on the

higher moments of the observed time series.

For sake of simplicity we restrict the following consideration mainly to univariate

processes

yt = ν (st) +

pj=1

αj (st)yt−j + ut, ut ∼ NID(0, σ2(st)).

Most of them are made for two-regimes. Thus, the process generating y t can be

rewritten as

yt = [ν 2 + (ν 1 − ν 2)ξ 1t] + p

j=1

[α2 + (α1 − α2)ξ 1t]yt−j + ut,

ut ∼ NID(0, [σ22 + (σ21 − σ22)ξ 1t]).

if the regime st is governed by a Markov chain, the MS(2)-AR( p) model ensues. It

will be shown that even such simple MS-AR models can encompass a wide spectrum

of modifications of the time-invariant normal linear time series model.

1.4.1 Non-Normality of the Distribution of the Observed Time

Series

As already seen the conditional densities p(yt|Y t−1) are a mixture of M normals p(yt|ξ t, Y t−1) with weights p(ξ t|Y t−1):

p(yt|Y t−1) =M

m=1

ξ̂ mt|t−1ϕ

σ−1(yt − ȳmt)

where ϕ(·) is a standard normal density and ȳmt = E[yt|ξ t = ιm, Y t−1]. Thereforethe distribution of the observed time series can be multi-modal. Relying on well-

known results, cf. e.g. TITTERINGTON et al. [1985, p. 162], we can notice for

M = 2:


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Example 1 An MS(2)-AR( p) process with a homoskedastic Gaussian inno-

vation process ut ∼ NID(0, σ2) generates bimodality of the conditional density p(yt|Y t−1) if

σ−1(ȳ1t − ȳ2t) > ∆ξ̄1 ≥ 2,where the critical value ∆ξ̄1 depends on the ergodic regime probability

ξ̄ 1 , e.g.

∆0.5 = 2 and ∆0.1 = ∆0.9 = 3.

In contrast to Gaussian VAR processes, MS-VAR models can produce skewness

(non-zero third-order cross-moments) and leptokurtosis (fat tails) in the distribution

of the observed time series. A simple model that generates leptokurtosis in the

distribution of the observed time series y t is provided by the MSH(2)-AR(0) model:

Example 2 Let yt be an MSH(2)-AR(0) process,

yt − µ = ut, ut ∼ NID(0, σ2

1 I (st = 1) + σ

2

2 I (st = 2)).

Then it can be shown that the excess kurtosis is given by

E[(yt − µ)4]E[(yt − µ)2]2 − 3 =

3ξ̄ 1ξ̄ 2(σ21 − σ22)2

(ξ̄ 1σ21 + ξ̄ 2σ22)

2 .

Thus, the excess kurtosis is different from zero if σ21 = σ22 and 0 < ξ̄ 1 < 1.

BOX AND TIAO [1968] have used such a model for the detection of outliers. In

order to generate skewness and excess kurtosis it is e.g. sufficient to assume an

MSI(2)-AR(0) model:

Example 3 Let yt be generated by an MSM(2)-AR(0) process:

yt − µ = (µ1 − µ)I (st = 1) + (µ2 − µ)I (st = 2) + ut, ut ∼ NID(0, σ2),

so that

yt − µ = (µ2 − µ) + (µ1 − µ2)ξ 1t + ut.Then it can be shown that the normalized third moment of y t is given by the skewness

E[(yt − µ)3]E[(yt − µ)2]3/2 =

(µ1 − µ2)3(1 − 2ξ̄ 1)ξ̄ 1(1 − ξ̄ 1)σ2 + (µ1 − µ2)2ξ̄ 1(1 − ξ̄ 1)

3/2 .if the regime i with the highest conditional mean µi > µj is less likely than the other

regime, ξ̄ i < ξ̄ j , then the observed variable is more likely to be far above the mean

than it is to be far below the mean.


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Furthermore the normalized fourth moment of y t is given by the excess kurtosis

E[(yt − µ)4]E[(yt − µ)2]2 − 3 =

(µ1 − µ2)4ξ̄ 1(1 − ξ̄ 1)

1 − 3ξ̄ 1(1 − ξ̄ 1)

σ2 + (µ1 − µ2)2ξ̄ 1ξ̄ 2

2

.

Since we have that maxξ̄1∈[0,1]{ξ̄ 1(1−ξ̄ 1)} = 1

4 < 1

3 , the excess kurtosis is positive,i.e. the distribution of y t has more mass in the tails than a Gaussian distribution

with the same variance.

The combination of regime switching means and variances in an MSIH(2)-AR(0)

process (cf. Example 4 ) is given in SOLA AND T IMMERMANN [1995]. The implic-

ations for option pricing are discussed in K ÄHLER AND MARNET [1994b]. For an

MSMH(2)-AR(4) model, the conditional variance of the one-step prediction error is

given by SCHWERT [1989] and PAGAN AND S CHWERT [1990].

1.4.2 Regime-dependent Variances and Conditional Heteroske-

dasticity

An MS(M )-AR( p) process is called conditional heteroskedastic if the conditional

variance of the prediction error y t − E[yt|Y t−1],

Var [yt|Y t−1] = E

(yt − E[yt|Y t−1])2

is a function of the information set Y t−1 . Conditional heteroskedasticity can be

induced by regime-dependent variances, autoregressive parameters or means.

In MS-AR models with regime-invariant autoregressive parameters, conditional

heteroskedasticity implies that the conditional variance of the prediction error

yt − E[yt|Y t−1], is a function of the filtered regime vector ξ̂ t−1|t−1. In general,an MS-AR process is called regime-conditional heteroskedastic if

Var [yt|ξ t−1, Y t−1] = E

(yt − E[yt|ξ t−1, Y t−1])2

is a function of ξ t−1. Interestingly, regime-dependent variances are neither neces-

sary nor sufficient for conditional heteroskedasticity. As stated in Chapter 3, a neces-

sary and sufficient condition for conditional heteroskedasticity in MS-VAR models

with regime-invariant autoregressive parameters is the serial dependence of regimes.

On the other hand, even if the white noise process u t is homoskedastic, σ2(st) = σ2,the observed process yt can be heteroskedastic. Consider the following example:


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Example 4 Let yt be an MSI(2)-AR(0) process

yt − µ = (µ1 − µ)I (st = 1) + (µ2 − µ)I (st = 2) + ut,

with ut ∼ NID(0, σ2) and serial correlation in the regimes according to the trans-ition matrix P. Employing the ergodic regime probability ξ̄ 1 , yt can be written as

yt − µ = (µ1 − µ2)(ξ 1t − ξ̄ 1) + ut.Thus E[yt|Y t−1] = µ + (µ1 − µ2)(ξ̂ 1t|t−1 − ξ̄ 1) and

Var [yt|Y t−1] = σ2 + (µ1 − µ2)2E

(ξ 1t − ξ̄ 1)2|Y t−1

= σ2 + (µ1 − µ2)2

ξ̂ 1t|t−1(1 − ξ̂ 1t|t−1)2 + (1 − ξ̂ 1t|t−1)(−ξ̂ 1t|t−1)2

= σ2 + (µ1 − µ2)2ξ̂ 1t|t−1(1 − ξ̂ 1t|t−1),

where ξ̂ 1t|t−1 = p11ξ̂ 1t−1|t−1 + p21(1 − ξ̂ 1t−1|t−1) = ( p11 + p22 − 1)ξ̂ 1t−1|t−1 +(1 − p22) is the predicted regime probability Pr(st = 1|Y t−1). Thus {yt} is aregime-conditional heteroskedastic process.

In contrast to ARCH models, the conditional variance in MS-VAR models (with

time-invariant autoregressive parameters) is a non-linear function of past squared

errors since the predicted regime probabilities generally are non-linear functions of

Y t−1.

Recently some approaches have been made to consider Markovian regime shifts in

variance generating processes. The class of autoregressive conditional heteroske-

dastic processes introduced by ENGLE [1982] is used to formulate the conditional

process; our assumption of an i.i.d. distributed error term is substituted by an ARCH

process ut, cf. inter alia HAMILTON AND LIN [1994], HAMILTON AND SUSMEL

[1994], CAI [1994] and HALL AND S OL A [1993b]. ARCH effects can be generated

by MSA-AR processes which will be considered in the next section.

1.4.3 Regime-dependent Autoregressive Parameters: ARCH

and Stochastic Unit Roots

Autoregressive conditional heteroskedasticity is known from random coefficient

models. Therefore it is not very surprising that also MSA-VAR models may lead to


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ARCH. This effect will be considered in the following simple example.

Example 5 Let yt be generated by an MSA(2)-MAR(1) process with i.i.d. regimes:

(yt

−µ) = α(st) (yt−1

−µ) + ut, ut

∼NID(0, σ2).

Serial independence of the regimes implies p11 = 1− p22 = ρ; the regime-dependent autoregressive parameters α1, α2 are restricted such that E[α] = α1ρ+α2(1−ρ) =0. Thus it can be shown that

E[yt|Y t−1] = µ + (α1ρ + α2(1 − ρ)) yt−1 = µ,E[(yt − µ)2|Y t−1] = σ2 + (α21ρ + α22(1 − ρ)) (yt−1 − µ)2.

Then yt possesses an ARCH representation yt = µ + et with

e2t = σ2 + γe2t−1 + εt

where γ = −α1α2 > 0 and εt is white noise. Thus, ARCH(1) models can beinterpreted as restricted MSA(2)-AR(1) models.

The theoretical foundations of MSA-VAR processes are laid in TJØSTHEIM

[1986b]. Some independent theoretical results are provided by BRANDT [1986].

As pointed out by TJØSTHEIM [1986b], the dynamic properties of models with

regime-dependent autoregressive parameters are quite complicated. Especially, if

the process is stationary for some regimes and mildly explosive for others, the prob-lems of stochastic unit root processes as introduced by GRANGER AND S WANSON

[1994] are involved.10

It is worth noting that the stability of each VAR sub-model and the ergodicity of

the Markov chain are sufficient stability conditions; they are however not neces-

sary to establish stability. Thus, the stability of MSA-AR models can be compatible

with AR polynomials containing in some regimes roots greater than unity in ab-

solute value and less than unity in others. Necessary and sufficient conditions for

the stability of stochastic processes as the MSA-VAR model have been derived in

10Models where the regime is switching between deterministic and stochastic trends are considered by

MCCULLOCH AND T SAY [1994a].


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KARLSEN [1990a], [1990b]. However in practice, their application has been found

to be rather complicated (cf. HOLST et al. [1994]).

In this study we will concentrate our analysis on modelling shifts in the (conditional)

mean and the variance of VAR processes which simplifies the analysis.

1.5 Conclusion and Outlook

In the preceding discussion of this chapter MS(M )-VAR( p) processes have been

introduced as doubly stochastic processes where the conditional stochastic process

is a Gaussian VAR( p) and the regime generating process is a Markov chain. As we

have seen in the discussion of the relationship of the MS-VAR model to other non-

linear models, the MS-VAR model can encompass many other time series models

proposed in the literature or replicates at least some of their features. In the fol-lowing chapter these considerations are formalized to state-space representations of

MS-VAR models where the measurement equation corresponds to the conditional

stochastic process and the transition equation reflects the regime generating process.

In Section 2.5 the MS-VAR model will be compared to time-varying coefficient

models with smooth variations in the parameters, i.e. an infinite number of regimes.


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1.A. Appendix: A Note on the Relation of SETAR to MS-AR Processes 27

1.A Appendix: A Note on the Relation of SETAR to

MS-AR Processes

While the presumptions of the SETAR and the MS-AR model seem to be quitedifferent, the relation between both model alternatives is rather close. Indeed, both

models can be observationally equivalent, as the following example demonstrates:

Example 6 Consider the SETAR model

yt = µ2 + (µ1 − µ2)I (yt−d ≤ r) + ut, ut ∼ NID(0, σ2). (1.19)

For d = 1 it has been shown by CARRASCO [1994, lemma 2.2] that (1.19) is a

particular case of the Markov-switching model

yt = µ2 + (µ1 − µ2)I (st = 1) + ut, ut ∼ NID(0, σ2),which is an MSI(2)-AR(0) model. For an unknown r , define the unobserved regime

variable st as the binary variable

st = I (yt−1 ≤ r) =

1 if yt−1 ≤ r2 if yt−1 > r

such that

Pr(st = 1

|st−1, Y ) = Pr(yt−1

≤r

|st−1, Y )

= Pr(µ2 + (µ1 − µ2)I (st−1 = 1) + ut−1 ≤ r)= Pr(ut−1 ≤ r − µ2 − (µ1 − µ2)I (st−1 = 1))

= Φ

r − µ2 − (µ1 − µ2)I (st−1 = 1)

σ

= Pr(st = 1|st−1).

Hence st follows a first order Markov process where the transition matrix is defined

as

P = p11 p12 p21 p22 = Φ(r−µ1

σ ) Φ(µ1−r

σ )

Φ( r−µ2σ ) Φ( µ2−rσ ) .


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if d > 1, the data can be considered as generated by d independent series which

are each particular Markov processes. A proof can be based on the property

Pr(st|{st−j}∞j=1, Y T ) = Pr(st|st−2, Y T ); thus st follows a second order Markovchain, which can be reparametrized as a higher dimensional first order Markov

chain.


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Chapter 2

The State-Space Representation

In the following chapters we will be concerned with the statistical analysis of

MS(M )-VAR( p) models. As a formal framework for these investigations we em-

ploy the state-space model which has been proven useful for the study of time series

with unobservable states. In order to motivate the introduction of state-space rep-resentations for MS(M )-VAR( p) models it might be helpful to sketch its use for the

three main tasks of statistical inference:

1. Filtering & smoothing of regime probabilities: Given the conditional dens-

ity function p(yt|Y t−1, ξ t), the discrete Markovian chain as regime generatingprocess ξ t, and some assumptions about the initial state y0 = (y

0, . . . , y

1− p)

of the observed variables and the unobservable initial state ξ 0 of the Markov

chain, the complete density function p(ξ, Y ) is specified. The statistical tools

to provide inference for ξ t given a specified observation set Y τ , τ ≤ T are thefilter and smoother recursions which reconstruct the time path of the regime,{ξ t}T t=1, under alternative information sets:

ξ̂ t|τ , τ < t predicted regime probabilities.

ξ̂ t|τ , τ = t filtered regime probabilities,

ξ̂ t|τ , t < τ ≤ T smoothed regime probabilities.

In the following, mainly the filtered regime probabilities, ξ̂ t|t and full-sample

smoothed regime probabilities, ξ̂ t|T , are considered. See Chapter 5 .

2. Parameter estimation & testing: If the parameters of the model are un-

known, classical Maximum Likelihood as well as Bayesian estimation meth-

ods are feasible. Here, the filter and smoother recursions provide the analyt-

29


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30 The State-Space Representation

ical tool to construct and evaluate the likelihood function. See Chapters 6 –

9 .

3. Forecasting: Given the state-space form, prediction of the system is a

straightforward task. See Chapter 4 and Section 8.5 .

The framework for the statistical analysis of MS-VAR models to be presented in the

next chapters is the state-space form. The advantage of viewing MS-VAR models

in this way is that general concepts can be introduced as the likelihood principle

(Chapter 6 ) and a recursive filter algorithm (Chapter 5 ) which corresponds to the

Kalman filter in Gaussian state-space models.

For particular MS-VAR processes, a state-space representation with ξ t as the state

vector has been introduced by H AMILTON [1994a]. 1 In the following section we

investigate some state-space representations of MS-VAR models. These representa-

tions are then used to work out general properties of MS-VAR processes, inter aliawe discuss the non-normality of the state-space form, we formulate conditions for

the linearity of the state-space representation and we show that the joint process of

observed variables and regimes, ( y t, ξ t), is Markovian. In Section 2.2 the specific-

ation of the state-space representation is discussed with regardto its adaptation to the

particular MS-VAR models proposed in Chapter 1. In the remaining sections, three

alternative state-space representations of MS-VAR processes are introduced which

will create new insights into the theory of MS-VAR processes and will be used later

on. In Section 2.3 the adding-up restriction on the state vector is eliminated by re-

ducing its dimension. Section 2.4 formulates the state-space representation in the

predicted state vector. Section 2.4 presents a state-space form in the vector of VARcoefficients which allows a comparison with other time-varying coefficient models.

2.1 A Dynamic Linear State-Space Representation

for MS-VAR Processes

The state-space model given in Table 2.1 consists of the set of measurement

and transition equations. The measurement equation (2.1) describes the relation

1HAMILTON [1994a] considers MSIA(M )-AR( p) and MSM(M )-AR( p) models. A similar approach

is taken in HALL AND S OL A [1993a), HALL AND S OL A [1993b] and FUNKE et al. [1994].


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2.1. A Dynamic Linear State-Space Representation for MS-V

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