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APPLIED TIME SERIES

E C O N O M E T R I C S

Edited by

HELMUT LÜTKEPOHL

European University Institute Florence

MARKUS KRÄTZIG

Humboldt University erlin

C A M B R I D G E

U N I V E R S I T Y P R E S S

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Contents

Preface page  xv

Notation and Abbreviations  xix

List of Con tributors  xxv

Initial Tasks and Overview

Helmut Lütkepohl

1.1 Introduction 1

1.2 Setting Up an Econom etric Project 2

1.3 Getting Data 3

1.4 Data Handling 5

1.5 Outline of Chapters 5

2 Univariate Time Series Analysis 8

Helmut Lütkepohl

2.1 Characteristics of

  ime

 Series 8

2.2 Stationary and Integrated Stochastic Processes 11

2.2.1 Stationarity 11

2.2.2 Sample Autocorrelations, Partial Autocorrelations,

and Spectral Densities 12

2.2.3 Data Transformations and Filters 17

2.3 Some Popular Time Series Models 22

2.3.1 Autoregressive Processes 22

2.3.2 Finite-Order Moving Average Processes 25

2.3.3 AR1MA Processes 27

2.3.4 Autoregressive Conditional Heteroskedasticity 28

2.3.5 Deterministic Terms 30

2.4 Parameter Estimation 30

2.4.1 Estimation of AR M odels 30

2.4.2 Estimation of ARM A Models 32

2.5 Model Specification 33

X

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X

Contents

2.5.1 AR Order Specification Criteria 33

2.5.2 Specifying More General Models 35

2.6 Model Checking 40

2.6.1 Descriptive Analysis of

 the

  Residuals 40

2.6.2 Diagnostic Tests of the Residuals 43

2.6.3 Stability Analysis 47

2.7 Unit Root Tests 53

2.7.1 Augm ented Dickey-Fuller ADF) Tests 54

2.7.2 Schm idt-Phillips Tests 57

2.7.3 A Test for Processes with Level Shift 58

2.7.4 KP SST est 63

2.7.5 Testing for Seasonal Unit Roots 65

2.8 Forecasting Univariate Time Series 70

2.9 Exam ples 73

2.9.1 German Consum ption 73

2.9.2 Polish Productivity 78

2.10 Where to Go from Here 85

3 Vector Autoregressive and Vector Error Correction M odels  86

Helmut Liitkepohl

3.1 Introduction 86

3.2 VA RsandV EC M s 88

3.2.1 The Models 88

3.2.2 Determ inistic Terms 91

3.2.3 Exogenous Variables 92

3.3 Estimation 93

3.3.1 Estimation of

 an

 Unrestricted VAR 93

3.3.2 Estimation of VECM s 96

3.3.3 Restricting the Error Correction Term 105

3.3.4 Estimation of Models with More General Restrictions

and Structural Forms 108

3.4 Model Specification 110

3.4.1 Determining the Autoregressive Order 110

3.4.2 Specifying the Co integra ting Rank 112

3.4.3 Cho ice of Determ inistic Term 120

3.4.4 Testing Restrictions Related to the Cointeg ration

Vectors and the Loading Matrix 121

3.4.5 Testing Restrictions for the Short-Run Parameters

and Fitting Subset Models 122

3.5 Model Checking 125

3.5.1 Descriptive Analysis of the Residuals 125

3.5.2 Diagnostic Tests 127

3.5.3 Stability Analysis 131

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Contents

XI

3.6 Forecasting VAR Processes and VE CM s 140

3.6.1 Known Processes 141

3.6.2 Estimated Processes 143

3.7 Granger-Causality Analysis 144

3.7.1 The Concept 144

3.7.2 Testing for Granger-Causality 148

3.8 An Example 150

3.9 Extensions 158

4 Structural Vector Autoregressive Modeling and Impulse

Responses

  159

Jörg

 Breitung

Ralf Brüggemann and Helmut Lütkepohl

4.1 Introduction 159

4.2 The Models 161

4.3 Impulse Response Analysis 165

4.3.1 Stationary VAR Processes 165

4.3.2 Impulse Response Analysis of Nonstationary VARs

and VEC M s 167

4.4 Estimation of Structural Parameters 172

4.4.1 SVAR Models 172

4.4.2 Structural VECM s 175

4.5 Statistical Inference for Impulse Responses 176

4.5.1 Asym ptotic Estimation Theory 176

4.5.2 Bootstrapping Impulse Responses 177

4.5.3 An Illustration 179

4.6 Forecast Error Variance Decomposition 180

4.7 Exam ples 183

4.7.1 A Simple AB-Model 183

4.7.2 The Blanch ard-Qu ah Model 185

4.7.3 An SVECM for Canadian Labor Market Data 188

4.8 Conclusions 195

5 Conditional Heteroskedasticity

  197

Helmut Herwartz

5.1 Stylized Facts of Em pirical Price Processes 197

5.2 Univariate GARCH Models 198

5.2.1 Basic Features of GARCH Processes 199

5.2.2 Estimation of GARCH Processes 201

5.2.3 Extensions 203

5.2.4 Blockdiagonality of

 the

  Information Matrix 206

5.2.5 Specification Testing 207

5.2.6 An Empirical Illustration with Exchange Rates 207

5.3 M ultivariate GARCH Models 212

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xii

  Contents

5.3.1 Alternative Model Specifications 214

5.3.2 Estimation of M ultivariate GA RCH M odels 217

5.3.3 Extensions 218

5.3.4 Continuing the Em pirical Illustration 220

6 Smooth Transition Regression Modeling

  222

Timo Teräsvirta

6.1 Introduction 222

6.2 The Model 222

6.3 The Modeling Cycle 225

6.3.1 Specification 225

6.3.2 Estimation of Param eters 228

6.3.3 Evaluation 229

6.4 Two Em pirical Exam ples 234

6.4.1 Chem ical Data 234

6.4.2 Demand for Money M l) in Germany 238

6.5 Final Rem arks 242

7 Nonparametric Time Series M odeling

  243

Rolf Tschernig

7.1 Introduction 243

7.2 Local Linear Estimation 245

7.2.1 The Estimators 245

7.2.2 Asymptotic Properties 248

7.2.3 Confidence Intervals 250

7.2.4 Plotting the Estimated Function 251

7.2.5 Forecasting 254

7.3 Bandwidth and Lag Selection 254

7.3.1 Bandw idth Estimation 256

7.3.2 Lag Selection 258

7.3.3 Illustration 261

7.4 Diagnostics 262

7.5 M odeling the Conditional Volatility 263

7.5.1 Estimation 264

7.5.2 Bandwidth Choice 265

7.5.3 Lag Selection 266

7.5.4 ARCH Errors 267

7.6 Local Linear Seasonal M odeling 268

7.6.1 The Seasonal Nonlinear Autoregressive Model 269

7.6.2 The Seasonal Dummy Nonlinear Autoregressive

Model 270

7.6.3 Seasonal Shift Nonlinear Autoregressive Model 271

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  ontents

х ш

7 7 Example I: Average Weekly W orking Hours in the United

States 272

7 8 Example II: XETRA Dax Index 280

The Software

  JMu lTi 289

rkus Krätzig

8.1 Introduction to JM ulT i 289

8.1.1 Software Concept 289

8.1.2 Operating JM ulT i 290

8.2 Numbers, Dates, and Variables in JM ulT i 290

8.2.1 Numb ers 290

8.2.2 Numbers in Tables 291

8.2.3 Dates 291

8.2.4 Variable Names 292

8.3 Handling Data Sets 292

8.3.1 Importing Data 292

8.3.2 Excel Format 292

8.3.3 ASCII Format 293

8.3.4 JM ulT i .dat Format 293

8.4 Selecting, Transforming, and Creating Time Series 293

8.4.1 Time Series Selector 293

8.4.2 Time Series Calculator 295

8.5 M anaging Variables in JM ulTi 296

8.6 Notes for Econom etric Software Developers 296

8.6.1 General Remark 296

8.6.2 T h e J S t a t C o m Framework 297

8.6.3 Com ponent Structure 297

8.7 Conclusion 299

References

Index

3 1

317