Multivariate Modelling of Non-Stationary Economic Time Series (eBook)

John Hunter, Simon P. Burke, Alessandra Canepa (Autoren)

eBook Download: PDF

2017 | 2nd ed. 2017
XIII, 502 Seiten
Palgrave Macmillan UK (Verlag)
978-1-137-31303-4 (ISBN)

Infos

This book examines conventional time series in the context of stationary data prior to a discussion of cointegration, with a focus on multivariate models. The authors provide a detailed and extensive study of impulse responses and forecasting in the stationary and non-stationary context, considering small sample correction, volatility and the impact of different orders of integration. Models with expectations are considered along with alternate methods such as Singular Spectrum Analysis (SSA), the Kalman Filter and Structural Time Series, all in relation to cointegration. Using single equations methods to develop topics, and as examples of the notion of cointegration, Burke, Hunter, and Canepa provide direction and guidance to the now vast literature facing students and graduate economists.

Simon P. Burke studied econometrics at the University of Reading, UK. He has published in the International Journal of Forecasting, Journal of Financial Econometrics and The Oxford Bulletin of Economics & Statistics. He has taught econometrics, mathematics and statistics at Reading and Surrey Universities.

John Hunter studied econometrics at the London School of Economics, UK, under Denis Sargan. He published recently in the International Review of Financial Analysis, Economic Modelling and developed the notion of Cointegrating Exogeneity. He taught econometrics and financial modelling at Brunel, City, Queen Mary, Southampton and Surrey. He has consulted for HM Treasury, Oftel, OFT and KPN Mobile.

Alessandra Canepa studied econometrics at Southampton University, UK. She has published in Statistics & Probability Letters, the European Journal of Operational Research and Oxford Economic Papers. She currently lectures in econometrics and Risk Management at Brunel University, UK, and is a member of CARISMA in the Department of Mathematics at Brunel.

 

Inhalt

Preface 6
Contents 9
1 Introduction 14
References 29
2 Multivariate Time Series 33
2.1 Introduction 33
2.2 Stationarity 34
2.2.1 Strict Stationarity 35
2.2.2 Strict (Joint Distribution) Stationarity 36
2.2.3 Describing Covariance Non-Stationarity: Parametric Models 36
2.2.4 The White Noise Process 37
2.2.4.1 White Noise 37
2.2.5 The Moving Average Process 38
2.2.6 Wold's Representation Theorem 40
2.2.7 The Autoregressive Process 40
2.2.8 Lag Polynomials and Their Roots 41
2.2.8.1 The Lag Operator and Lag Polynomials 41
2.2.9 Non-Stationarity and the Autoregressive Process 43
2.2.9.1 Stationarity of an Autoregressive Process 43
2.2.10 The Random Walk and the Unit Root 43
2.2.10.1 The Random Walk Process 43
2.2.10.2 Differencing and Stationarity 44
2.2.10.3 The Random Walk as a Stochastic Trend 45
2.2.10.4 The Random Walk with Drift 47
2.2.11 The Autoregressive Moving Average Process and Operator Inversion 47
2.2.11.1 Illustration of Operator Inversion 49
2.2.12 Testing Stationarity in Single Series 50
2.2.12.1 Reparameterizing the Autoregressive Model 50
2.2.12.2 Semi-parametric Methods 52
2.3 Multivariate Time Series Models 54
2.3.1 The VAR and VECM Models 54
2.3.2 The VMA Model 56
2.3.3 Estimation 57
2.3.4 The Procedure 59
2.4 Persistence 61
2.4.1 Reparameterizing the VAR 62
2.4.2 Long-Run Growth Models 62
2.5 Impulse Responses 66
2.5.1 Impulse Responses and VAR Models 67
2.5.2 Orthogonality and the IRF 71
2.5.3 The Choleski Decomposition 72
2.5.4 IRFs in the General VAR Case 74
2.5.4.1 IRFs and Time Series Identification 76
2.6 Variance Decomposition 77
2.6.1 Prediction Errors and Forecasts 79
2.7 Conclusion 82
References 84
3 Cointegration 88
3.1 Cointegration of the VMA, VAR and VECM 90
3.1.1 The Granger Representation Theorem: Systems Representation of Cointegrated Variables 91
3.1.1.1 Cointegration Starting from a VMA and Deriving VAR and VECM Forms 91
3.1.2 VARMA Representation of CI(1,1) Variables 94
3.2 The Smith-McMillan-Yoo Form 97
3.2.1 Using the Smith Form to Reparameterize a Finite Order VMA 99
3.2.1.1 Reparameterizing a VMA in Differences 101
3.2.2 The SM Form in General Applied to a Rational VMA: The SMY Form 103
3.2.2.1 The SMY Form and Cointegration of Order (1,1) 107
3.2.3 Cointegrating Vectors in the VMA and VAR Representations of CI(1,1) 111
3.2.3.1 A(L) as Partial Inverse of C(L) in the CI(1,1) Case 113
3.2.4 Equivalence of VAR and VMA Representations in the CI(1,1) Case 114
3.3 Johansen's VAR Representation of Cointegration 115
3.3.1 Cointegration Assuming Integration of Order 1 116
3.3.1.1 Cointegrated VARs with I(1) Processes 117
3.3.2 Conditions for the VAR Process to be I(1) and Cointegrated 117
3.3.2.1 Discussion 124
3.3.3 The MA Representation 124
3.4 Cointegration with Intercept and Trend 127
3.4.1 Levels Process for the VECM with Intercept 128
3.4.2 Levels Process for the VECM with Higher Order Trends and Other Deterministic Terms 130
3.5 Alternative Representations of the Cointegrating VAR, VMA and VARMA 132
3.5.1 The Sargan-Bézout Factorization 133
3.5.2 A VAR(1) Representation of a VMA(1) Model Under Cointegration 139
3.6 Single Equation Implications and Examples 142
3.6.1 Cointegration: Static Equilibrium with I(1) Variables 143
3.6.2 ADL Models, Cointegration and Equilibrium 146
3.6.2.1 ADL Models, Cointegration and Equilibrium 147
3.6.2.2 Example 148
3.7 Conclusion 153
References 154
4 Testing for Cointegration: Standard and Non-Standard Conditions 156
4.1 Introduction 156
4.2 Maximum Likelihood Estimation 158
4.3 Johansen's Approach to Testing for Cointegration in Systems 158
4.3.1 Testing for Reduced Rank and Estimating Cointegrating Vectors 159
4.3.1.1 Review of Source of Reduced Rank in Cointegrated Systems 159
4.3.1.2 Using Eigenvalues and Eigenvectors in Cointegration Analysis 159
4.3.2 The Removal of Nuisance Parameters 160
4.3.3 Estimating Potentially Cointegrating Relations 161
4.3.4 Testing Cointegrating Rank 164
4.4 Performing Tests of Cointegrating Rank in the Presence of Deterministic Components 170
4.4.1 Intercepts and Trends and the Preliminary Regressions to Remove Nuisance Parameters 171
4.5 Examples of Tests of Cointegration in VAR Models 173
4.5.1 Special Cases of the Johansen Test 176
4.5.2 Empirical Examples of the Johansen Test 177
4.6 The VMA and VARMA Form 186
4.6.1 The Removal of Nuisance Parameters 187
4.6.2 The Impact of the VMA Structure on the Tests of Cointegration 190
4.6.3 A Simple Multi-Cointegration Extension 195
4.7 Quasi-Maximum Likelihood Estimator (QMLE) and Non-Gaussianity 200
4.7.1 Further Evidence on the Performance of the Johansen Test 200
4.7.2 Breaks in Structure 203
4.7.3 Outliers in the Mean Equation and the Johansen Trace Test 208
4.8 Conclusion 209
References 210
5 Structure and Evaluation 216
5.1 An Introduction to Exogeneity 217
5.1.1 Conditional Models and Testing for Cointegration and Exogeneity 218
5.1.2 Cointegration and Exogeneity 220
5.1.3 Tests of Long-Run Exogeneity 223
5.2 Identification 228
5.2.1 I(0) Systems and Some Preliminaries 230
5.2.1.1 The Cointegration Case 235
5.2.2 A Simple Indirect Procedure for Generic Identification 237
5.2.3 Johansen Identification Conditions 238
5.2.4 Boswijk Conditions and Observational Equivalence 243
5.2.5 Hunter's Conditions for Identification 244
5.2.6 An Example of Empirical and Generic Identification 248
5.3 Exogeneity and Identification 251
5.3.1 Empirical Examples 255
5.4 Impulse Response Functions 258
5.4.1 The Cointegration Case 258
5.4.2 IRF of a Bivariate VAR(1) 259
5.4.3 Lütkepohl's Method 261
5.5 Forecasting in Cointegrated Systems 266
5.5.1 VMA Analysis 266
5.5.2 Forecasting from the VAR 271
5.5.3 The Mechanics of Forecasting from a VECM 273
5.5.4 Forecast Performance 275
5.5.4.1 Lin and Tsay 277
5.5.4.2 Forecast Evaluation 282
5.5.4.3 Other Issues Relevant to Forecasting Performance in Practice 283
5.6 Conclusion 285
References 287
6 Testing in VECMs with Small Samples 291
6.1 Introduction 291
6.2 Testing for Cointegrating Rank in Finite Samples 292
6.2.1 Bartlett Correction Factor for the Trace Test 294
6.2.2 The Bootstrap p-Value Test 296
6.3 Testing Linear Restrictions on ? 298
6.3.1 A Monte Carlo Experiment 302
6.3.1.1 Some Simulation Results 304
6.3.1.2 The Probability of a Type II Error 308
6.4 An Empirical Application 310
6.5 Conclusion 312
References 313
7 Heteroscedasticity and Multivariate Volatility 315
7.1 Introduction 315
7.2 VAR Models for Multivariate Heteroscedasticity 317
7.2.1 The MGARCH-VECM in Systems Form 317
7.2.1.1 The Model 317
7.2.1.2 The Disturbance Variance-Covariance Matrix 319
7.2.2 The VAR-GARCH FIML (Full Information Maximum Likelihood) Approach 321
7.2.3 The Optimization Problem 324
7.2.4 An Example Estimating the Variance by BEKK 324
7.2.4.1 Cointegration Testing and the Mean Specification 325
7.2.5 BEKK Estimation of the Variance Equation 326
7.3 Estimation of the Transformed Mean Equation 329
7.3.1 The Stacked GLS Problem 329
7.4 The FWL Simplification to the Vectorized System 332
7.4.1 Purging the Data Equation by Equation 332
7.4.1.1 Adding Lagged Differences 333
7.5 Testing for Cointegration Using the GLS Transformed Data 336
7.6 Dynamic Heteroscedasticity and Market Imperfection 340
7.7 Conclusion 345
References 346
8 Models with Alternative Orders of Integration 349
8.1 Introduction 349
8.2 Cointegration Mixing I(0) and I(1) Series 350
8.2.1 Mixing I(0) and I(1) Variables 350
8.3 Some Examples 353
8.4 Inference and Estimation When Series Are Not I(1) 357
8.4.1 Relations Between I(1) and I(2) Variables 358
8.4.2 Cointegration When Series Are I(2) 359
8.4.2.1 The Johansen Procedure for Testing Cointegrating Rank with I(2) Variables 361
8.4.2.2 An Example of I(2) 367
8.5 Modified Estimators and Fractional Cointegration 375
8.5.1 Fractional Integration 375
8.5.2 Fractional Cointegration 376
8.5.3 Cointegration Testing and Selection of the Difference Order 380
8.6 Conclusion 388
References 390
9 The Structural Analysis of Time Series 393
9.1 Introduction 393
9.2 Cointegration and Models of Expectations 394
9.2.1 Linear Quadratic Adjustment Cost Models 396
9.2.2 Cointegration Solutions to Forward Behaviour with n2 Weakly Exogenous Variables 400
9.2.3 Estimation and Inference 403
9.2.4 The Effect of Cointegration on Solutions to Rational Expectations Models 405
9.2.4.1 Cointegration, Exogeneity and the VARMA 406
9.2.4.2 Rational Expectations and Smith-McMillan Forms 409
9.3 Singular Spectral Analysis 416
9.3.1 The Relation Between SSA and TSA 417
9.3.2 The Singular Spectral Analysis 418
9.3.3 Multivariate Singular Spectral Analysis 420
9.3.4 Difference Stationarity, Cointegration and Economic Time Series 421
9.3.5 Forecasting, Missing Data and Structural Change 424
9.4 Structural Time Series Models 425
9.4.1 State Space Form 425
9.4.2 Structural Time Series 427
9.4.3 The Multivariate Case 428
9.4.4 Stochastic Trends and Cointegration 428
9.4.5 Further Developments 432
9.5 Further Methods 432
9.5.1 Factor Models 432
9.5.2 Non-Linear Error Correction Model 436
9.5.3 Wavelets 439
9.6 Conclusion 440
References 442
Appendix A Matrix Preliminaries 450
A.1 Elementary Row Operations and Elementary Matrices 450
A.2 Unimodular Matrices 452
Appendix B Matrix Algebra for B:Engle and Granger1987 Representation 454
B.1 Determinant/Adjoint Representation of a Polynomial Matrix 454
B.2 Expansions of the Determinant and Adjoint About z[ 0,1 ] 455
B.3 Drawing Out a Factor of z from a Reduced Rank Matrix Polynomial 456
Application to Lag Polynomial to Draw Out Unit Root Factor 456
Appendix C Johansen's Procedure as a Maximum Likelihood Procedure 458
Appendix D The Maximum Likelihood Procedure in Terms of Canonical Correlations 466
Appendix E Distribution Theory 469
E.1 Some Univariate Theory 469
E.2 Vector Processes and Cointegration 472
E.3 Testing the Null Hypothesis of Non-Cointegration 473
E.4 Testing a Null Hypothesis of Non-Zero Rank 475
E.5 Distribution Theory When There Are Deterministic Trends in the Data 482
E.5.1 Tables of Approximate Asymptotic and Finite Sample Distributions 483
E.6 Other Issues 485
E.6.1 The Maximal Eigenvalue Statistic 485
E.6.2 Sequential Testing and Model Selection 486
E.6.3 Partial Systems 486
Appendix F Estimation Under General Restrictions 487
Appendix G Proof of Identification Based on an Indirect Solution 490
Appendix H Generic Identification of Long-Run Parameters in Sect.5.3 493
Appendix I IRF MA Parameters for the Case in Sect.5.4.3 495
References 497
Bibliography 499
Index 501

Produkt-Details

• Erscheint lt. Verlag: 8.5.2017
• Reihe/Serie: Palgrave Texts in Econometrics
• Reihe/Serie: Palgrave Texts in Econometrics
• Zusatzinfo: XIII, 502 p.
• Verlagsort: London
• Sprache: englisch
• Themenwelt: Wirtschaft ► Allgemeines / Lexika
• Themenwelt: Wirtschaft ► Volkswirtschaftslehre ► Ökonometrie
• Schlagworte: Cointegration • conventional time series • Econometrics • Economics • impulse responses • Integration • Kalman Filter • Modelling • Singular Spectrum Analysis (SSA) • small sample correction • stationary data • Structural Time Series • volatililty
• ISBN-10: 1-137-31303-X / 113731303X
• ISBN-13: 978-1-137-31303-4 / 9781137313034