ISBN: 3540262383
TITLE: New Introduction
AUTHOR: Luetkepohl
TOC:

1 Introduction 1
1.1 Objectives of Analyzing Multiple Time Series 1
1.2 Some Basics 2
1.3 Vector Autoregressive Processes 4
1.4 Outline of the Following Chapters 5
Part I Finite Order Vector Autoregressive Processes
2 Stable Vector Autoregressive Processes 13
2.1 Basic Assumptions and Properties of VAR. Processes 13
2.1.1 Stable VAR(p) Processes 13
2.l.2 The Moving Average Representation of a VAR Process 18
2.1.3 Stationary Processes 24
2.1.4 Computation of Autocovariances and Autocorrelations of Stable VAR Processes 26
2.2 Forecasting 31
2.2.1 The Loss Function 32
2.2.2 Point Forecasts 33
2.2.3 Interval Forecasts and Forecast Regions 39
2.3 Structural Analysis with VAR Models 41
2.3.1 Granger-Causality, Instantaneous Causality, and Multi-Step Causality 41
2.3.2 Impulse Response Analysis 51
2.3.3 Forecast Error Variance Decomposition 63
2.3.4 Remarks an the Interpretation of VAR l\Models 66
2.4 Exercises 66
3 Estimation of Vector Autoregressive Processes 69
3.1 Introduction G9
3.2 Multivariate Least: Squares Estimation 69
3.2.1 The Estimator 70
3.2.2 Asymptotic Properties of the Least Squares Estimator 73
3.2.3 An Example 77
3.2.4 Small Sample Properties of the LS Estimator 80
3.3 Least Squares Estimation with Mean-Adjusted Data and Yule-Walker Estimation 82
3.3.1 Estimation when the Process Mean Is Known 82
3.3.2 Estimation of the Process Mean 83
3.3.3 Estimation with Unknown Process \Man 85
3.3.4 The Yule-Walker Estimator 85
3.3.5 An Example 87
3.4 Maximum Likelihood Estimation 87
3.4.1 The Likelihood Function 87
3.4.2 The NIL Estimators 89
3.4.3 Properties of the EIL Estimators 90
3.5 Forecasting will Estimated Processes 94
3.5.1 General Assumptions and Results 94
3.5.2 The Approximate MSE Matrix 96
3.5.3 Air Example 98
3.5.4 A Small Sample Investigation 100
3.6 Testing for Causality 102
3.6.1 A Wald Test for Granger-Causality 102
3.6.2 An Example 103
3.6.3 Testing for Instantaneous Causality 104
3.6.4 Testing for Multi-Step Causality 106
3.7 The Asymptotic Distributions of Impulse Responses and Forecast Error Variance Decompositions 109
3.7.1 The Main Results 109
3.7.2 Proof of Proposition
3.6 116
3.7.3 An Example 118
3.7.4 Investigating the, Distributions of the Impulse Responses by Simulation Techniques 126
3.8 Exercises 130
3.8.1 Algebraic Problems 130
3.8.2 Numerical Problems 132
4 VAR Order Selection and Checking the Model Adequacy 13,5
4.1 Introduction 1'.35
4.2 A Sequence of Tests for Determining the VAR Order 136
4.2.1 The Impact. of the Fitted V.UR Order an the Forecast NISE 136
4.2.2 The Likelihood Ratio Test Statistic 138
4.2.3 A Testing Scheme for VAR Order Determination 143
4.2.4 An Example 145
4.3 Criteria for WIR Order Selection 146
4.3.1 Minimizing the Forecast NISE 146
4.3.2 Consistent Order Selection 148
4.3.3 Comparison of Order Selection Criteria 151
4.3.4 Some Small Sample Simulation Results 153
4.4 Checking the Whiteness of the Residuals 157
4.4.1 The Asymptotic Distributions of the Autocovariances and Autocorrelations of a White Noise Process 157
4.4.2 The Asymptotic Distributions of the Residual Autocovariances and Autocorrelations of an Estimated VAR. Process 161
4.4.3 Portinanteau Tests 169
4.4.4 Lagrange Multiplier Tests 171
4.5 Testing for Nonnormality 174
4.5.1 Tests for Nonnormality of a Vector White Noise Process 174
4.5.2 Tests for Nonnormnality of a VAR. Process 177
4.6 Tests for Structural Change 181
4.6.1 Chow '-Tests 182
4.6.2 Forecast Tests for Structural Change(184
4.7 Exercises 189
4.7.1 Algebraic Problems 189
4.7.2 Numerical Problems 191
5 VAR Processes with Parameter Constraints 193
5.1 Introduction 193
5.2 Linear Constraints 194
5.2.1 The Model and the Constraints 194
5.2.2 LS, GLS, an(] EGLS Estimation 195
5.2.3 Maximum Likelihood Estimation 200
5.2.4 Constraints for Individual Equations 201
5.2.5 Restrictions for the White Noise Covariance Matrix 202
5.2.6 Forecasting 204
5.2.7 Impulse Response Analysis and Forecast Error Variance Decomposition 205
5.2.8 Specification of Subset VAR Models 206
5.2.9 Model Checking 212
5.2.10 An Example 217
5.3 VAR Processes with Nonlinear Parameter Restrictions 221
5.4 Bayesian Estimation 222
5.4.1 Basic Terms and Notation 222
5.4.2 Normal Priors for the Parameters of a Gaussian VAR. Process 223
5.1.3 The Minnesota or Litterman Prior 225
5.4.4 Practical Considerations 227
5.4.5 An Example 227
5.4.6 Classical versus Bayesian Interpretation of x in Forecasting and Structural Analysis 228 Exercises 230
5.5.1 Algebraic Exercises 230
5.5.2 Numerical Problems 231
Part II Cointegrated Processes
6 Vector Error Correction Models 237
6.1 Integrated Processes 238
6.2 VAR Processes with Integrated Variables 243
6.3 Cointegrated Processes, Common Stochastic Trends, and Vector Error Correction Models 244
6.4 Deterministic Terms in Cointegrated Processes 256
6.5 Forecasting Integrated and Cointegrated Variables 258
6.6 Causality Analysis 261
6.7 Impulse Response Analysis 262
6.8 Exercises 265
7 Estimation of Vector Error Correction Models 269
7.1 Estimation of a Simple Special Case VECNI 269
7.2 Estimation of General VECMs 286
7.2.1 LS Estimation 287
7.2.2 EGLS Estimation of the Cointegration Parameters 291
7.2.3 NIL Estimation 294
7.2.4 Including Deterministic Terms 299
7.2.5 Other Estimation Methods for Cointegrated Systems 300
7.2.6 An Example 302
7.3 Estimating VECMs with Parameter Restrictions 305
7.3.1 Linear Restrictions for the Cointegration Matrix 305
7.3.2 Linear Restrictions for the Short-Run and Loading Parameters 307
7.3.3 An Example 309
7.4 Bayesian Estimation of Integrated1 Systems 309
7.4.1 The Model Setup 310
7.4.2 The Minnesota or Litterman Prior 310
7.4.3 An Example 312
7.5 Forecasting Estimated Integrated and Cointegrated Systems 315
7.6 Testing for Granger-Causality 316
7.6.1 The Noncausality Restrictions 316
7.6.2 Problems Related to Standard Wald Tests 317
7.6.3 A Wald Test Based an a Lag Augmented VAR 318
7.6.4 An Exarnple 320
7.7 Impulse Response Analysis 321
7.8 Exercises 323
7.8.1 Algebraic Exercises 323
7.8.2 Numerical Exercises 324
8 Specification of VECMs :325
8.1 Lag Order Selection 325
8.2 Testing for the Rank of Cointegration 327
8.2.1 A VECM without Deterministic Terms 28
8.2.2 A Nonzero Mean Term 330
8.2.3 A Linear Trend 3:31
8.2.4 A Linear Trend in the Variables and Not in the Cointegration Relations 331
8.2.5 Summary of Results and Other Deterministic Terms 332
8.2.6 An Example 335
8.2.7 Prior Adjustment for Deterministic Terms 337
8.2.8 Choice of Deterministic Terms 341
8.2.9 Other Approaches to Testing for the Cointegrating Rank342
8.3 Subset VECMs :343
8.4 1 Model Diagnostics 345
8.4.1 Checking for Residual Autocorrelation 345
8.4.2 Testing for Nonnormality 348
8.4.3 Tests for Structural Change 348
8.5 Exercises 351
8.5.1 Algebraic Exercises 351
8.5.2 Numerical Exercises 352
Part 111 Structural and Conditional Models
9 Structural VARs and VECMs 357
9.1 Structural Vector Autoregressions 358
9.1.1 The A-Model 358
9.1.2 The B-Model 362
9.1.3 The AB-Model 364
9.1.4 Long-Run Restrictions  la Blancharcl-Quak 367
9.2 Structural Vector Error Correction Models 368
9.3 Estimation of Structural Parameters 372
9.3.1 Estimating SVAR Models 372
9.3.2 Estimating Structural VECMs 376
9.4 Impulse Response Analysis and Forecast Error Variance Decomposition 377
9.5 Further Issues 383
9.6 Exercises 384
9.6.1 Algebraic Problems 384
9.6.2 Numerical Problems 385
10 Systems of Dynamic Simultaneous Equations 387
10.1 Background 387
10.2 Systems with Unmodelled Variables 388
10.2.1 Types of Variables 388
10.2.2 Structural Form, Reduced Form, Final Form 390
10.2.3 Models with Rational Expectations 393
10.2.4 Cointegrated Variables 394
10.3 Estimation 395
10.3.1 Stationary Variables 396
10.3.2 Estimation of Models with I(1) Variables 398
10.4 Remarks an Model Specification and Model Checking 400
10.5 Forecasting 401
10.5.1 Unconditional and Conditional Forecasts 401
10.5.2 Forecasting Estimated Dynamic SEMs 405
10.6 Multiplier Analysis 406
10.7 Optimal Control 108
10.8 Concluding Remarks an Dynamic SEMs 411
10.9 Exercises 412
Part IV Infinite Order Vector Autoregressive Processes
11 Vector Autoregressive Moving Average Processes 419
11.1 Introduction 419
11.2 Finite Order Moving Average Processes 420
11.3 VARMA Processes 423
11.3.1 The Pure MA and Pure VAR Representations of a VARMA Process 423
11.3.2 A VAR(1) Representation of a VARMA Process 426
11.4 The Autocovariances and Autocorrelations of a VARMA (p, q) Process 429
11.5 Forecasting VARMA Processes 432
11.6 Transforming and Aggregating VARMA Processes 434
11.6.1 Linear Transformations of VARMA Processes 435
11.6.2 Aggregation of VARMA Processes 440
11.7 Interpretation of VARMA Models 442
11.7.1 Granger-Causality 442
11.7.2 Impulse Response Analysis 444
11.8 Exercises 444
12 Estimation of VARMA Models 447
12.1 The Identification Problem 447
12.1.1 Nonuniqueness of VARMA Representations 447
12.1.2 Final Equations Form and Echelon Form 452
12.1.3 Illustrations 455
12.2 The Gaussian Likelihood Function 459
12.2.1 The Likelihood Function of au MA(1) Process 459
12.2.2 The MA(q) Case 461
12.2.3 The VAR.MA(1,1) Case 463
12.2.4 The General VARMA(p, q) Case 464
12.3 Computation of the ML Estimates 467
12.3.1 The Normal Equations 468
12.3.2 Optimization Algorithms 470
12.3.3 The Information Matrix 473
12.3.4 Preliminary Estimation 474
12.3.5 An Illustration 477
12.4 Asymptotic Properties of the NIL Estimators 479
12.4.1 Theoretical Results 479
12.4.2 A Real Data Example 486
12.5 Forecasting Estimated VARMA Processes 487
12.6 Estimated Impulse Responses 490
12.7 Exercises 491
13 Specification and Checking the Adequacy of VARMA Models 493
13.1 Introduction 493
13.2 Specification of the Final Equations Form 494
13.2.1 A Specification Procedure 494
13.2.2 An Example 497
13.3 Specification of Echelon Forms 498
13.3.1 A Procedure for Small Systems 499
13.3.2 A Full Search Procedure Based an Linear Least Squares Computations 501
13.3.3 Hannan-Kavalieris Procedure 503
13.3.4 Poskitt's Procedure 505
13.4 Remarks an Other Specification Strategies for VARMA Models 507
13.5 Model Checking 508
13.5.1 LM Tests 508
13.5.2 Residual Autocorrelations and Portmanteau Tests 510
13.5.3 Prediction Tests for Structural Change 511
13.6 Critique of VARMA Model Fitting 511
13.7 Exercises 512
14 Cointegrated VARMA Processes 515
14.1 Introduction 515
14.2 The VARMA Framework for I(1) Variables 516
14.2.1 Levels VARA Models 516
14.2.2 The Reverse Echelon Form 518
14.2.3 The Error Correction Echelon Form 519
14.3 Estimation 521
14.3.1 Estimation of VARMA_RE Models 521
14.3.2 Estimation of EC-ARMA_RE Models 522
14.4 Specification of EC-ARMA_RE Models 523
14.4.1 Specification of Kronecker Indices 523
14.4.2 Specification of the Cointegrating Rank 525
14.5 Forecasting Cointegrated VARMA Processes 526
14.6 An Example 526
14.7 Exercises 528
14.7.1 Algebraic Exercises 528
14.7.2 Numerical Exercises 529
15 Fitting Finite Order VARMA Models to Infinite Order Processes 531
15.1 Background 531
15.2 Multivariate Least Squares Estimation 532
15.3 Forecasting 536
15.3.1 Theoretical Results 536
15.3.2 An Example 538
15.4 Impulse Response Analysis and Forecast Error Variance Decompositions 540
15.4.1 Asymptotic Theory 540
15.4.2 An Example 513
15.5 Cointegrated Infinite Order VARs 545
15.5.1 The Model Setup 546
15.5.2 Estimation 549
15.5.3 Testing for the Cointegrating Rank 551
15.6 Exercises 552
Part V Time Series Topics
16 Multivariate ARCH and GARCH Models 557
16.1 Background 557 16.2 Univariate GARCH Models 559
16.2.1 Definitions 559
16.2.2 Forecasting 561
16.3 Multivariate GARCH Models 562
16.3.1 Multivariate ARCH 563
16.3.2 MGARCH 564
16.3.3 Other Multivariate ARCH and GARCH Models 567
16.4 Estimation 569
16.4.1 Theory 569
16.4.2 An Example 571
16.5 Checking NIGARCH Models 576
16.5.1 ARCH-LM and ARCH-Portmanteau Tests 576
16.5.2 LM add Portmanteau Tests for Remaining ARCH 577
16.5.3 Other Diagnostic Tests 578
16.5.4 An Example 578
16.6 Interpreting GARCH Models 579
16.6.1 Causality in Variance 579
16.6.2 Conditional Moment Profiles and Generalized Impulse Responses 580
16.7 Problems and Extensions 582
16.8 Exercises 584
17 Periodic VAR Processes and Intervention Models 585
17.1 Introduction 585
17.2 The VAR(p) Model with Time Varying Coefficients 87
17.2.1 General Properties 587
17.2.2 ML Estimationi 589
17.3 Periodic Processes 591
17.3.1 A VAR. Representation with Tune Invariant Coefficients 592
17.3.2 ML Estimation and Testing for Time Varying Coefficients 595
17.3.3 An Example 602
17.3.4 Bibliographical Notes and Extensions 604
17.4 Intervention Models 604
17.4.1 Interventions in the Intercept Model 605
17.4.2 A Discrete Change in the Mean 606
17.4.3 An Illustrative Example 608
17.4.4 Extensions and References 609
17.5 Exercises 609
18 State Space Models 611
18.1 Background 611
18.2 State Space Models 613
18.2. The Model Setup 613
18.2.2 More General State Space Models 624
18.3 The Kalman Filter 625
18.3.1 The Kalman Filter Recursions 626
18.3.2 Proof of the Kalman Filter Recursions 630
18.4 Maximum Likelihood Estimation of State Space Models 631
18.4.1 The Log-Likelihood Function 632
18.4.2 The Identification Problem 633
18.4.3 Maximization of The Log-Likelihood Function 634
18.4.4 Asymptotic Properties of the ML Estimator 636
18.5 A Real Data Example 637
18.6 Exercises 641
Appendix
A Vectors and Matrices 645
A.1 Basic Definitions 645
A.2 Basic Matrix Operation- 646
A.3 The Determinant 647
A.4 The Inverse, the Adjoint, and Generalized Inverses 649
A.4.1 Inverse and Adjoint of a Square Matrix 649
A.4.2 Generalized Inverses 650
A.5 The Rank 651
A.6 Eigenvalties and -vectors - Characteristic Values and Vectors 652
A.7 The Trace 653
A.8 Some Special Matrices and Vectors 653
A.8.1 Idempotent and Nilpotent 'Matrices 653
A.8.2 Orthogonal Matrices and Vectors and Orthogonal Complements 654
A.8.3 Definite Matrices and Quadratic Forms 655
A.9 Decomposition and Diagonalization of Matrices 656
A.9.1 The Jordan Canonical Form 656
A.9.2 Decomposition of Symmetric Matrices 658
A.9.3 The Choleski Decomposition of a Positive Definite Matrix 658
A.10 Partitioned Matrices 659
A.11 The Kronecker Product 660
A.12 The vec and vech Operators and Related Matrices 661
A.12.1 The Operators 661
A.12.2 Elimination, Duplication, and Commutation Matrices 662
A.13 Vector and Matrix Differentiation 664
A.14 Optimization of Vector Functions 671
A.15 Problems 675
B Multivariate Normal and Related Distributions 677
B.1 Multivariate Normal Distributions 677
B.2 Related Distributions 678
C Stochastic Convergence and Asymptotic Distributions 681
C.1 Concepts of Stochastic Convergence 681
C.2 Order in Probability 684
C.3 Infinite Sums of Random Variables 685
C.4 Laws of Large Numbers and Central Limit; Theorems 689
C.5 Standard Asymptotic Properties of Estimators and Test Statistics 692
C.6 Maximum Likelihood Estimation 693
C.7 Likelihood Ratio, Lagrange Multiplier, and Wald Tests 694
C.8 Unit Root Asymptotics 698
C.8.1 Univariate Processes 698
C.8.2 Multivariate Processes 703
D Evaluating Properties of Estimators and Test Statistics by Simulation and Resampling Techniques 707
D.1 Simulating a Multiple Time Series with VAR Generation Process 707
D.2 Evaluating Dlstributions of Functions of Mltiple Time Series by Simulation 708
D.3 Resampling Methods 709
References 713
Index of Notation 733
Author Index 741
Subject Index 747
END

