Cover......Page 1
Title......Page 3
Copyright Page......Page 4
Preface......Page 6
Contents......Page 10
1.1 Objectives of Analyzing Multiple Time Series......Page 21
1.2 Some Basics......Page 22
1.3 Vector Autoregressive Processes......Page 24
1.4 Outline of the Following Chapters......Page 25
Part I Finite Order Vector Autoregressive Processes......Page 30
2.1.1 Stable VAR(p) Processes......Page 33
2.1.2 The Moving Average Representation of a VAR Process......Page 38
2.1.3 Stationary Processes......Page 44
2.1.4 Computation of Autocovariances and Autocorrelations of Stable VAR Processes......Page 46
2.2 Forecasting......Page 51
2.2.1 The Loss Function......Page 52
2.2.2 Point Forecasts......Page 53
2.2.3 Interval Forecasts and Forecast Regions......Page 59
2.3.1 Granger- Causality, Instantaneous Causality, and Multi-Step Causality......Page 61
2.3.2 Impulse Response Analysis......Page 71
2.3.3 Forecast Error Variance Decomposition......Page 83
2.4 Exercises......Page 86
3.2 Multivariate Least Squares Estimation......Page 89
3.2.1 The Estimator......Page 90
3.2.2 Asymptotic Properties of the Least Squares Estimator......Page 93
3.2.3 An Example......Page 97
3.2.4 Small Sample Properties of the LS Estimator......Page 100
3.3.1 Estimation when the Process Mean Is Known......Page 102
3.3.2 Estimation of the Process Mean......Page 103
3.3.4 The Yule-Walker Estimator......Page 105
3.4.1 The Likelihood Function......Page 107
3.4.2 The ML Estimators......Page 109
3.4.3 Properties of the ML Estimators......Page 110
3.5.1 General Assumptions and Results......Page 114
3.5.2 The Approximate MSE Matrix......Page 116
3.5.3 An Example......Page 118
3.5.4 A Small Sample Investigation......Page 120
3.6.1 A Wald Test for Granger- Causality......Page 122
3.6.2 An Example......Page 123
3.6.3 Testing for Instantaneous Causality......Page 124
3.6.4 Testing for Multi-Step Causality......Page 126
3.7.1 The Main Results......Page 129
3.7.2 Proof of Proposition 3.6......Page 136
3.7.3 An Example......Page 138
3.7.4 Investigating the Distributions of the Impulse Responses by Simulation Techniques......Page 146
3.8.1 Algebraic Problems......Page 150
3.8.2 Numerical Problems......Page 152
4.1 Introduction......Page 155
4.2.1 The Impact of the Fitted VAR Order on the Forecast MSE......Page 156
4.2.2 The Likelihood Ratio Test Statistic......Page 158
4.2.3 A Testing Scheme for VAR Order Determination......Page 163
4.2.4 An Example......Page 165
4.3.1 Minimizing the Forecast MSE......Page 166
4.3.2 Consistent Order Selection......Page 168
4.3.3 Comparison of Order Selection Criteria......Page 171
4.3.4 Some Small Sample Simulation Results......Page 173
4.4.1 The Asymptotic Distributions of the Autocovariances and Autocorrelations of a White Noise Process......Page 177
4.4.2 The Asymptotic Distributions of the Residual Autocovariances and Autocorrelations of an Estimated VAR Process......Page 181
4.4.3 Portmanteau Tests......Page 189
4.4.4 Lagrange Multiplier Tests......Page 191
4.5.1 Tests for Nonnormality of a Vector White Noise Process......Page 194
4.5.2 Tests for Nonnormality of a VAR Process......Page 197
4.6 Tests for Structural Change......Page 201
4.6.1 Chow Tests......Page 202
4.6.2 Forecast Tests for Structural Change......Page 204
4.7.1 Algebraic Problems......Page 209
4.7.2 Numerical Problems......Page 211
5.1 Introduction......Page 213
5.2.1 The Model and the Constraints......Page 214
5.2.2 LS, GLS, and EGLS Estimation......Page 215
5.2.3 Maximum Likelihood Estimation......Page 220
5.2.4 Constraints for Individual Equations......Page 221
5.2.5 Restrictions for the White Noise Covariance Matrix......Page 222
5.2.6 Forecasting......Page 224
5.2.7 Impulse Response Analysis and Forecast Error Variance Decomposition......Page 225
5.2.8 Specification of Subset VAR Models......Page 226
5.2.9 Model Checking......Page 232
5.2.10 An Example......Page 237
5.3 VAR Processes with Nonlinear Parameter Restrictions......Page 241
5.4.1 Basic Terms and Notation......Page 242
5.4.2 Normal Priors for the Parameters of a Gaussian VAR Process......Page 243
5.4.3 The Minnesota or Litterman Prior......Page 245
5.4.5 An Example......Page 247
5.4.6 Classical versus Bayesian Interpretation of $\overline{\alpha}$ in Forecasting and Structural Analysis......Page 248
5.5.1 Algebraic Exercises......Page 250
5.5.2 Numerical Problems......Page 251
Part II Cointegrated Processes......Page 254
6 Vector Error Correction Models......Page 257
6.1 Integrated Processes......Page 258
6.2 VAR Processes with Integrated Variables......Page 263
6.3 Cointegrated Processes, Common Stochastic Trends, and Vector Error Correction Models......Page 264
6.4 Deterministic Terms in Cointegrated Processes......Page 276
6.5 Forecasting Integrated and Cointegrated Variables......Page 278
6.6 Causality Analysis......Page 281
6.7 Impulse Response Analysis......Page 282
6.8 Exercises......Page 285
7.1 Estimation of a Simple Special Case VECM......Page 289
7.2 Estimation of General VECMs......Page 306
7.2.1 LS Estimation......Page 307
7.2.2 EGLS Estimation of the Cointegration Parameters......Page 311
7.2.3 ML Estimation......Page 314
7.2.4 Including Deterministic Terms......Page 319
7.2.5 Other Estimation Methods for Cointegrated Systems......Page 320
7.2.6 An Example......Page 322
7.3.1 Linear Restrictions for the Cointegration Matrix......Page 325
7.3.2 Linear Restrictions for the Short- Run and Loading Parameters......Page 327
7.4 Bayesian Estimation of Integrated Systems......Page 329
7.4.2 The Minnesota or Litterman Prior......Page 330
7.4.3 An Example......Page 332
7.5 Forecasting Estimated Integrated and Cointegrated Systems......Page 335
7.6.1 The Noncausality Restrictions......Page 336
7.6.2 Problems Related to Standard Wald Tests......Page 337
7.6.3 A Wald Test Based on a Lag Augmented VAR......Page 338
7.6.4 An Example......Page 340
7.7 Impulse Response Analysis......Page 341
7.8.1 Algebraic Exercises......Page 343
7.8.2 Numerical Exercises......Page 344
8.1 Lag Order Selection......Page 345
8.2 Testing for the Rankof Cointegration......Page 347
8.2.1 A VECM without Deterministic Terms......Page 348
8.2.2 A Nonzero Mean Term......Page 350
8.2.4 A Linear Trend in the Variables and Not in the Cointegration Relations......Page 351
8.2.5 Summary of Results and Other Deterministic Terms......Page 352
8.2.6 An Example......Page 355
8.2.7 Prior Adjustment for Deterministic Terms......Page 357
8.2.8 Choice of Deterministic Terms......Page 361
8.2.9 Other Approaches to Testing for the Cointegrating Rank......Page 362
8.3 Subset VECMs......Page 363
8.4.1 Checking for Residual Autocorrelation......Page 365
8.4.3 Tests for Structural Change......Page 368
8.5.1 Algebraic Exercises......Page 371
8.5.2 Numerical Exercises......Page 372
Part III Structural and Conditional Models......Page 374
9 Structural VARs and VECMs......Page 377
9.1.1 The A-Model......Page 378
9.1.2 The B-Model......Page 382
9.1.3 The AB-Model......Page 384
9.1.4 Long-Run Restrictions ? la Blanchard-Quah......Page 387
9.2 Structural Vector Error Correction Models......Page 388
9.3.1 Estimating SVAR Models......Page 392
9.3.2 Estimating Structural VECMs......Page 396
9.4 Impulse Response Analysis and Forecast Error Variance Decomposition......Page 397
9.5 Further Issues......Page 403
9.6.1 Algebraic Problems......Page 404
9.6.2 Numerical Problems......Page 405
10.1 Background......Page 407
10.2.1 Types of Variables......Page 408
10.2.2 Structural Form, Reduced Form, Final Form......Page 410
10.2.3 Models with Rational Expectations......Page 413
10.2.4 Cointegrated Variables......Page 414
10.3 Estimation......Page 415
10.3.1 Stationary Variables......Page 416
10.3.2 Estimation of Models with I(1) Variables......Page 418
10.4 Remarks on Model Specification and Model Checking......Page 420
10.5.1 Unconditional and Conditional Forecasts......Page 421
10.5.2 Forecasting Estimated Dynamic S E Ms......Page 425
10.6 Multiplier Analysis......Page 426
10.7 Optimal Control......Page 428
10.8 Concluding Remarks on Dynamic S E Ms......Page 431
10.9 Exercises......Page 432
Part IV Infinite Order Vector Autoregressive Processes......Page 436
11.1 Introduction......Page 439
11.2 Finite Order Moving Average Processes......Page 440
11.3.1 The Pure MA and Pure VAR Representations of a VARMA Process......Page 443
11.3.2 A VAR(1) Representation of a VARMA Process......Page 446
11.4 The Autocovariances and Autocorrelations of a VARMA(p, q) Process......Page 449
11.5 Forecasting VARMA Processes......Page 452
11.6 Transforming and Aggregating VARMA Processes......Page 454
11.6.1 Linear Transformations of VARMA Processes......Page 455
11.6.2 Aggregation of VARMA Processes......Page 460
11.7.1 Granger-Causality......Page 462
11.8 Exercises......Page 464
12.1.1 Nonuniqueness of VARMA Representations......Page 467
12.1.2 Final Equations Form and Echelon Form......Page 472
12.1.3 Illustrations......Page 475
12.2.1 The Likelihood Function of an MA(1) Process......Page 479
12.2.2 The MA(q) Case......Page 481
12.2.3 The VARMA(1, 1) Case......Page 483
12.2.4 The General VARMA(p, q) Case......Page 484
12.3 Computation of the ML Estimates......Page 487
12.3.1 The Normal Equations......Page 488
12.3.2 Optimization Algorithms......Page 490
12.3.3 The Information Matrix......Page 493
12.3.4 Preliminary Estimation......Page 494
12.3.5 An Illustration......Page 497
12.4.1 Theoretical Results......Page 499
12.4.2 A Real Data Example......Page 506
12.5 Forecasting Estimated VARMA Processes......Page 507
12.6 Estimated Impulse Responses......Page 510
12.7 Exercises......Page 511
13.1 Introduction......Page 513
13.2.1 A Specification Procedure......Page 514
13.2.2 An Example......Page 517
13.3 Specification of Echelon Forms......Page 518
13.3.1 A Procedure for Small Systems......Page 519
13.3.2 A Full Search Procedure Based on Linear Least Squares Computations......Page 521
13.3.3 Hannan-Kavalieris Procedure......Page 523
13.3.4 Poskitt's Procedure......Page 525
13.4 Remarks on Other Specification Strategies for VARMA Models......Page 527
13.5.1 LM Tests......Page 528
13.5.2 Residual Autocorrelations and Portmanteau Tests......Page 530
13.6 Critique of VARMA Model Fitting......Page 531
13.7 Exercises......Page 532
14.1 Introduction......Page 535
14.2.1 Levels VARMA Models......Page 536
14.2.2 The Reverse Echelon Form......Page 538
14.2.3 The Error Correction Echelon Form......Page 539
14.3.1 Estimation of ARMA_{RE} Models......Page 541
14.3.2 Estimation of EC-ARMA_{RE} Models......Page 542
14.4.1 Specification of Kronecker Indices......Page 543
14.4.2 Specification of the Cointegrating Rank......Page 545
14.6 An Example......Page 546
14.7.1 Algebraic Exercises......Page 548
14.7.2 Numerical Exercises......Page 549
15.1 Background......Page 551
15.2 Multivariate Least Squares Estimation......Page 552
15.3.1 Theoretical Results......Page 556
15.3.2 An Example......Page 558
15.4.1 Asymptotic Theory......Page 560
15.4.2 An Example......Page 563
15.5 Cointegrated Infinite Order VARs......Page 565
15.5.1 The Model Setup......Page 566
15.5.2 Estimation......Page 569
15.5.3 Testing for the Cointegrating Rank......Page 571
15.6 Exercises......Page 572
Part V Time Series Topics......Page 576
16.1 Background......Page 577
16.2.1 Definitions......Page 579
16.2.2 Forecasting......Page 581
16.3 Multivariate GARCH Models......Page 582
16.3.1 Multivariate ARCH......Page 583
16.3.2 MGARCH......Page 584
16.3.3 Other Multivariate ARCH and GARCH Models......Page 587
16.4.1 Theory......Page 589
16.4.2 An Example......Page 591
16.5.1 ARCH-LM and ARCH-Portmanteau Tests......Page 596
16.5.2 LM and Portmanteau Tests for Remaining ARCH......Page 597
16.5.4 An Example......Page 598
16.6.1 Causality in Variance......Page 599
16.6.2 Conditional Moment Profiles and Generalized Impulse Responses......Page 600
16.7 Problems and Extensions......Page 602
16.8 Exercises......Page 604
17.1 Introduction......Page 605
17.2.1 General Properties......Page 607
17.2.2 ML Estimation......Page 609
17.3 Periodic Processes......Page 611
17.3.1 A VAR Representation with Time Invariant Coefficients......Page 612
17.3.2 ML Estimation and Testing for Time Varying Coefficients......Page 615
17.3.3 An Example......Page 622
17.4 Intervention Models......Page 624
17.4.1 Interventions in the Intercept Model......Page 625
17.4.2 A Discrete Change in the Mean......Page 626
17.4.3 An Illustrative Example......Page 628
17.5 Exercises......Page 629
18.1 Background......Page 631
18.2.1 The Model Setup......Page 633
18.2.2 More General State Space Models......Page 644
18.3 The Kalman Filter......Page 645
18.3.1 The Kalman Filter Recursions......Page 646
18.3.2 Proof of the Kalman Filter Recursions......Page 650
18.4 Maximum Likelihood Estimation of State Space Models......Page 651
18.4.1 The Log- Likelihood Function......Page 652
18.4.2 The Identification Problem......Page 653
18.4.3 Maximization of the Log- Likelihood Function......Page 654
18.4.4 Asymptotic Properties of the ML Estimator......Page 656
18.5 A Real Data Example......Page 657
18.6 Exercises......Page 661
Appendix......Page 663
A.1 Basic Definitions......Page 665
A.2 Basic Matrix Operations......Page 666
A.3 The Determinant......Page 667
A.4.1 Inverse and Adjoint of a Square Matrix......Page 669
A.4.2 Generalized Inverses......Page 670
A.5 The Rank......Page 671
A.6 Eigenvalues and -vectors. Characteristic Values and Vectors......Page 672
A.8.1 Idempotent and Nilpotent Matrices......Page 673
A.8.2 Orthogonal Matrices and Vectors and Orthogonal Complements......Page 674
A.8.3 Definite Matrices and Quadratic Forms......Page 675
A.9.1 The Jordan Canonical Form......Page 676
A.9.3 The Choleski Decomposition of a Positive Definite Matrix......Page 678
A.10 Partitioned Matrices......Page 679
A.11 The Kronecker Product......Page 680
A.12.1 The Operators......Page 681
A.12.2 Elimination, Duplication, and Commutation Matrices......Page 682
A.13 Vector and Matrix Differentiation......Page 684
A.14 Optimization of Vector Functions......Page 691
A.15 Problems......Page 695
B.1 Multivariate Normal Distributions......Page 697
B.2 Related Distributions......Page 698
C.1 Concepts of Stochastic Convergence......Page 701
C.2 Order in Probability......Page 704
C.3 Infinite Sums of Random Variables......Page 705
C.4 Laws of Large Numbers and Central Limit Theorems......Page 709
C.5 Standard Asymptotic Properties of Estimators and Test Statistics......Page 712
C.6 Maximum Likelihood Estimation......Page 713
C.7 Likelihood Ratio, Lagrange Multiplier, and Wald Tests......Page 714
C.8.1 Univariate Processes......Page 718
C.8.2 Multivariate Processes......Page 723
D.1 Simulating a Multiple Time Series with VAR Generation Process......Page 727
D.2 Evaluating Distributions of Functions of Multiple Time Series by Simulation......Page 728
D.3 Resampling Methods......Page 729
References......Page 733
Index of Notation......Page 753
Author Index......Page 761
Subject Index......Page 767