Time Series Analysis: Nonstationary and Noninvertible Distribution Theory, 2nd Edition

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thumbnail image: Time Series Analysis: Nonstationary and Noninvertible Distribution Theory, 2nd Edition

Reflects the developments and new directions in the field since the publication of the first successful edition and contains a complete set of problems and solutions

This revised and expanded edition reflects the developments and new directions in the field since the publication of the first edition. In particular, sections on nonstationary panel data analysis and a discussion on the distinction between deterministic and stochastic trends have been added. Three new chapters on long-memory discrete-time and continuous-time processes have also been created, whereas some chapters have been merged and some sections deleted. The first eleven chapters of the first edition have been compressed into ten chapters, with a chapter on nonstationary panel added and located under Part I: Analysis of Non-fractional Time Series. Chapters 12 to 14 have been newly written under Part II: Analysis of Fractional Time Series. Chapter 12 discusses the basic theory of long-memory processes by introducing ARFIMA models and the fractional Brownian motion (fBm). Chapter 13 is concerned with the computation of distributions of quadratic functionals of the fBm and its ratio. Next, Chapter 14 introduces the fractional Ornstein–Uhlenbeck process, on which the statistical inference is discussed. Finally, Chapter 15 gives a complete set of solutions to problems posed at the end of most sections. This new edition features:

• Sections to discuss nonstationary panel data analysis, the problem of differentiating between deterministic and stochastic trends, and nonstationary processes of local deviations from a unit root

• Consideration of the maximum likelihood estimator of the drift parameter, as well as asymptotics as the sampling span increases

• Discussions on not only nonstationary but also noninvertible time series from a theoretical viewpoint

• New topics such as the computation of limiting local powers of panel unit root tests, the derivation of the fractional unit root distribution, and unit root tests under the fBm error

Time Series Analysis: Nonstationary and Noninvertible Distribution Theory, Second Edition, is a reference for graduate students in econometrics or time series analysis.

Katsuto Tanaka, PhD, is a professor in the Faculty of Economics at Gakushuin University and was previously a professor at Hitotsubashi University. He is a recipient of the Tjalling C. Koopmans Econometric Theory Prize (1996), the Japan Statistical Society Prize (1998), and the Econometric Theory Award (1999). Aside from the first edition of Time Series Analysis (Wiley, 1996), Dr. Tanaka had published five econometrics and statistics books in Japanese.

Preface to the Second Edition xi

Preface to the First Edition xiii

Part I Analysis of Non Fractional Time Series 1

1 Models for Nonstationarity and Noninvertibility 3

1.1 Statistics from the One-Dimensional Random Walk 3

1.1.1 Eigenvalue Approach 4

1.1.2 Stochastic Process Approach 11

1.1.3 The Fredholm Approach 12

1.1.4 An Overview of the Three Approaches 14

1.2 A Test Statistic from a Noninvertible Moving Average Model 16

1.3 The AR Unit Root Distribution 23

1.4 Various Statistics from the Two-Dimensional Random Walk 29

1.5 Statistics from the Cointegrated Process 41

1.6 Panel Unit Root Tests 47

2 Brownian Motion and Functional Central Limit Theorems 51

2.1 The Space L2 of Stochastic Processes 51

2.2 The Brownian Motion 55

2.3 Mean Square Integration 58

2.3.1 The Mean Square Riemann Integral 59

2.3.2 The Mean Square Riemann–Stieltjes Integral 62

2.3.3 The Mean Square Ito Integral 66

2.4 The Ito Calculus 72

2.5 Weak Convergence of Stochastic Processes 77

2.6 The Functional Central Limit Theorem 81

2.7 FCLT for Linear Processes 87

2.8 FCLT for Martingale Differences 91

2.9 Weak Convergence to the Integrated Brownian Motion 99

2.10 Weak Convergence to the Ornstein–Uhlenbeck Process 103

2.11 Weak Convergence of Vector-Valued Stochastic Processes 109

2.11.1 Space Cq 109

2.11.2 Basic FCLT for Vector Processes 110

2.11.3 FCLT for Martingale Differences 112

2.11.4 FCLT for the Vector-Valued Integrated Brownian Motion 115

2.12 Weak Convergence to the Ito Integral 118

3 The Stochastic Process Approach 127

3.1 Girsanov’s Theorem: O-U Processes 127

3.2 Girsanov’s Theorem: Integrated Brownian Motion 137

3.3 Girsanov’s Theorem: Vector-Valued Brownian Motion 142

3.4 The Cameron–Martin Formula 145

3.5 Advantages and Disadvantages of the Present Approach 147

4 The Fredholm Approach 149

4.1 Motivating Examples 149

4.2 The Fredholm Theory: The Homogeneous Case 155

4.3 The c.f. of the Quadratic Brownian Functional 161

4.4 Various Fredholm Determinants 171

4.5 The Fredholm Theory: The Nonhomogeneous Case 190

4.5.1 Computation of the Resolvent – Case 1 192

4.5.2 Computation of the Resolvent – Case 2 199

4.6 Weak Convergence of Quadratic Forms 203

5 Numerical Integration 213

5.1 Introduction 213

5.2 Numerical Integration: The Nonnegative Case 214

5.3 Numerical Integration: The Oscillating Case 220

5.4 Numerical Integration: The General Case 228

5.5 Computation of Percent Points 236

5.6 The Saddlepoint Approximation 240

6 Estimation Problems in Nonstationary Autoregressive Models 245

6.1 Nonstationary Autoregressive Models 245

6.2 Convergence in Distribution of LSEs 250

6.2.1 Model A 251

6.2.2 Model B 253

6.2.3 Model C 255

6.2.4 Model D 257

6.3 The c.f.s for the Limiting Distributions of LSEs 260

6.3.1 The Fixed Initial Value Case 261

6.3.2 The Stationary Case 265

6.4 Tables and Figures of Limiting Distributions 267

6.5 Approximations to the Distributions of the LSEs 276

6.6 Nearly Nonstationary Seasonal AR Models 281

6.7 Continuous Record Asymptotics 289

6.8 Complex Roots on the Unit Circle 292

6.9 Autoregressive Models with Multiple Unit Roots 300

7 Estimation Problems in Noninvertible Moving Average Models 311

7.1 Noninvertible Moving Average Models 311

7.2 The Local MLE in the Stationary Case 314

7.3 The Local MLE in the Conditional Case 325

7.4 Noninvertible Seasonal Models 330

7.4.1 The Stationary Case 331

7.4.2 The Conditional Case 333

7.4.3 Continuous Record Asymptotics 335

7.5 The Pseudolocal MLE 337

7.5.1 The Stationary Case 337

7.5.2 The Conditional Case 339

7.6 Probability of the Local MLE at Unity 341

7.7 The Relationship with the State Space Model 343

8 Unit Root Tests in Autoregressive Models 349

8.1 Introduction 349

8.2 Optimal Tests 350

8.2.1 The LBI Test 352

8.2.2 The LBIU Test 353

8.3 Equivalence of the LM Test with the LBI or LBIU Test 356

8.3.1 Equivalence with the LBI Test 356

8.3.2 Equivalence with the LBIU Test 358

8.4 Various Unit Root Tests 360

8.5 Integral Expressions for the Limiting Powers 362

8.5.1 Model A 363

8.5.2 Model B 364

8.5.3 Model C 365

8.5.4 Model D 367

8.6 Limiting Power Envelopes and Point Optimal Tests 369

8.7 Computation of the Limiting Powers 372

8.8 Seasonal Unit Root Tests 382

8.9 Unit Root Tests in the Dependent Case 389

8.10 The Unit Root Testing Problem Revisited 395

8.11 Unit Root Tests with Structural Breaks 398

8.12 Stochastic Trends Versus Deterministic Trends 402

8.12.1 Case of Integrated Processes 403

8.12.2 Case of Near-Integrated Processes 406

8.12.3 Some Simulations 409

9 Unit Root Tests in Moving Average Models 415

9.1 Introduction 415

9.2 The LBI and LBIU Tests 416

9.2.1 The Conditional Case 417

9.2.2 The Stationary Case 419

9.3 The Relationship with the Test Statistics in Differenced Form 424

9.4 Performance of the LBI and LBIU Tests 427

9.4.1 The Conditional Case 427

9.4.2 The Stationary Case 430

9.5 Seasonal Unit Root Tests 434

9.5.1 The Conditional Case 434

9.5.2 The Stationary Case 436

9.5.3 Power Properties 438

9.6 Unit Root Tests in the Dependent Case 444

9.6.1 The Conditional Case 444

9.6.2 The Stationary Case 446

9.7 The Relationship with Testing in the State Space Model 447

9.7.1 Case (I) 449

9.7.2 Case (II) 450

9.7.3 Case (III) 452

9.7.4 The Case of the Initial Value Known 454

10 Asymptotic Properties of Nonstationary Panel Unit Root Tests 459

10.1 Introduction 459

10.2 Panel Autoregressive Models 461

10.2.1 Tests Based on the OLSE 463

10.2.2 Tests Based on the GLSE 471

10.2.3 Some Other Tests 475

10.2.4 Limiting Power Envelopes 480

10.2.5 Graphical Comparison 485

10.3 Panel Moving Average Models 488

10.3.1 Conditional Case 490

10.3.2 Stationary Case 494

10.3.3 Power Envelope 499

10.3.4 Graphical Comparison 502

10.4 Panel Stationarity Tests 507

10.4.1 Limiting Local Powers 508

10.4.2 Power Envelope 512

10.4.3 Graphical Comparison 514

10.5 Concluding Remarks 515

11 Statistical Analysis of Cointegration 517

11.1 Introduction 517

11.2 Case of No Cointegration 519

11.3 Cointegration Distributions: The Independent Case 524

11.4 Cointegration Distributions: The Dependent Case 532

11.5 The Sampling Behavior of Cointegration Distributions 537

11.6 Testing for Cointegration 544

11.6.1 Tests for the Null of No Cointegration 544

11.6.2 Tests for the Null of Cointegration 547

11.7 Determination of the Cointegration Rank 552

11.8 Higher Order Cointegration 556

11.8.1 Cointegration in the I(d) Case 556

11.8.2 Seasonal Cointegration 559

Part II Analysis of Fractional Time Series 567

12 ARFIMA Models and the Fractional Brownian Motion 569

12.1 Nonstationary Fractional Time Series 569

12.1.1 Case of d = ½ 570

12.1.2 Case of d > ½ 572

12.2 Testing for the Fractional Integration Order 575

12.2.1 i.i.d. Case 575

12.2.2 Dependent Case 581

12.3 Estimation for the Fractional Integration Order 584

12.3.1 i.i.d. Case 584

12.3.2 Dependent Case 586

12.4 Stationary Long-Memory Processes 591

12.5 The Fractional Brownian Motion 597

12.6 FCLT for Long-Memory Processes 603

12.7 Fractional Cointegration 608

12.7.1 Spurious Regression in the Fractional Case 609

12.7.2 Cointegrating Regression in the Fractional Case 610

12.7.3 Testing for Fractional Cointegration 614

12.8 The Wavelet Method for ARFIMA Models and the fBm 614

12.8.1 Basic Theory of the Wavelet Transform 615

12.8.2 Some Advantages of the Wavelet Transform 618

12.8.3 Some Applications of the Wavelet Analysis 625

13 Statistical Inference Associated with the Fractional Brownian Motion 629

13.1 Introduction 629

13.2 A Simple Continuous-Time Model Driven by the fBm 632

13.3 Quadratic Functionals of the Brownian Motion 641

13.4 Derivation of the c.f. 645

13.4.1 Stochastic Process Approach via Girsanov’s Theorem 645

13.4.2 Fredholm Approach via the Fredholm Determinant 647

13.5 Martingale Approximation to the fBm 651

13.6 The Fractional Unit Root Distribution 659

13.6.1 The FD Associated with the Approximate Distribution 659

13.6.2 An Interesting Moment Property 664

13.7 The Unit Root Test Under the fBm Error 669

14 Maximum Likelihood Estimation for the Fractional Ornstein–Uhlenbeck Process 673

14.1 Introduction 673

14.2 Estimation of the Drift: Ergodic Case 677

14.2.1 Asymptotic Properties of the OLSEs 677

14.2.2 The MLE and MCE 679

14.3 Estimation of the Drift: Non-ergodic Case 687

14.3.1 Asymptotic Properties of the OLSE 687

14.3.2 The MLE 687

14.4 Estimation of the Drift: Boundary Case 692

14.4.1 Asymptotic Properties of the OLSEs 692

14.4.2 The MLE and MCE 693

14.5 Computation of Distributions and Moments of the MLE and MCE 695

14.6 The MLE-based Unit Root Test Under the fBm Error 703

14.7 Concluding Remarks 707

15 Solutions to Problems 709

References 865

Author Index 879

Subject Index 883

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