Time Series Analysis

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