Time Series Analysis

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.

Inhalt

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 RiemannStieltjes 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 OrnsteinUhlenbeck 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 CameronMartin 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....