Time Series Analysis: Nonstationary and Noninvertible Distribution Theory

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 first Edition

Preface to Second Edition

Part I: Analysis of Non–Fractional Time Series

Chapter 1: Models for Nonstationarity and Noninvertibility

1.1 Statistics from the One–Dimensional Random Walk

1.2 A Test Statistic from a Noninvertible Moving Average Model

1.3 The AR Unit Root Distribution

1.4 Various Statistics from the Two–Dimensional Random Walk

1.5 Statistics from the Cointegrated Process

1.6 Panel Unit Root Tests

Chapter 2: Brownian Motions and Functional Central Limit Theorems

2.1 The Space L2 of Stochastic Processes

2.2 The Brownian Motion

2.3 Mean Square Integration

2.4 The Ito Calculus

2.5 Weak Convergence of Stochastic Processes

2.6 The Functional Central Limit Theorem

2.7 FCLT for Linear Processes

2.8 FCLT for Martingale Differences

2.9 Weak Convergence to the Integrated Brownian Motion

2.10 Weak Convergence to the Ornstein–Uhlenbeck Process

2.11 Weak Convergence of Vector–Valued Stochastic Processes

2.12 Weak Convergence to the Ito Integral

Chapter 3: The Stochastic Process Approach

3.1 Girsanov′s Theorem: O–U Processes

3.2 Girsanov′s Theorem: Integrated Brownian Motion

3.3 Girsanov′s Theorem: Vector–Valued Brownian Motion

3.4 The Cameron{Martin Formula

3.5 Advantages and Disadvantages of the Present Approach

Chapter 4: The Fredholm Approach

4.1 Motivating Examples

4.2 The Fredholm Theory: The Homogeneous Case

4.3 The c.f. of the Quadratic Brownian Functional

4.4 Various Fredholm Determinants

4.5 The Fredholm Theory: The Nonhomogeneous Case

4.6 Weak Convergence of Quadratic Forms

Chapter 5: Numerical Integration

5.1 Introduction

5.2 Numerical Integration: The Nonnegative Case

5.3 Numerical integration: The Oscillating Case

5.4 Numerical integration: The General Case

5.5 Computation of Percent Points

5.6 The Saddlepoint Approximation

Chapter 6: Estimation Problems in Nonstationary Autoregressive Models

6.1 Nonstationary Autoregressive Models

6.2 Convergence in Distribution of LSEs

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

6.4 Tables and Figures of Limiting Distributions

6.5 Approximations to the Distributions of the LSEs

6.6 Nearly Nonstationary Seasonal AR Models

6.7 Continuous Record Asymptotics

6.8 Complex Roots on the Unit Circle

6.9 Autoregressive Models with Multiple Unit Roots

Chapter 7: Estimation Problems in Noninvertible Moving Average Models

7.1 Noninvertible Moving Average Models

7.2 The Local MLE in the Stationary Case

7.3 The Local MLE in the Conditional Case

7.4 Noninvertible Seasonal Models

7.5 The Pseudolocal MLE

7.6 Probability of the Local MLE at Unity

7.7 The Relationship with the State Space Model

Chapter 8: Unit Root Tests in Autoregressive Models

8.1 Introduction

8.2 Optimal Tests

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

8.4 Various Unit Root Tests

8.5 Integral Expressions for the Limiting Powers

8.6 Limiting Power Envelopes and Point Optimal Tests

8.7 Computation of the Limiting Powers

8.8 Seasonal Unit Root Tests

8.9 Unit Root Tests in the Dependent Case

8.10 The Unit Root Testing Problem Revisited

8.11 Unit Root Tests with Structural Breaks

8.12 Stochastic Trends versus Deterministic Trends

Chapter 9: Unit Root Tests in Moving Average Models

9.1 Introduction

9.2 The LBI and LBIU Tests

9.3 The Relationship with the Test Statistics in Differenced Form

9.4 Performance of the LBI and LBIU Tests

9.5 Seasonal Unit Root Tests

9.6 Unit Root Tests in the Dependent Case

9.7 The Relationship with Testing in the State Space Model

Chapter 10: Asymptotic Properties of Nonstationary Panel Unit Root Tests

10.1 Introduction

10.2 Panel Autoregressive Models

10.3 Panel Moving Average Models

10.4 Panel Stationarity Tests

10.5 Concluding Remarks

Chapter 11: Statistical Analysis of Cointegration

11.1 Introduction

11.2 Case of No Cointegration

11.3 Cointegration Distributions: The Independent Case

11.4 Cointegration Distributions: The Dependent Case

11.5 The Sampling Behavior of Cointegration Distributions

11.6 Testing for Cointegration

11.7 Determination of the Cointegration Rank

11.8 Higher Order Cointegration

Part II: Analysis of Fractional Time Series

Chapter 12: ARFIMA Models and the Fractional Brownian Motion

12.1 Nonstationary Fractional Time Series

12.2 Testing for the Fractional Integration Order

12.3 Estimation for the Fractional Integration Order

12.4 Stationary Long–Memory Processes

12.5 The Fractional Brownian Motion

12.6 FCLT for Long–Memory Processes

12.7 Fractional Cointegration

12.8 The Wavelet Method for ARFIMA Models and the fBm

Chapter 13: Statistical Inference Associated with the Fractional Brownian Motion

13.1 Introduction

13.2 A Simple Continuous–Time Model Driven by the fBm

13.3 Quadratic Functionals of the Brownian Motion

13.4 Derivation of the c.f.

13.5 Martingale Approximation to the fBm

13.6 The Fractional Unit Root Distribution

13.7 The Unit Root Test under the fBm Error

Chapter 14: Maximum Likelihood Estimation for the Fractional Ornstein–Uhlenbeck Process

14.1 Introduction

14.2 Estimation of the Drift: Ergodic Case

14.3 Estimation of the Drift: Non–Ergodic Case

14.4 Estimation of the Drift: Boundary Case

14.5 Computation of Distributions and Moments of the MLE and MCE

14.6 The MLE–Based Unit Root Test under the fBm Error

14.7 Concluding Remarks

Chapter 15: Solutions to Problems

References

Author Index

Subject Index

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.