##
**Time series analysis. Nonstationary and noninvertible distribution theory.**
*(English)*
Zbl 0861.62062

Wiley Series in Probability and Mathematical Statistics. New York, NY: Wiley. x, 623 p. (1996).

The book is devoted to the theory of linear time series models when the assumptions of stationarity or invertibility are not fulfilled. The first part of the book contains theoretical background and introductory material. Stochastic integrals are defined and important types of stochastic processes (Brownian motion, Brownian bridge, Ornstein-Uhlenbeck process) are discussed. Weak convergence of processes is used to obtain functional central limit theorems (or invariance principles). It is shown how to use such tools to get asymptotic distributions of statistics arising in nonstationary models. Numerical computations are based either on the stochastic process approach, which uses Girsanov’s theorem and the Cameron-Martin formula, or on the Fredholm approach.

In the second part of the book theoretical results are applied to specific problems in time series. The author considers regression models with the error term following a near integrated process or a process with roots on the unit circle. Asymptotic properties of ML estimators are derived for noninvertible MA models. Unit root tests in autoregressive models having certain optimality properties are introduced and compared with tests suggested in the literature. Another unit root testing problem concerns the MA part of linear models.

The last part of the book deals with cointegration. Estimation and testing problems associated with cointegration are discussed and a general procedure for determining the algebraic structure of cointegration is described. The last chapter contains complete solutions to problems posed at the end of sections of each chapter. Most of the problems concern some parts of proofs in the main text.

Contents of the book: 1. Motivating examples; 2. Stochastic calculus in mean square; 3. Functional central limit theorems; 4. The stochastic process approach; 5. The Fredholm approach; 6. Numerical integration; 7. Estimation problems in nonstationary autoregressive models; 8. Estimation problems in noninvertible moving average models; 9. Unit root tests in autoregressive models; 10. Unit root tests in moving average models; 11. Statistical analysis of cointegration; 12. Solutions of problems.

The book is carefully and clearly written. It can be recommended as a textbook for graduate students with a general knowledge of mathematical statistics and classical time series analysis.

In the second part of the book theoretical results are applied to specific problems in time series. The author considers regression models with the error term following a near integrated process or a process with roots on the unit circle. Asymptotic properties of ML estimators are derived for noninvertible MA models. Unit root tests in autoregressive models having certain optimality properties are introduced and compared with tests suggested in the literature. Another unit root testing problem concerns the MA part of linear models.

The last part of the book deals with cointegration. Estimation and testing problems associated with cointegration are discussed and a general procedure for determining the algebraic structure of cointegration is described. The last chapter contains complete solutions to problems posed at the end of sections of each chapter. Most of the problems concern some parts of proofs in the main text.

Contents of the book: 1. Motivating examples; 2. Stochastic calculus in mean square; 3. Functional central limit theorems; 4. The stochastic process approach; 5. The Fredholm approach; 6. Numerical integration; 7. Estimation problems in nonstationary autoregressive models; 8. Estimation problems in noninvertible moving average models; 9. Unit root tests in autoregressive models; 10. Unit root tests in moving average models; 11. Statistical analysis of cointegration; 12. Solutions of problems.

The book is carefully and clearly written. It can be recommended as a textbook for graduate students with a general knowledge of mathematical statistics and classical time series analysis.

Reviewer: J.Anděl (Praha)

### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

60F17 | Functional limit theorems; invariance principles |