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**Introduction to multiple time series analysis.**
*(English)*
Zbl 0729.62085

Berlin etc.: Springer-Verlag. xxi, 545 p. with 34 fig. DM 78.00/pbk (1991).

This work is an excellent textbook for graduate courses on multivariate (or vector) time series analysis and forecasting. Although written with the economics or business student in mind, the book is definitely adequate for many other categories of readers interested in time series analysis. The book covers most of the dynamic discrete-time models in use:

VAR models in Chapters 2 to 5 (here “V” stands for “vector”);

VARMA models in Chapters 6 to 8;

VAR models whose orders depend on the sample length, in Chapter 9;

Difference equation models with exogeneous variables, in Chapter 10;

Integrated and cointegrated VAR models, in Chapter 11;

Periodic VAR models and intervention models, in Chapter 12;

State-space models, in Chapter 13.

For the most part of these models, the discussion follows the following pattern:

Derivation of basic properties, such as conditions for stability, stationarity, invertibility, causality; expressions for autocovariances and autocorrelations, h-step ahead prediction; and impulse response analysis.

Estimation of parameters, with a focus on ML techniques and their simplification to LS methods whenever possible. In relation to the estimation stage, the problem of using parameter constraints and special parametrizations (e.g., for VARMA models) is discussed whenever necessary.

Derivation of asymptotic distributions of the estimated quantities, such as parameters, impulse responses, forecasts and others.

Structure determination (for instance, order selection), residual checking, testing for nonnormality, structural change or causality and related subjects.

Background material required is included in 4 appendices that contain a collection of results on matrix algebra, multivariate normal and related distributions, stochastic convergence and asymptotic theory. A final appendix contains the data used in the examples and exercises throughout the book.

In conclusion, this book may become for the analysis of multivariate time series what the book by Box and Jenkins is for the univariate case.

VAR models in Chapters 2 to 5 (here “V” stands for “vector”);

VARMA models in Chapters 6 to 8;

VAR models whose orders depend on the sample length, in Chapter 9;

Difference equation models with exogeneous variables, in Chapter 10;

Integrated and cointegrated VAR models, in Chapter 11;

Periodic VAR models and intervention models, in Chapter 12;

State-space models, in Chapter 13.

For the most part of these models, the discussion follows the following pattern:

Derivation of basic properties, such as conditions for stability, stationarity, invertibility, causality; expressions for autocovariances and autocorrelations, h-step ahead prediction; and impulse response analysis.

Estimation of parameters, with a focus on ML techniques and their simplification to LS methods whenever possible. In relation to the estimation stage, the problem of using parameter constraints and special parametrizations (e.g., for VARMA models) is discussed whenever necessary.

Derivation of asymptotic distributions of the estimated quantities, such as parameters, impulse responses, forecasts and others.

Structure determination (for instance, order selection), residual checking, testing for nonnormality, structural change or causality and related subjects.

Background material required is included in 4 appendices that contain a collection of results on matrix algebra, multivariate normal and related distributions, stochastic convergence and asymptotic theory. A final appendix contains the data used in the examples and exercises throughout the book.

In conclusion, this book may become for the analysis of multivariate time series what the book by Box and Jenkins is for the univariate case.

Reviewer: P.Stoica (Bucureşti)

### MSC:

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

62M20 | Inference from stochastic processes and prediction |

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

91B84 | Economic time series analysis |

93E12 | Identification in stochastic control theory |