Time Series Analysis

Preface

This book attempts to describe nonstandard theory for linear time series models which are nonstationary and/or noninvertible. Nonstandard aspects of the departure from stationarity or invertibiltity have attracted much attention in the field of time series econometrics during the last ten years. Since there seem few books concerned with the present theory, I have been at liberty to choose my way. Throughout this book, attention is oriented toward the most interesting theoretical issue; that is, the asymptotic distributional aspect of nonstandard statistics. The subtitle of the book reflects this.

Chapter 1 is a prelude to the main theme. By using simple examples, various asymptotic distributions of nonstandard statistics are derived by a classical approach which I call the eigenvalue approach. It turns out that, if more complicated problems are to be dealt with, the eigenvalue approach breaks down and the introduction of notions such as the Brownian motion, the Ito integral, the functional central limit theorem and so on is inevitable. These notions are now developed very deeply in probability theory. In this book, however, a knowledge of such probability notions is required only at a moderate level, which I explain in Chapters 2 and 3 in an easily accessible way.

Probability theory, in particular the functional central limit theorem, enables us to establish weak convergence of nonstandard statistics, and to realize that limiting forms can be expressed by functionals of the Brownian motion. However, more important from a statistical point of view is how to compute limiting distributions of those statistics. For this purpose I do not simply resort to simulations, but employ numerical integration. To make the computation possible we first need to derive limiting characteristic functions of nonstandard statistics. To this end two approaches are presented. Chapter 4 discusses one approach which I call the stochastic process approach, while Chapter 5 the other which I call the Fredholm approach. The two approaches originate from quite different mathematical theories, which I explain fully, indicating the advantage and disadvantage of each approach for judicious use.

Chapter 6 discusses and illustrates numerical integration for computing distribution functions via inversion of characteristic functions. This chapter is necessitated because a direct application of any computer package for integration cannot do a proper job. We overcome the difficulty by employing Simpson's rule, which can be executed on a desktop computer. The necessity for accurate computation based on numerical integration is recognized, for instance, when close comparison has to be made between limiting local powers of competing nonstandard tests.

Chapters 7 through 11 deal with statistical and econometric problems to which the nonstandard theory discussed in previous chapters applies. Chapter 7 considers the estimation problems associated with nonstationary autoregressive models, while Chapter 8 considers those with noninvertible moving average models. The corresponding testing problems, called the unit root tests are discussed in Chapters 9 and 10, respectively. Chapter 11 is concerned with cointegration, which is a stochastic collinearity relationship among multiple nonstationary time series. The problems discussed in these chapters originate in time series econometrics. I describe in detail how to derive and compute limiting nonstandard distributions of various estimators and test statistics.

Chapter 12, the last chapter, gives a complete set of solutions to problems posed at the end of most sections of each chapter. Most of the problems are concerned with corroborating the results described in the text, so that one can gain a better understanding of details of the discussions.

There are about 90 figures and 50 tables. Most of these are of limiting distributions of nonstandard statistics. These are all produced by the methods described in this book, and include many distributions that have never appeared in the literature. Among these are limiting powers and power envelopes of various nonstandard tests under a sequence of local alternatives.

This book may be used as a textbook for graduate students majoring in econometrics or time series analysis. A general knowledge of mathematical statistics, including the theory of stationary processes, is presupposed, although the necessary material is offered in the text and problems of this book. Some knowledge of programming language like FORTRAN and computerized algebra like REDUCE is also useful.

The late Professor E. J. Hannan gave me valuable comments on the early version of my manuscript. I would like to thank him for his kindness and for pleasant memories extending over years since my student days. This book grew out of joint work with Professor S. Nabeya, another respected teacher of mine. He read substantial parts of the manuscript and corrected a number of errors in its preliminary stages, for which I am most grateful. I am also grateful to Professors C. W. Helstrom, S. Kusuoka, and P. Saikkonen for helpful discussions, and to Professor G. S. Watson for help of various kinds. Most of the manuscript was keyboarded, many times over, by Ms. M. Yuasa, and some parts were done by Ms. Y. Fukushima, to both of whom I am greatly indebted. Finally, I thank my wife, Yoshiko, who has always been a source of encouragement.

Tokyo, Japan