Introduction to Multiple Time Series Analysis

556 Pages·1991·12.27 MB·English

by  Prof. Dr. Helmut Lütkepohl (auth.)

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Helmut Lütkepohl
Introduction
to Multiple
Time Series
Analysis
With 34 Figures
Springer-Verlag Berlin Heidelberg GmbH
Prof. Dr. Helmut Lütkepohl
Institute of Statistics and Econometrics
University ofKiel
Olshausenstraße 40
2300 Kiel, FRG
Library of Congress Cataloging-in-Publication Data
Lütkepohl, Helmut.
Introduction to multiple time series analysis / Helmut Lütkepohl.
p.cm.
Includes bibliographical references and indexes.
1. Time-series analysis. I. TitJe.
QA280.L87 1991
519.5'5--dc20
This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is
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fall under the prosecution act ofthe German Copyright Law.
ISBN 978-3-540-53194-4 ISBN 978-3-662-02691-5 (eBook)
DOI 10.1007/978-3-662-02691-5
© Springer-Verlag Berlin Heidelberg 1991
Originally published by Springer-Verlag Berlin Heidelberg New York in 1991.
The use ofr egistered names, trademarks, etc. in this publication does not imply, even in the absence
of a specific statement, that such names are exempt from the relevant protective laws and regulati
ons and therefore free for general use.
214217130-543210
To Sabine
Preface
A few years ago, when I started teaching a course on multiple time series
analysis for graduate students in business and economics at the University of
Kiel, Germany, a suitable textbook was not available. Therefore I prepared
lecture notes from which the present text evolved. There are a number of books
on time series analysis that contain portions on multiple time series models and
methods. However, none of them covers the fuH range of models and methods
that are now commonly used in the economics and econometrics literature. The
present text also neglects some tools which are currently in widespread use. It
goes beyond other time series books, however. Unlike advanced books on the
topic such as Hannan & Deistler (1988) this text is written at a level which should
be accessible to graduate students in business and economics. The issues and
models covered in detail reflect my personal interests. ladmit that some impor
tant models and methods are omitted just because I havn't used them much in
the past. EspeciaHy the omission of spectral methods may perhaps be regreted
by some instructors.
Although multiple time series analysis is applied in many disciplines I have
prepared the text with economics and business students in mind. The examples
and exercises are chosen accordingly. Despite this orientation I hope that the
book will also serve multiple time series courses in other fields.1t contains enough
material for a one semester course on multiple time series analysis. It mayaIso
be combined with univariate time series books such as Schlittgen & Streitberg
(1984) and Pankratz (1983) or with texts like FuHer (1976) to form the basis of a
one or two semester course on univariate and multivariate time series analysis.
Alternatively, it is also possible to select some of the chapters or sections for a
special topic of a graduate level econometrics course.
Chapters 1-4 contain an introduction to the vector !lutoregressive methodol
ogy. Chapters 1-9 may be used as an introduction to vector autoregressive and
mixed autoregressive moving average models. Chapter 10 briefly reviews econo
metric dynamic simultaneous equations models; Chapter 11 considers the curr
ently popular cointegration topic; in Chapter 12 models with systematicaHy
varying coefficients are treated, and state space models are discussed in Chapter
13. In a course or special topic on vector autoregressive models all or some of
Chapters 11-13 may be treated right after Chapter 5. It is also possible to cover
Chapters 1-5 and 11 first and then proceed with Chapters 6-9 and, if time
permits, conclude a course with Chapters 12 and 13.
VIII Preface
The students participating in my multiple time series course typically have
knowledge of matrix algebra. They also have been introduced to mathematical
statistics, for instance, based on textbooks like Mood, Graybill & Boes (1974),
Hogg & Craig (1978), or Rohatgi (1976). Moreover, many ofthem have a working
knowledge of the Box-Jenkins approach and other univariate time series tech
niques. Although, in principle, it may be possible to use the present text without
any prior knowledge of univariate time series analysis if the instructor provides
the required motivation, it is clearly an advantage to have some time series
background. Also, a previous introduction to econometrics will be helpful.
Matrix algebra and an introductory mathematical statistics course plus the
multiple regression model are necessary prerequisites.
Since this is meant to be an introductory exposition I am not striving for
utmost generality. For instance, quite often I use the normality assumption
although the considered results hold under more general conditions. The em
phasis is on explaining the underlying ideas and not on generality. In Chapters
2-5 a number of results are proven to illustrate some of the techniques that are
often used in the multiple time series arena. Most proofs may be skipped without
loss of continuity. Therefore the beginning and the end of a proof are usually
clearly marked. Many results are summarized in propositions for easy reference.
A listing of the propositions is provided at the end of the book.
The appendices contain a collection ofuseful results on matrix algebra, multi
variate normal distributions, asymptotic theory and so on. Some results and
topics were added in response to questions and comments by students. It is not
necessary to know all of these results before going through the chapters of the
book. It may be useful, however, to have them nearby because many of them are
used and referred to in the main body of the text. Therefore they are included in
the appendix.
Exercises are given at the end of each chapter with the exception of Chapter
1. Some ofthe problems may be too difficult for students without a good formal
training. Some are just included to avoid details of proofs given in the text. In
most chapters empirical exercises are provided in addition to algebraic problems.
Solving the empirical problems requires the use of a computer. Matrix oriented
software such as GA USS, MA TLAB, or SAS will be most helpful. A menu driven
program based on GAUSS is now being developed which can do most of the
examples and exercises.
Many persons have contributed directly or indirectly to this book and I
am very grateful to all of them. Many students have commented on my lecture
notes. Thereby they have helped to improve the presentation and to correct
errors. A number of colleagues have commented on portions of the manuscript
and have been available for discussions on the topics covered. These comments
and discussions have been very helpful for my own understanding of the subject
and have resulted in improvements to the manuscript.
Although the persons who have contributed to the project in some way or
other are too numerous to be listed here I wish to express my special gratitude
to some of them. Special thanks go to Theo Dykstra who read and commented
Preface IX
on a large part of the manuscript during his visit in Kiel in the summer of 1990.
Hans-Eggert Reimers read the entire manuscript, suggested many improvements,
and pointed out numerous errors. Wolfgang Schneider helped with the example
in Chapter 13 and also commented on parts of the manuscript. Bernd Theilen
prepared the final versions of most figures, Knut Haase and Holger Claessen
performed the computations for the examples. I deeply appreciate the help of all
these collaborators. Last but not least I owe an obligation to Mrs. Milda
Tauchert who performed the difficult technical typing job. She skillfully typed
and retyped numerous versions of the manuscript. Of course, I assume full
responsibility for any remaining errors and I welcome any comments by readers.
Kiel, August 1990 Helmut Lütkepohl
Table of Contents
Chapter 1. Introduction ....................................... 1
1.1 Objectives of Analyzing Multiple Time Series ................ 1
1.2 Some Basics ............................................ 2
1.3 Vector Autoregressive Processes ........................... 3
1.4 Outline of the Following Chapters ......................... 5
Part I. Finite Order Vector Autoregressive Processes 7
Chapter 2. Stable Vector Autoregressive Processes ................. 9
2.1 Basic Assumptions and Properties of VA R Processes .......... 9
2.1.1 Stable VA R(p) Processes ........................... 9
2.1.2 The Moving Average Representation of a VAR Process. 13
2.1.3 Stationary Processes ............................... 19
2.1.4 Computation of Autocovariances and Autocorrelations of
Stable VA R Processes .............................. 21
2.1.4a Autocovariances of a VAR(1) Process .......... 21
2.1.4b Autocovariances of a Stable VAR(p) Process 23
2.1.4c Autocorrelations of a Stable VA R(p) Process ... 25
2.2 Forecasting ............................................ 27
2.2.1 The Loss Function ................................ 27
2.2.2 Point Forecasts ................................... 28
2.2.2a Conditional Expectation ..................... 28
2.2.2b Linear Minimum MSE Predictor ............. 30
2.2.3 Interval Forecasts and Forecast Regions .............. 33
2.3 Structural Analysis with VAR Models ...................... 35
2.3.1 Granger-Causality and Instantaneous Causality ........ 35
2.3.1a Definitions of Causality ..................... 35
2.3.1 b Characterization of Granger-Causality ......... 37
2.3.1c Characterization ofInstantaneous Causality .... 40
2.3.1d Interpretation and Critique of Instantaneous and
Granger-Causality .......................... 41
XII Table of Contents
2.3.2 Impulse Response Analysis .......................... 43
2.3.2a Responses to Forecast Errors ................. 43
2.3.2b Responses to Orthogonal Impulses ............ 48
2.3.2c Critique of Impulse Response Analysis ......... 55
2.3.3 Forecast Error Variance Decomposition .............. 56
2.3.4 Remarks on the Interpretation of VA R Models ......... 58
2.4 Exercises ............................................... 59
Cbapter 3. Estimation of Vector Autoregressive Processes .......... 62
3.1 Introduction............................................ 62
3.2 Multivariate Least Squares Estimation ..................... 62
3.2.1 The Estimator .................................... 62
3.2.2 Asymptotic Properties of the Least Squares Estimator ... 65
3.2.3 An Example ...................................... 70
3.2.4 Small Sam pie Properties of the LS Estimator .......... 73
3.3 Least Squares Estimation with Mean-Adjusted Data and Yule-
Walker Estimation ...................................... 75
3.3.1 Estimation when the Process Mean Is Known ......... 75
3.3.2 Estimation of the Process Mean ..................... 76
3.3.3 Estimation with Unknown Process Mean ............. 78
3.3.4 The Yule-Walker Estimator ......................... 78
3.3.5 An Example ...................................... 79
3.4 Maximum Likelihood Estimation .......................... 80
3.4.1 The Likelihood Function ........................... 80
3.4.2 The ML Estimators ................................ 81
3.4.3 Properties of the ML Estimators ..................... 82
3.5 Forecasting with Estimated Processes ...................... 85
3.5.1 General Assumptions and Results .................... 85
3.5.2 The Approximate MSE Matrix ...................... 87
3.5.3 An Example ...................................... 89
3.5.4 A Small Sampie Investigation ....................... 91
3.6 Testing for Granger-Causality and Instantaneous Causality .... 93
3.6.1 A Wald Test for Granger-Causality .................. 93
3.6.2 An Example ...................................... 94
3.6.3 Testing for Instantaneous Causality .................. 95
3.7 The Asymptotic Distributions ofImpulse Responses and Forecast
Error Variance Decompositions ........................... 97
3.7.1 The Main Results ................................. 97
3.7.2 Proof of Proposition 3.6 ............................ 103