Modelling Non-Stationary Time Series: A Multivariate Approach

261 Pages·2005·1.713 MB·English

by  Simon P. Burke, John Hunter (auth.)

Modelling Non-Stationary Time Series
Palgrave Texts in Econometrics
Series Editor: Kerry Patterson
Titles include:
Simon P. Burke and John Hunter
MODELLING NON-STATIONARY TIME SERIES
Michael P. Clements
EVALUATING ECONOMETRIC FORECASTS OF ECONOMIC AND FINANCIAL
VARIABLES
Terence C.Mills
MODELLING TRENDS AND CYCLES IN ECONOMIC TIME SERIES
Kerry Patterson
UNIT ROOTS IN ECONOMIC TIME SERIES
Jan Podivinsky
MODELLING VOLATILITY
Palgrave Texts in Econometrics
Series Standing Order ISBN 978-1-4039-0172-9 Hardcover
Series Standing Order ISBN 978-1-4039-0173-6 Paperback
(outside North America only)
You can receive future titles in this series as they are published by placing a standing order. Please
contact your bookseller or, in case of difficulty, write to us at the address below with your name and
address, the title of the series and the ISBN quoted above.
Customer Services Department, Macmillan Distribution Ltd, Houndmills, Basingstoke, Hampshire
RG21 6XS, England.
Modelling Non-Stationary
Time Series:
A Multivariate Approach
Simon P. Burke and John Hunter
© Simon P. Burke and John Hunter 2005
All rights reserved. No reproduction, copy or transmission of this
publication may be made without written permission.
No paragraph of this publication may be reproduced, copied or transmitted
save with written permission or in accordance with the provisions of the
Copyright, Designs and Patents Act 1988, or under the terms of any licence
permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham
Court Road, London W1T 4LP.
Any person who does any unauthorized act in relation to this publication
may be liable to criminal prosecution and civil claims for damages.
The authors have asserted their rights to be identified as the authors of this work in
accordance with the Copyright, Designs and
Patents Act 1988.
First published 2005 by
PALGRAVE MACMILLAN
Houndmills, Basingstoke, Hampshire RG21 6XS and
175 Fifth Avenue, New York, N. Y. 10010
Companies and representatives throughout the world
PALGRAVE MACMILLAN is the global academic imprint of the Palgrave
Macmillan division of St. Martin’s Press, LLC and of Palgrave Macmillan Ltd.
Macmillan® is a registered trademark in the United States, United Kingdom
and other countries. Palgrave is a registered trademark in the European
Union and other countries.
ISBN 978-1-4039-0203-0 ISBN 978-0-230-00578-5 (eBook)
DOI 10.1057/9780230005785
This book is printed on paper suitable for recycling and made from fully managed and
sustained forest sources.
A catalogue record for this book is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Burke, Simon P.
Modelling non-stationary economic time series : a multivariate approach / by
Simon P. Burke and John Hunter.
p. cm. – (Palgrave texts in econometrics)
Includes bibliographical references and index.
1. Econometric models. 2. Time-series analysis. I. Title. II. Series.
HB141.B866 2004
330¢.01¢51955–dc22 2004056896
10 9 8 7 6 5 4 3 2 1
14 13 12 11 10 09 08 07 06 05
Contents
Preface vii
1 Introduction: Cointegration, Economic Equilibrium and the 1
Long Run
2 Properties of Univariate Time Series 8
2.1 Introduction 8
2.2 Non-stationarity 8
2.3 Univariate statistical time series models and
non-stationarity 15
2.4 Testing for non-stationarity in single series 31
2.5 Conclusion 37
3 Relationships Between Non-Stationary Time Series 38
3.1 Introduction 38
3.2 Equilibrium and equilibrium correction 38
3.3 Cointegration and equilibrium 47
3.4 Regression amongst cointegrated variables 62
3.5 Conclusion 66
4 Multivariate Time Series Approach to Cointegration 69
4.1 Introduction 69
4.2 The VMA, the VAR and VECM 71
4.3 The Smith–McMillan–Yoo form 78
4.4 Johansen’s VAR representation of cointegration 89
4.5 Johansen’s approach to testing for cointegration in systems 97
4.6 Tests of cointegration in VAR models 105
4.7 Alternative representations of cointegration 118
4.8 Conclusion 126
5 Exogeneity and Identification 128
5.1 An introduction to exogeneity 129
5.2 Identification 137
5.3 Exogeneity and identification 151
54 Empirical examples 154
5.5 Conclusion 156
6 Further Topics in the Analysis of Non-Stationary Time Series 159
6.1 Introduction 159
6.2 Inference and estimation when series are not I(1) 160
v
vi Contents
6.2 Forecasting in cointegrated systems 173
6.3 Models with short-run dynamics induced by expectations 188
6.4 Conclusion 198
7 Conclusions: Limitations, Developments and Alternatives 200
7.1 Approximation 200
7.2 Alternative models 201
7.3 Structural breaks 201
7.4 Last comments 202
Notes 203
Appendix A Matrix Preliminaries 215
A.1 Elementary Row Operations and Elementary Matrices 215
A.2 Unimodular Matrices 216
A.3 Roots of a Matrix Polynomial 216
Appendix B Matrix Algebra for Engle and Granger (1987) Representation 217
B.1 Determinant/Adjoint Representation of a Polynomial Matrix 217
B.2 Expansions of the Determinant and Adjoint about z ∈[0, 1] 217
B.3 Drawing out a Factor of z from a Reduced Rank Matrix Polynomial 218
Appendix C Johansen’s Procedure as a Maximum Likelihood Procedure 219
Appendix D The Maximum Likelihood Procedure in Terms of Canonical 223
Correlations
Appendix E Distribution Theory 225
E.1 Some Univariate Theory 225
E.2 Vector Processes and Cointegration 226
E.3 Testing the Null Hypothesis of Non-Cointegration 227
E.4 Testing a Null Hypothesis of Non-zero Rank 228
E.5 Distribution Theory when there are Deterministic Trends in the Data 231
E.6 Other Issues 233
Appendix F Estimation Under General Restrictions 235
Appendix G Proof of Identification based on an Indirect Solution 237
Appendix H Generic Identification of Long-run Parameters in Section 5.5 239
References 240
Index 250
Preface
This book deals with an analysis of non-stationary time series that has been
very influential in applied research in econometrics, economics and finance.
The notion that series are non-stationary alters the way in which series are
grouped and may even prove to be relevant to some aspects of regulation and
competition policy when the definition of market becomes an economic issue.
The later might also apply to any discussion of the nature of globalized
financial markets. In terms of econometric and statistical theory an enormous
literature has grown up to handle the behaviour of the different forms of per-
sistence and non-stationary behaviour that economic and financial data
might exhibit. This is emphasized by the Nobel Prize that has been presented
to Clive Granger and Robert Engle in relation to their extension of our under-
standing of the way in which non-stationary series behave. However, the
requirement to analyze non-stationary behaviour has spawned a wide range of
approaches that relate to and interrelate with the notion that series are non-
stationary and/or cointegrated.
It has been our privilege to in some part be involved in these developments
and to have learned much from our colleagues and teachers alike. We must
acknowledge our debt of gratitude to those who taught us and supervised us
over the years. We would also like to thank participants at various Econo-
metrics Society Conferences, EC2, Econometrics Study Group Conferences
held at the Burwells campus of Bristol University and participants in the
Econometrics workshop for their incites, comments and stimulating research.
Dr. Lindsey Anne Gillan also provided us with some guidance through the
potential minefied that is academic publishing. However, all errors are our
own.
SIMONP. BURKE
JOHNHUNTER
vii
1
Introduction: Cointegration, Economic
Equilibrium and the Long Run
The econometrician or statistician might be viewed as a forensic scientist,
trying to detect from the splatter (of blood), a line through space from which
it may be determined, how and by whom a crime was committed. The tools
available to calculate and describe this evidence are estimators and tests, and
then – conditional on the model selected – identification of the cause or the
perpetrator of the crime.
At the very core of econometrics lies measurement, the quality of measure-
ment and the existence of the measure. When a measure is considered then
there is the practical question of whether measurement is feasible or not.
Conventional statistical measurement and inference considered the behaviour
of processes that are associated with distributions that are generally viewed as
being fixed across the sample. When economists started to apply statistical
measurement to economic data then the notion that the data were identically
and independently distributed (IID) had to be rejected. Regression was used to
measure the heterogeneity by estimating a mean conditional on exogenous
information while the assumption that the data are independently and identi-
cally distributed (IID), was used to give structure to the unknown error in the
model. Essentially some form of least squares regression became the method
generally applied to explain economic phenomena, but in the early literature
it is hard to find reference to the notion of non-stationarity. One exception is
the book written by Herman Wold with Lars Jureen on the subject of demand
analysis, which does consider the behaviour of stationary economic time
series. However, Wold and Jureen (1953) analyzed data for the inter-war years,
a period when price series fell in relative terms and growth of output was rela-
tively stagnant. Hence, any question of how demand models might be derived
when time series are non-stationary was apart from some exceptions ignored.
It is of interest to note that, in a study of the demand for food, James Tobin
estimated both a logarithmic inverse demand curve and in an attempt to
remove serial correlation the same relationship in differences. The latter
1
2 Modelling Non-Stationary Time Series
equation became the basis of the Rotterdam model developed by Theil (1965)
and Barten (1969). In the early 1970s, Box and Jenkins wrote a book that
became highly influential in the statistical analysis of time series data. Box
and Jenkins set out a methodology for building time series models, that firstly
considers the appropriate degree of differencing required to render a series sta-
tionary, and then discusses the type of alternative models autoregressive (AR)
or moving average (MA), or ARMA that might be used to estimate univariate
time series and then considered the method of estimation. Fama (1970) sug-
gests that the observation that financial time series follow random walks is
consistent with the idea that markets were efficient. The random walk model
implies that financial time series are non-stationary and, following Box and
Jenkins, need to be differenced to make them stationary. The difference in the
log of the share price approximates a return and when the financial market is
efficient then returns are not supposed to be predictable.
The structure of time series models pre-dates Box and Jenkins. Yule (1927)
first estimated AR processes and in 1929 Kolmogorov considered the behav-
iour of sums of independent random variables (see the discussion in Wold and
Jureen (1953)). In the regression context, Sargan (1964) applied an MA error
structure to a dynamic model of UK wage inflation. The Sargan model became
the basis of most of the UK wage equations used in the large macroeconomic
models (Wallis et al. 1984). In demand analysis, approximation rather than
non-stationarity was behind differencing and developments in economic
theory related to the structure of demand equations was more interested in
issues of aggregation as compared with the possible time series structure of the
data (Deaton and Muellbauer 1980). To difference time series became
common practice in modelling univariate time series and this approach was
also applied in finance where it was common to consider returns of different
assets rather than share prices. The market model relates the return on a share
to the return on the market. There was now a discrepancy between the
methods applied in statistics and finance to time series data and the approach
predominantly used by economists.
However, the first oil shock precipitated a crisis in macroeconomic model
building. Most of the world’s large macroeconomic models were unable to
resolve many of the problems that ensued from this shock. Forecasts and
policy simulations that provide the governments’ predictions of the future
and a practical tool for understanding the impact of policy on the economy
were unable to explain what had happened and what policies might remedy
the situation (Wallis et al. 1984). The UK Treasury’s inability to forecast the
balance of payments position led to the ludicrous situation of a developed
economy being forced to borrow from the IMF – a remedy that would not
have been sought had reasonable estimates been available of the true pay-
ments position. The whole approach to the econometric modelling of eco-
nomic time series was in doubt.