Modelling Non-Stationary Time Series: A Multivariate Approach PDF
Preview Modelling Non-Stationary Time Series: A Multivariate Approach
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.
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