Lecture Notes in Economics and Mathematical Systems

Transcription

1 Lecture Notes in Economics and Mathematical Systems 407 Founding Editors: M. Beckmann H. P. Kiinzi Editorial Board: H. Albach, M. Beckmann, G. Feichtinger, W. Hildenbrand, W. Krelle H. P. Kiinzi, K. Ritter, U. Schittko, P. Schonfeld, R. Selten Managing Editors: Prof. Dr. G. Fandel Fachbereich Wirtschaftswissenschaften Fernuniversitat Hagen Feithstr. l40/avz II, D Hagen, FRG Prof. Dr. W. Trockel Institut fiir Mathematische Wirtschaftsforschung (IMW) Universitat Bielefeld Universitatsstr. 25, D-336l5 Bielefeld, FRG

2 Marius Ooms Empirical Vector Autoregressive Modeling Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

3 Author Dr. Marius Ooms Department of Econometrics Erasmus University Rotterdam P. O. Box 1738 NL-3000 DR Rotterdam The Netherlands The Foundation for the Promotion of Research in Economic Sciences (ECOZOEK), which is part of the Netherlands Organization for Scientific Research (NWO), supported this study financially. ISBN-l3: e-isbn-13: DOl: / Library of Congress Cataloging-in-Publication Data. Ooms, Marius, Empirical vector autoregressive modeling / Marius. p. cm. - (Lecture notes in economics and mathematical systems; 407) (Springer-Verlag New York Berlin Heidelberg: acid-free paper) - (Springer-Verlag Berlin Heidelberg New York: acid-free paper). I. Econometric models. 2. Autoregression (Statistics) I. Title. II. Series. HB '.01 '5195- dc This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1994 Typesetting: Camera ready by author SPIN: / Printed on acid-free paper

4 1 O.q. 'Segdu pat it nfunda, allz pik svinnan kveda ok pil, VafprUdnir, vitir: hvadan vindr um k\ilmr, sa er ferr vag yfir? re madr hann sialfan um ser.' V.q. 'Hrresvelgr heitir, i9tunn, f arnar ham; af hans vrengiom alia menn yfir.' er sitr a himins enda, kveda vind koma ISource: VafprUdnismal, Edda (1927, pp ). (O.q.: 6dinn kvad) 6dinn spoke 'Say this ninth, if your sagacity is sufficient and you, Vafprildnir, know: where does the wind come from, which wanders over the waves? never does a human see him himself.' (V.q.: VafprUdnir kvad) VafprUdnir spoke 'His name is Hrresvelgr, who sits at the end of heaven, a giant, in an eagle's shape; from his wings one says the wind comes over all men.'

5 ACKNOWLEDGEMENTS Several people deserve special credit for their assistance in the creation of this book. First I have to thank my supervisor Teun Kloek for formulating the research project which formed the basis of this study, for his advice and for his careful time-consuming reading effort. His sense for quality and perfection and his patience have been a great help. I thank the members of the reading committee of my Ph.D. thesis, especially Franz Palm, for their participation and for their comments on the manuscript. Jan Kiviet gave me the idea to embark on this project. I learned a lot from my colleagues at the econometrics department, who were so friendly to put up with me in this period. Herman van Dijk enthusiastically arranged numerous contacts with inspiring econometricians from all over the world. He was co-author of a paper which encompasses 6.5. Philip-Hans Franses introduced an effective sense for organization and management. Peter Schotman kept me in touch with fashions in theory and practice of empirical econometrics during many valuable conversations. He made a respectable career, which opened up my current job. My computer skills significantly benefited from advice and tools of Jurgen Doornik, Gerrit Draisma and Arjen Merckens. I am grateful to the Foundation for the Promotion of Research in Economic Sciences (ECOZOEK) for financial support during four years.

6 CONTENTS List of figures List of tables Chapter 1 Introduction 1.1 Integrating results 1.2 Goal of the study 1.3 Data and measurement model 1.4 Baseline model and methodology 1.5 Outline of the study 1.6 What is new? XII XIV page The Unrestricted V AR and its components Introduction The model Univariate processes and unit roots Integrated processes Definitions and notation MA representation, autocorrelation and pseudo spectrum Alternative models for nonstationarity, long memory and persistence Nonstationarity Long memory, the variance time function and adjusted range analysis Persistence 32 Appendix A2.l MA representation integrated process 34 A2.1.1 MA representations 34 A2.1.2 Pseudo autocorrelation functions 35 Appendix A2.2 Univariate testing for unit root nonstationarity 37 A2.2.l The pure unit root case without deterministic terms 37 A Notation and model 37 A Discussion 39 A2.2.2 Deterministic terms and unknown residual autocorrelation 40 A2.2.2.l Generalization of the test regression 40 A Interesting null hypotheses, alternatives and tests 43 A The parameters 0; and Q; in (A2.2.11) and (A2.2.12) 46 A Test statistics and distributions 50

7 x Chapter A Evaluation of methods A Other approaches and some extensions Contents page Data Analysis by Vector Autoregression Introduction Data-oriented measures of influence Goal of the influence analysis Influence measures in regression Influence measures for dynamic and multiple equation models Other influence measures from multivariate analysis Diagnostic checking Choosing test statistics Theoretical consideration for choosing tests Practical considerations for choosing tests Dynamic specification of the mean Distribution of the disturbances Parameter constancy of dynamic and covariance parameters An alternative test for parameter stability Multivariate diagnostics A diagnostic for multivariate unit roots Consequences of "rejection" of the model 94 Appendix A3.1 Influence measures for the normal linear model 95 A3.1.1 Global influence measures 95 A3.1.2 Local influence measures 98 Appendix A3.2 Influence measures for the multivariate general linear model 100 Appendix A3.3 Influence measures in principal component analysis Seasonality Introduction Application of the idea of unobserved components Application of linear filters to estimate unobserved components Optimal extraction in multivariate series Optimal extraction in nonstationary series Specification of low dimensional univariate models Optimal extraction in a finite sample Optimal extraction in the presence of outliers 120

8 Contents XI Chapter page 4.4 Data analysis of the seasonal component Application of the Census X-ll filter in a VAR 123 Appendix 4.1 Trigonometric seasonal processes in regression 128 A4.Ll Notation and underlying model 128 A4.1.2 Zero correlation between seasonal patterns 128 A4.1.3 Circularity: Unit correlation between seasonal patterns 130 Appendix 4.2 Backforecasts and deterministic changes in mean 131 A4.2.1 Introduction 131 A4.2.2 Backforecasting and deterministic changes in mean with linear trends 131 A4.2.3 Backforecasting and deterministic changes in mean with seasonal dummies 133 A4.2.4 Changes in mean in multivariate model with unit roots Outliers Introduction The outlier model Some effects of outliers on VAR estimates Effect of outliers on unit root tests Effect of outliers on estimates of ~ Derivation of the LM-statistics Case of known parameters and timing Case of estimated parameters and unknown timing Distinguishing between outlier types Distinguishing between outliers in different equations An artificial example Application to macroeconomic series Two simple ways to study the influence of outliers 185 Appendix 5.1 Some proofs concerning outlier test statistics 193 A5.Ll Derivation simultaneous test 193 A5.1.2 Finite sample alternatives for I test procedure 195 Appendix 5.2 Subsample analysis outlier influence 196 Appendix 5.3 Robust estimation by extraction of additive outliers Restrictions on the V AR Introduction Cointegration, the number of unit roots, and common trends 206

9 XII Contents Chapter page Vector error correction Other parameterizations Straightforward transformation formulae From Campbell-Shiller to vector error correction From vector error correction to Campbell-Shiller, mean 215 growth From vector error correction to common trends Examples Conditions for VECM, I(2)-ness, and explosive systems Trend stationary processes and quadratic trends Estimating pushing trends and pulling equilibria Deterministic trends Estimating the stochastic part of the trend Estimating pulling equilibria Multivariate tests for unit roots Models with p = 1 and zero mean Deterministic terms and serial correlation in AR(l) residuals 239 Appendix 6.1 Computation and distribution multivariate unit root test statistics A6.I.1 Computation A6.1.2 Distribution Applied V AR Analysis for Aggregate Investment Introduction The variable of interest and some of its supposed relationships Theoretical relationships Empirical models Measurement model Investment in the national accounts Definition of investment Other macroeconomic price indexes Univariate analysis The variables Graphs and influence analysis Representations of the autocorrelation function Adjusted range techniques 266

10 Contents Chapter Application Results Outliers Autocorrelations Long memory analysis Data analysis seasonal components Variance time functions Statistical unit root analysis Parameter stability Summary of univariate results 7.5 Multivariate analysis Predictions and seasonality in the unrestricted VAR Unit root analysis Detecting a structural break The final model Appendix 7.1 Data sources and construction Appendix 7.2 Results of final VECM model Appendix 7.3 Open economy stochastic dynamic general equilibrium models Summary References Name index Subject index XIII page