Table Of ContentLecture Notes in Economics
and Mathematical Systems 407
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Marius Ooms
Empirical Vector
Autoregressive Modeling
Springer-Verlag
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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: 978-3-540-57707-2 e-ISBN-13: 978-3-642-48792-7
DOl: 10.1007/ 978-3-642-48792-7
Library of Congress Cataloging-in-Publication Data. Ooms, Marius, 1961- . Em
pirical vector autoregressive modeling / Marius. p. cm. - (Lecture notes in econom
ics 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 141.055 1994 330'.01 '5195-
dc20 93-46882
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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, er sitr a himins enda,
i9tunn, f arnar ham;
af hans vrengiom kveda vind koma
alia menn yfir.'
ISource: VafprUdnismal, Edda (1927, pp. 49-50).
(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.'
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.
CONTENTS
List of figures XII
List of tables XIV
Chapter page
1 Introduction 1
1.1 Integrating results 1
1.2 Goal of the study 2
1.3 Data and measurement model 3
1.4 Baseline model and methodology 4
1.5 Outline of the study 7
1.6 What is new? 9
2 The Unrestricted VA R and its components 11
2.1 Introduction 11
2.2 The model 12
2.3 Univariate processes and unit roots 15
2.4 Integrated processes 18
2.4.1 Definitions and notation 18
2.4.2 MA representation, autocorrelation and pseudo spectrum 20
2.5 Alternative models for nonstationarity, long memory and persistence 23
2.5.1 Nonstationarity 23
2.5.2 Long memory, the variance time function and adjusted range
analysis 24
2.5.3 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
A2.2.1.1 Notation and model 37
A2.2.1.2 Discussion 39
A2.2.2 Deterministic terms and unknown residual autocorrelation 40
A2.2.2.l Generalization of the test regression 40
A2.2.2.2 Interesting null hypotheses, alternatives and tests 43
A2.2.2.3 The parameters 0; and Q; in (A2.2.11) and (A2.2.12) 46
A2.2.2.4 Test statistics and distributions 50
x Contents
Chapter page
A2.2.2.5 Evaluation of methods 55
A2.2.2.6 Other approaches and some extensions 56
3 Data Analysis by Vector Autoregression 59
3.1 Introduction 59
3.2 Data-oriented measures of influence 59
3.2.1 Goal of the influence analysis 60
3.2.2 Influence measures in regression 61
3.2.3 Influence measures for dynamic and multiple equation models 67
3.2.4 Other influence measures from multivariate analysis 70
3.3 Diagnostic checking 71
3.3.1 Choosing test statistics 72
3.3.2 Theoretical consideration for choosing tests 73
3.3.3 Practical considerations for choosing tests 76
3.3.4 Dynamic specification of the mean 78
3.3.5 Distribution of the disturbances 80
3.3.6 Parameter constancy of dynamic and covariance parameters 82
3.3.7 An alternative test for parameter stability 88
3.3.8 Multivariate diagnostics 89
3.3.9 A diagnostic for multivariate unit roots 91
3.3.10 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 107
4 Seasonality 109
4.1 Introduction 109
4.2 Application of the idea of unobserved components 110
4.3 Application of linear filters to estimate unobserved components 113
4.3.1 Optimal extraction in multivariate series 113
4.3.2 Optimal extraction in nonstationary series 115
4.3.3 Specification of low dimensional univariate models 117
4.3.4 Optimal extraction in a finite sample 120
4.3.5 Optimal extraction in the presence of outliers 120
Contents XI
Chapter page
4.4 Data analysis of the seasonal component 120
4.5 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 136
5 Outliers 139
5.1 Introduction 139
5.2 The outlier model 140
5.3 Some effects of outliers on VAR estimates 142
5.3.1 Effect of outliers on unit root tests 142
5.3.2 Effect of outliers on estimates of ~ 144
5.4 Derivation of the LM-statistics 146
5.4.1 Case of known parameters and timing 146
5.4.2 Case of estimated parameters and unknown timing 149
5.4.3 Distinguishing between outlier types 150
5.4.4 Distinguishing between outliers in different equations 152
5.5 An artificial example 153
5.6 Application to macroeconomic series 171
5.7 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 198
6 Restrictions on the VA R 204
6.1 Introduction 204
6.2 Cointegration, the number of unit roots, and common trends 206
XII Contents
Chapter page
6.2.3 Vector error correction 209
6.2.4 Other parameterizations 211
6.3 Straightforward transformation formulae 212
6.3.1 From Campbell-Shiller to vector error correction 213
6.3.2 From vector error correction to Campbell-Shiller, mean 215
growth
6.3.3 From vector error correction to common trends 218
6.3.4 Examples 221
6.3.5 Conditions for VECM, I(2)-ness, and explosive systems 222
6.4 Trend stationary processes and quadratic trends 225
6.5 Estimating pushing trends and pulling equilibria 227
6.5.1 Deterministic trends 227
6.5.2 Estimating the stochastic part of the trend 228
6.5.3 Estimating pulling equilibria 232
6.6 Multivariate tests for unit roots 237
6.6.1 Models with p = 1 and zero mean 237
6.6.2 Deterministic terms and serial correlation in AR(l)
residuals 239
Appendix 6.1 Computation and distribution multivariate unit root test
statistics 242
A6.I.1 Computation 242
A6.1.2 Distribution 244
7 Applied VA R Analysis for Aggregate Investment 248
7.1 Introduction 248
7.2 The variable of interest and some of its supposed relationships 249
7.2.1 Theoretical relationships 249
7.2.2 Empirical models 251
7.3 Measurement model 256
7.3.1 Investment in the national accounts 257
7.3.2 Definition of investment 257
7.3.3 Other macroeconomic price indexes 259
7.4 Univariate analysis 260
7.4.1 The variables 260
7.4.2 Graphs and influence analysis 260
7.4.3 Representations of the autocorrelation function 265
7.4.4 Adjusted range techniques 266
Contents XIII
Chapter page
7.4.6 Application 269
7.4.7 Results 270
7.4.7.1 Outliers 270
7.4.7.2 Autocorrelations 271
7.4.7.3 Long memory analysis 274
7.4.7.4 Data analysis seasonal components 275
7.4.7.5 Variance time functions 276
7.4.7.6 Statistical unit root analysis 277
7.4.7.7 Parameter stability 279
7.4.7.8 Summary of univariate results 281
7.5 Multivariate analysis 281
7.5.1 Predictions and seasonality in the unrestricted VAR 281
7.5.2 Unit root analysis 290
7.5.3 Detecting a structural break 299
7.5.4 The final model 303
Appendix 7.1 Data sources and construction 325
Appendix 7.2 Results of final VECM model 326
Appendix 7.3 Open economy stochastic dynamic general equilibrium
models 328
Summary 329
References 333
Name index 350
Subject index 355
