Table Of ContentLecture Notes in Economies
and Mathematical Systems 488
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Springer-Verlag Berlin Heidelberg GmbH
B jöm Schmo1ck
Omitted Variable Tests
and
Dynamic Specification
An Application to Demand Homogeneity
Springer
Author
Dr. Björn Schmolck
Swiss Reinsurance Company
Economic Research & Consulting
Mythenquai 50/60
CH-8022 Zurich, Switzerland
E-mail: [email protected]
Library of Congress Cataloging-in-Publication Data
Schmolck, Bjöm, 1968-
Omitted variable tests and dynamic specification : an application to demand
homogeneity / Bjöm Schmolck.
p. cm. -- (Lecture notes in economics and mathematical systems, ISSN 0075-8442 ; 488)
Thesis (ph.D.)--University ofFribourg, Switzerland, 2000?
Includes bibliograpbical references.
ISBN 978-3-540-67358-3 ISBN 978-3-642-58324-7 (eBook)
DOI 10.1007/978-3-642-58324-7
1. Demand (Economic theory) 2. Time-series analysis. 3. Regression analysis. I. Title.
ll. Series.
HB842 .S36 2000
519.5'36--dc21
00-038753
ISSN 0075-8442
ISBN 978-3-540-67358-3
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Preface
This book deals with the omitted variable tests for a multivariate time-series
regression model. What are the consequences of testing for the omission of
a variable when the model is dynamically misspecified? What is the small
sample bias of the omitted variable test when the model dynamics is correctly
specificfied? The answers to these questions are proposed in this book. As an
empirical illustration, the analysis is applied to the homogeneity test of a
demand system.
I particularly thank Professor Dr. Philippe J. Deschamps who draw my
attention on this subject and who made very helpful comments and sugges
tions. Additionally, I would like to thank Professor Dr. Reiner Wolff for his
comments especially on the chapter dealing with consumer theory. Special
thanks go to Maria Jose Redondo, who read this book several times and for
the inspiring discussions with her. I would also like to thank Dr. Ali Vak
ili (always ready to answer any questions in mathematics), Prof. Dr. Hans
Wolfg ang Brachinger, Curzio De Gottardi, Peter Mantsch, Dr. Paul-Andre
Monney, Dr. Uwe Steinhauser, Leon Stroeks and Dr. Peter Windlin. Frances
Angell improved the English of this work.
The research for this book had been financially suppürted by the Univer
site de Fribourg (Switzerland).
Finally, I appreciated the support from Springer-Verlag and I thank Dr.
Werner A. Müller and Ruth Milewski für their friendly collaboration. An
anonymous referee from Springer-Verlag made helpful comments.
Abstract
The purpose of this work is to study the omitted variable test for a multivari
ate regression model. The empirical motivation for this is the homogeneity
test for a demand system; the hypothesis of homogeneity can be formulated
as the hypothesis of an omitted variable.
The exact distribution of the test statistic for homogeneity is only known
for static demand systems. The static models are, however, very restrictive
for time-series models. It is, therefore, interesting to study the consequences
of dynamic misspecification for the omitted variable test. In most of our
examples the homogeneity test is biased substantially towards rejection. This
result lllustrates the importance of specifying the dynamics of the demand
system correctly.
In order to take the dynamics of the demand system into account, we
analyse two classes of dynamically correctly specified tests for homogeneity:
robust Wald tests and versions of the likelihood ratio test. Since the null dis
tribution of the related test statistics is only known asymptotically, the small
sampie performance of these tests is studied by Monte Carlo experimenta
tion. In our examples, the robust Wald test and the usual likelihood-ratio
test perform badly and the null hypothesis of homogeneity is rejected too
often in most of the cases. As a remedy we propose a bootstrap version of
the likelihood ratio test which has an excellent performance.
Table of Contents
1. Introduction.............................................. 1
2. The t-statistic under dynamic misspecmcation . . . . . . . . . . . . 5
2.1 The model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Properties of the estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 The distribution of the quasi t-statistic . . . . . . . . . . . . . . . . . . .. 10
2.4 Invariance results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14
2.5 Monte Carlo experimentation . .... .. .... .. .... .... .. ..... 14
2.5.1 Simulating the distribution of the t-statistic ......... 15
2.5.2 Presentation of the results . . . . . . . . . . . . . . . . . . . . . . . .. 17
3. Consumer theory and the Rotterdam model. . . . . . . . . . . . .. 23
3.1 Commodity space and budget set......................... 23
3.2 Preferences, direct utility function and Marshallian demand .. 23
3.3 Cost function and Hicksian demand ...................... 24
3.4 The Rotterdam model .................................. 25
3.5 The Rotterdam model in matrix form. . . . . . . . . . . . . . . . . . . .. 29
4. Robust estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31
4.1 Quasi-maximum likelihood estimation.. .. . . .... .... .. ..... 31
4.2 Estimation of the covariance matrix of the quasi-maximum
likelihood estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36
4.2.1 Estimation of ~ if the errors are homoskedastic and
serially uncorrelated . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37
4.2.2 Estimation of ~ if the errors are dependent and het-
erogeneous ...................................... 39
5. Testing for homogeneity .................................. 47
5.1 Anderson's U test if the errors are
time-independent (LRU) ................................ 47
5.2 Functional equivalence between the LRU and Laitinen's statis-
tic.. . .. . . .. .. .... .... .. .... . . .. .. . . .. .. .... .. ...... . .. 51
5.3 Likelihood ratio test if the errors are VAR(p) .............. 55
5.3.1 The likellhood ratio test (LR) ... .... ...... ... ... ... 55
X Table of Contents
5.3.2 Small sampie corrected likelihood ratio test (LRC). . .. 57
5.3.3 A Monte Carlo test (LR-MC) . . . . . . . . . . . . . . . . . . . . .. 59
5.4 The distribution of the LRU statistic under dynamic mis-
specification ........................................... 61
5.5 The robust Wald test ................................... 71
5.6 Summary.............................................. 73
6. Monte Carlo experimentation. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75
6.1 Data .................................................. 75
6.2 The datar-generating process .. . ....... ...... .... ...... ... 77
6.2.1 Definition of the datar-generating processes . . . . . . . . . .. 78
6.2.2 Computational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80
6.3 Experiments........................................... 82
6.3.1 Estimating the test's true type I error by simulation.. 83
6.3.2 Bias of the misspecified LRU test. . . . . . . . . . . . . . . . . .. 84
6.3.3 The small sampie bias of the correctly specified likeli-
hood ratio test .................................. 87
6.3.4 The small sampie bias of the robust Wald test ....... 89
6.3.5 Summary of the experimental design. . . . . . . . . . . . . . .. 90
6.4 Simulation results ...................................... 93
6.4.1 Aspects of the presentation of the simulation results .. 93
6.4.2 Bias of the LRU test under dynamic misspecification.. 96
6.4.3 The small sampie bias of the correctly specified LR test 100
6.4.4 The small sampie bias of the robust Wald test ....... 104
6.4.5 Summary ........................................ 111
7. Conclusions ............................................... 113
A. Proof of proposition 5.4 .................................. 115
B. Data ...................................................... 119
C. Values of the population parameters ...................... 127
D. Algorithm for the MA(l) parameters ..................... 133
List of Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 135
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 135
Index ......................................................... 143
1. Introdu ction
This book deals with the omitted variable test for the multivariate time-series
regression model. The empirieal motivation for this is the homogeneity test
for a demand system; the hypothesis of homogeneity ean be formulated as the
hypothesis of an omitted variable. We are interested in studying the omitted
variable test under dynamic misspeeification and under eorreet speeifieation.
The exact distribution of the test statistic for homogeneity is only known
for statie demand systems. This is Laitinen's statistic whieh ean be inter
preted as the F-version of the Wald statistic. Sinee the statie models are very
restrictive for time-series models, it is interesting to study the eonsequenees
of dynamic misspeeification for the omitted variable test.
The analysis is applied to a restricted form of dynamie models which
faeilitates the interpretation of the results. We are interested in the omitted
variable test for the multivariate linear regression model of the form Yt =
BXt + Ut, where Yt is an x 1 veetor of endogenous variables, B is an x k
eoefficient matrix, Xt is a k x 1 veetor of exogenous regressors and Ut is a n x 1
veetor of serially eorrelated errors. The null hypothesis is that one eolumn
of the eoefficient matrix B equals zero. This is equivalent to testing that
one variable ean be omitted. Many empirical applieations ean be analysed
in this framework. Here we will study the homogeneity test for a demand
system. The demand system under eonsideration is the Rotterdam model.
More generally, an omitted variable test for the redueed form of an equation
system fits into this framework.
For univariate regression models, the distributions of some test statisties
have been analysed when. the errors are dynamieally misspecified. For the
multivariate regression model, however, the omitted variable test has not yet
been analysed under dynamie speeifieation to our knowledge. The test under
eonsideration is the F-version of the Wald test for a multivariate regression
model. This test is exaet if the regressors are exogenous and the errors are
normal, homoskedastie and independent in time. We study the type I error of
this Wald test when the error proeess is dynamically misspecified, for example
if the errors follow a veetor autoregressive or moving average proeess. It will
be shown that the Wald statistie is distributed asymptotically as a qUadratie
form in normal variables under quite general assumptions. The asymptotic
2 1. Introduction
distribution depends on the data-generating process and its functional form
is based on Imhof's formula.
In order to illustrate the bias of the omitted variable test under dynamic
misspecification, some examples will be presented. Here, the interest is to
study the bias of Laitinen's test under dynamic misspecification. For some
specific data-generating processes the true type I error is estimated by sim
ulation. The true data-generating processes are based on the homogeneity
constrained estimation of the Rotterdam model with annual and seasonal
data from the UK. The errors are assumed to follow a vector autoregressive
or moving average process. In our examples, the homogeneity test is biased
substantially towards rejection. The estimated rejection frequencies vary be
tween 4% and 46% for a given nominal type I error of 5%. This illustrates the
importance of using a dynamically correctly specified test for homogeneity.
It is, furthermore, one plausible reason why homogeneity has been rejected
in many empirical applications.
We analyse the small sampie performance of two dasses of correctly spec
ified tests for homogeneity: robust Wald tests and versions of the likelihood
ratio test. The small sampie performance of these tests will be analysed by
Monte Carlo experimentation.
The dass of robust Wald tests is based on the "heteroskedasticity and au
tocorrelation consistent" (HAC) variance-covariance matrix estimators which
have become quite popular in the recent years. These Wald tests are robust
in the sense that the Wald statistics have an asymptotic chi-square distribu
tion under fairly general conditions. The small sampie performance of these
tests will be investigated by simulation. In the literature, the small sampie
properties of these robust tests have only been studied for univariate and not
for multivariate regression models to the author's knowledge. Monte Carlo
experimentation suggests that the bias of these tests is extremely large and
increases with the number of equations of the model. The small sampie prop
erties of these tests, therefore, differ for univariate and multivariate regression
models. The condusion from the Monte Carlo experiments is that the robust
Wald tests should not be used when testing demand homogeneity in a time-
series models if working with annual or seasonal data. .
The other dass of tests for homogeneity is related to the likelihood ratio
test. In order to take the small sampie problem into account, two possible
remedies are proposed: a modified likelihood ratio statistic with a small sam
pIe correction factor and a Monte Carlo test. In the simulation study we have
found that the likelihood ratio test does not perform weIl. The small sampie
corrected version of this test works better but not satisfactorily. The bias
increases dramatically with the number of equations. The performance of the
Monte Carlo test, however, is excellent and, therefore, recommended.
Organisation of the book:
In chapter 2, a simple model is presented in order to study the omitted
variable test under dynamic misspecification. The model is a linear regression
