Readings in econometric theory and practice : a volume in honor of George Judge

371 Pages·1992·18.868 MB·English

by  George G Judge, William E Griffiths, Helmut Lütkepohl, M E Bock

CONTRIBUTIONS
TO
ECONOMIC ANALYSIS
209
Honorary Editor:
J. TINBERGEN
Editors:
D. W. JORGENSON
J.J.-LAFFONT
NORTH-HOLLAND
AMSTERDAM · LONDON · NEW YORK · TOKYO
READINGS IN ECONOMETRIC
THEORY AND PRACTICE
A Volume in Honor of George Judge
Edited by:
W. E. GRIFFITHS
University of New England
Armidale, NSW
Australia
H. LÜTKEPOHL
Christian-Albrechts-Universität
Kiel
Germany
M. E. BOCK
Purdue University
West Lafayette, IN
USA
1992
NORTH-HOLLAND
AMSTERDAM · LONDON · NEW YORK · TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V.
Sara Burgerhartstraat 25
RO. Box 2n, 1000 AE Amsterdam
The Netherlands
Library of Congress Catalog1ng-in-Publ1cat1on Data
Readings In econometric theory and practice : a volume In honor of
George Judge / edited by W.E. Griffiths, H. L٧tkepohl, M.E. Bock,
p. cm. — (Contributions to economic analysis ; 209)
Includes bibliographical references and Index.
ISBN 0-444-89574-4 (hardback : a Ik. paper)
1. Econometrics. I. Judge, George G. II. Griffiths, William E.
III. L٧tkepohl, Helmut. IV. Bock, M. E. (Mary Ellen) V. Series.
HB139.R43 1992
330· .or 5195~dc20 92-15019
CIP
ISBN: 0 444 89574 4
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Chapter 4 was previously published in Journal of Econometrics, Vol. 37, No. 1, January 1988, pp. 27-50.
© 1988 Elsevier Science Publishers B.V. Reproduced with permission.
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v
INTRODUCTION TO THE SERIES
This series consists of a number of hitherto unpubhshed studies, which are
introduced by the editors in the beUef that they represent fresh contributions
to economic science.
The term "economic analysis" as used in the title of the series has been
adopted because it covers both the activities of the theoretical economist and
the research worker.
Although the analytical methods used by the various contributors are not the
same, they are nevertheless conditioned by the common origin of their
studies, namely theoretical problems encountered in practical research. Since
for this reason, business cycle research and national accounting, research
work on behalf of economic policy, and problems of planning are the main
sources of the subjects dealt with, they necessarily determine the manner of
approach adopted by the authors. Their methods tend to be "practical" in
the sense of not being too far remote from application to actual economic
conditions. In addition they are quantitative.
It is the hope of the editors that the publication of these studies will help to
stimulate the exchange of scientific information and to reinforce interna­
tional cooperation in the field of economics.
The Editors
vii
PREFACE
The purpose of this book is to honor George Judge and his many, varied
and outstanding contributions to econometrics, statistics, mathematical pro­
gramming and spatial equilibrium modeling. George Judge was born on May
2, 1925, he obtained his B.S. from the University of Kentucky in 1948, and his
M.S. (1949) and Ph.D. (1952) from Iowa State University. His first position
was as Assistant Professor in the Department of Agricultural Economics at
the University of Connecticut (1951-55). From there he went, as Professor,
to the Department of Agricultural Economics at Oklahoma State University
(1955-58); he was a Research Fellow in Economics at Yale University in
1958-59, and was Professor of Economics and Agricultural Economics at the
University of Illinois from 1959 to 1986. He moved to his present position of
Professor in the Department of Agricultural and Resource Economics at the
University of California, Berkeley in 1986. He has also held visiting positions
at London School of Economics, University of California at Berkeley, Purdue
University, University of Georgia and Institut National de la Statistique et
des Etudes Economiques, Paris. He is a Fellow of the Econometric Society
and has served on the editorial boards of several of our leading journals.
During his long and distinguished career, George has published over 60
research papers and 11 books. In each of his chosen research areas he has
made a significant and lasting impact. He was first to apply the limited
information maximum likelihood estimator to what was then the new area of
simultaneous equations modeling and estimation. In subsequent applications
he integrated this new methodology with the decision modeling necessary for
econometric results to be used in an economic framework. His work in the
60s on partial and general equilibrium models for price and quantity alloca­
tion over time and space provided the foundation stone upon which today's
research is based. Related work involved solutions for mathematical pro­
gramming problems, applications of and estimation techniques for Markov
processes, and the use of programming techniques for econometric estima­
tion. All this work was new, innovative and forward looking. It provided
inspiration and direction for many future researchers.
A long-standing practice in much applied econometric research is to try a
multitude of models until we find one 'that works', one where all the
statistical measures of performance are favorable. Unfortunately, such favor­
able performance measures can often be a measure of the researcher's
vii
perseverance, rather than a measure of how good the model actually is. They
ignore the preliminary testing procedures that lead to the final model
selection. George's research contributions in the 70s and 80s have done much
to alert the profession to these dangers. He has derived the properties of
many pre-test estimators and has suggested other alternative biased estima­
tors that belong to the Stein and Bayesian families of estimators.
George Judge's research contributions have been great. However, to
mention only his research contributions is to leave much unsaid. As a
colleague, he is dynamic and stimulating. He has a real talent for motivating
those around him so that they too share in his vision, enthusiasm and
productivity. Much of his time is given unselfishly to young researchers who
are trying to establish themselves. We editors can attest to the enormous
positive impact he has on our lives and we feel sure our words would be
echoed by his other friends and colleagues. This collection of papers is an
opportunity for us and others to say thank you.
The papers in the volume have been grouped into four parts, each part
representing an area in which George has made a significant contribution.
The authors have all benefited in some way, directly or indirectly, through an
association with George and his work. The three papers in Part I are
concerned with various aspects of pre-test and Stein-rule estimation. Part II
contains applications of Bayesian methodology, new developments in Bayesian
methodology, and an overview of Bayesian econometrics. The papers in Part
III comprise new developments in time-series analysis, improved estimation
and Markov chain analysis. The final part on spatial equilibrium modeling
contains papers that had their origins from George's pioneering work in the
60s.
We would like to thank North-Holland for giving us the opportunity to
honor George in this way. To those who used their valuable time refereeing
the papers, we also say thank you.
William Griffiths
Helmut Lütkepohl
Mary Ellen Bock
LIST OF CONTRIBUTORS
Aigner, Dennis J. University of California, Irvine, CA, USA
Beesley, P.A. University of New England, Armidale, NSW, Australia
Bera, Anil K. University of Illinois, Champaign, IL, USA
and Indiana University, Bloomington, IN, USA
Bernstein, David Massachusetts Institute of Technology, Cambridge,
MA, USA
Bohrer, Robert University of Illinois, Champaign, IL, USA
Chalfant, James A. University of California, Davis, CA, USA
Doran, H.E. University of New England, Armidale, NSW, Australia
Fomby, Thomas B. Southern Methodist University, Dallas, TX, USA
Friesz, Terry L. George Mason University, Fairfax, VA, USA
Ghali, Khalifa Faculté de Sousse, Sousse, Tunisia
Giles, David E.A. University of Canterbury, Christchurch, New Zealand
Griffiths, W.E. University of New England, Armidale, NSW, Australia
Hill, R. Carter Louisiana State University, Baton Rouge, LA, USA
Lee, Tsoung-Chao University of Connecticut, Storrs, CT, USA
Lütkepohl, Helmut Christian-Albrechts-Universität, Kiel, Germany
MacAulay, T.G. University of Sydney, Sydney, NSW, Australia
Machado, José A.F. Universidade Nova de Lisboa, Lisbon, Portugal
and University of Illinois, Champaign, IL, USA
Mellin, Illka University of Helsinki, Helsinki, Finland
Nagurney, Anna University of Massachusetts, Amherst, MA, USA
Racine, Jeff York University, Toronto, Ont., Canada
Terasvirta, Timo Research Institute of the Finnish Economy, Helsinki,
Finland
Ullah, Aman University of California, Riverside, CA, USA and
University of Western Ontario, London, Ont., Canada
Wallace, Nancy A. University of California, Berkeley, CA, USA
Yancey, T.A. University of Illinois, Champaign, IL, USA
Zellner, Arnold University of Chicago, Chicago, IL, USA
Readings in Econometric Theory and Practice
W. Griffiths, H. Lütkepohl and M.E. Bock (Editors)
© 1992 Elsevier Science Publishers B.V. All rights reserved
CHAPTER 1
THE EFFECTS OF EXTRAPOLATION ON MINIMAX
STEIN-RULE PREDICTION *
R. Carter Hill
Economics Department, Louisiana State University, Baton Rouge, LA 70803, USA
Thomas B. Fomby
Economics Department, Southern Methodist University, Dallas, TX 75275, USA
The purpose of this paper is to evaluate the performance of a variety of improved estimators
under an out-of-sample mean square error of prediction criterion. Estimators that are minimax
with respect to the criterion function are compared to estimators that are minimax with respect
to other loss functions as well as several other biased estimators. Out-of-sample mean square
error improvement is sensitive to multicollinearity conditions in both the sample data and
out-of-sample regressor values. This multicollinearity difference is characterized by differences
in characteristic root spectra, data ellipsoid rotation and distance between regressor means.
1. Introduction
The purpose of this paper is to consider the performance of a variety of
improved estimators for the classical normal linear regression model when
the measure of performance is out-of-sample mean square error of prediction
and where the data are multicollinear. In particular we consider the perfor­
mance of several Stein-like estimators that combine sample and non-sample
information in a way that dominates ordinary least squares (LS) when certain
design related conditions are met. Furthermore, we define data extrapolation
* The authors would like to thank Minbo Kim and Shengyi Guo for their valuable research
assistance on this research project. The authors received useful comments from Roger Koenker,
Mary Ellen Bock and Jim Chalfant. Any errors are, of course, the responsibility of the authors.
Fomby is grateful for the partial financial support provided by the Federal Reserve Bank of
Dallas while this research was in progress, especially during the summer of 1989. Of course, the
opinions expressed herein are those of the authors and should not be attributed to the Federal
Reserve Bank of Dallas or the Federal Reserve System.
4 R.C. Hill and T.B. Fomby
in the muhivariate context so that we can examine the effects on prediction
when the out-of-sample data are different from the data used for parameter
estimation. Data extrapolation takes into account not only distance between
two data sets but also the directions of the major and minor axes of the data
ellipsoids and the amounts of variation in each direction.
Our work may be thought of as extensions of several works. Friedman and
Montgomery (1985) discuss the performance of two biased estimation proce­
dures, ridge regression and principal components regression, with respect to
out-of-sample mean square error of prediction when the data are multi­
collinear. They develop conditions under which ridge and principal compo­
nents regression can improve upon the least squares estimator. Unfortu­
nately, the conditions they develop depend upon the true but unknown
parameters of the regression model and are not made operational. It is well
known that usual procedures for operationalizing these biased estimators
lead to rules that are not better than LS over substantial portions of the
parameter space.
Hill and Judge (1987) examine the performance of a variety of Stein-like
estimation rules with respect to in-sample mean square error of prediction.
Hill, Cartwright and Arbaugh (1991) examine the performance of Stein-like
and ridge rules when forecasting within the context of marketing data. Their
results reveal that out-of-sample mean square error of prediction gains can
be substantial when non-minimax rules are used. Zellner and Hong (1989)
use Bayesian Shrinkage rules to forecast international growth rates.
The plan of the paper is as follows: In Section 2 we present the statistical
model and describe the alternative estimators we will consider. In Section 3
we define multivariate measures of data extrapolation and use these concepts
in designing a Monte Carlo experiment, described in Section 4. We then use
this experiment to explore estimator performance. In Section 5 we report the
results of our Monte Carlo experiment and Section 6 contains concluding
remarks.
2. The Statistical Model and Estimators
Consider the linear statistical model
y=Xß-^e (2.1)
where y is a (Γ X 1) random vector. Λ" is a (ΓΧ A^) non-stochastic matrix of
rank K^T, β is a(Kxl) vector of unknown parameters and e is a (Γ X 1)
vector of random disturbances distributed as MO, σ^Ι).
The Effects of Extrapolation on Minimax Stein-Rule Prediction 5
The vector of regression parameters is unknown and it is our purpose to
estimate β using an estimator δ under a weighted squared error loss measure
L(ß,8,Q) = {8-ß)'Qi8-ß) (2.2)
where β is a positive definite and symmetric matrix. The sampling perfor­
mance of 8 is evaluated by its risk function
R(ß,8,Q)=E[L{ß,8,Q)]. (2.3)
The most common choices for the weight matrix Q in (2.2) are Q = I, which
defines the risk of estimation to be the mean square error, and Q=X'X
which corresponds to in-sample mean square error of prediction risk. As we
are interested in out-of-sample mean square error of prediction we let XQ be
an (mXK) matrix of regressor values of rank Κ such that m^K. The
corresponding values of the dependent variable are
yo=Xoß^e, (2.4)
where CQ MO, σ^Ι^) and Ε[β6^] = 0. The weight matrix for (2.2) we wish to
consider is Q =X¿Xo so that
Riß, δ, X¿X,) =E{8-ß)'XiX,(8-ß)
= E{X,8-X,ß)\x,8-X,ß)
= E{y,-Ey,)\y,-Ey,) (2.5)
which is the out-of-sample mean square error of prediction.
For the linear statistical model (2.1) the BLU and MVU estimator is
b = {X'Xy'X'y -N{ß, σ^{Χ'Χ)-'),
The estimator
&^ = {y-Xb)\y-Xb)/{T-K)=s/{T-K)
is MVU for (7^ and ^ Χ(τ-κγ Under the loss measure (2.2) the risk of b
is
R{ß,b,Q)=a' tviX'Xy'Q.