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 ® 1992 ELSEVIER SCIENCE PUBLISHERS B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior writ­ ten permission of the publisher, Elsevier Science Publishers B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. 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. Special regulations for readers in the USA - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. PRINTED IN THE NETHERLANDS 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.