THEODORE W. ANDERSON Studies in Econometrics, Time Series, and Multivariate Statistics EDITED BY SAMUEL KARLIN Department of Mathematics Stanford University Stanford, California TAKESHI AMEMIYA Department of Economics Stanford University Stanford, California LEO A. GOODMAN Departments of Statistics and Sociology University of Chicago Chicago, Illinois ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers San Diego New York Berkeley Boston London Sydney Tokyo Toronto Academic Press Rapid Manuscript Reproduction COPYRIGHT © 1983, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 1250 Sixth Avenue, San Diego, California 92101 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 7DX Library of Congress Cataloging in Publication Data Main entry under title: Studies in econometrics, time series, and multivariate statistics· "In commemoration of T.W. Anderson's 65th birthday." Includes bibliograpical references. 1. Econometrics—Addresses, essays, lectures. 2. Time-series analysis—Addresses, essays, lectures. 3. Multivariate analysis—Addresses, essays, lectures. 4. Social sciences—Statistical methods—Addresses, essays, lectures. 5. Anderson, T. W. (Theodore Wilbur), Date · I. Karlin, Samuel, Date II. Amemiya, Takeshi. III. Goodman, Leo A. IV. Ander- son, T. W. (Theodore Wilbur), Date HB139.S83 1983 330'.028 83-15523 ISBN 0-12-398750-4 PRINTED IN THE UNITED STATES OF AMERICA 88 89 90 91 92 10 9 8 7 6 5 4 3 Contributors Numbers in parentheses indicate the pages on which the authors' contributions begin. TAKESHI AMEMIYA (3), Department of Economics, Stanford University, Stanford, California 94305 DAVID R. BRILLÏNGER (31), Department of Statistics, University of California, Berkeley, California 94720 B. W. BROWN (131), Department of Economics, Princeton University, Princeton, New Jersey 08544 ARTHUR COHEN (379), Department of Statistics, Rutgers University, New Brunswick, New Jersey 08903 SOMESH DAS GUPTA (407), School of Statistics, University of Minnesota, Minneapolis, Minnesota 55455 J. DURBIN (243), Department of Statistical and Mathematical Sciences, The London School of Economics and Political Sciences, London, England C. FANG (419), Center for Multivariate Analysis, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 CONSTANTINE GATSONIS (379), Department of Statistics, Rutgers University, New Brunswick, New Jersey 08903 ARTHUR S. GOLDBERGER (67), Department of Economics, University of Wisconsin, Madison, Wisconsin 53706 LEO A. GOODMAN (85), Departments of Statistics and Sociology, University of Chicago, Chicago, Illinois 60637 C. W. J. GRANGER (255), Department of Economics, University of California at San Diego, La Jolla, California 92093 ix X Contributors LARRY V. HEDGES (437), Department of Education, University of Chicago, Chicago, Illinois 60637 CHENG HSIAO (93), Institute for Policy Analysis, University of Toronto, Toronto, Ontario M5S 1A1, Canada A. T. JAMES (455), Department of Statistics, University of Adelaide, Adelaide, South Australia 5001 SAMUEL KARLIN (465), Department of Mathematics, Stanford University, Stanford, California 94305 P. R. KRISHNAIAH (419), Center for Multivariate Analysis, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 E. L. LEHMANN (491), Department of Statistics, University of California, Berkeley, California 94720 JOHN MARDEN (379), Department of Mathematics, University of Illinois, Urbana, Illinois 61801 ROBERTO S. MARIANO (131), Department of Economics, University of Pennsylvania, Philadelphia, Pennsylvania 19104 INGRAM OLKIN (437), Department of Statistics, Stanford University, Stanford, California 94305 EMANUEL PARZEN (279), Institute of Statistics, Texas A & M University, College Station, Texas 77843 MICHAEL D. PERLMAN (505), Department of Statistics, University of Washington, Seattle, Washington 98195 JAMES L. POWELL (3), Department of Economics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 HAIGANOUSH K. PREISLER (31), Northern California Sickle Cell Center, University of California at San Francisco, San Francisco, California 94117 C. R. RAO (529), Center for Multivariate Analysis, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 YOSEFRINOTT (465), Department of Statistics, Hebrew University of Jerusalem, Jerusalem 91905, Israel MURRAY ROSENBLATT (299), Department of Mathematics, University of California at San Diego, La Jolla, California 92093 THOMAS J. ROTHENBERG (153), Department of Economics, University of California, Berkeley, California 94720 J. D. SARGAN (169), Department of Economics, London School of Economics and Political Science, London, England Contributors xi STANLEY L. SCLOVE (311), Quantitative Methods Department, College of Business Administration, University Center, University of Illinois, Chicago, Illinois 60680 PAUL SHAMAN (331), Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania 19104 GEORGE P. H. STYAN (545), Department of Mathematics, McGill University, Montreal, Quebec, Canada AKIMICHI TAKEMURA (545), Department of Statistics, Stanford University, Stanford, California 94305 JOHN B. TAYLOR (207), Department of Economics, Princeton University, Princeton, New Jersey 08544 A. M. WALKER (343), Department of Probability and Statistics, The University of Sheffield, Sheffield S3 74H, England G. S. WATSON (559), Department of Statistics, Princeton University, Princeton, New Jersey 08544 A. A. WEISS (255), Department of Economics, University of California at San Diego, La Jolla, California 92093 ARNOLD ZELLNER (227), Graduate School of Business, University of Chicago, Chicago, Illinois 60637 Biographical Note In commemoration of Ted Anderson's 65th birthday, his many friends, students, and scientific associates offer him their affection, esteem, and best wishes for everlasting good health and for continued discoveries of statistical and econometric significance. The contributions of this Festschrifi, centering on the themes of multivariate statistical analysis, time series processes, econ- ometrics, and quantitative social science studies, attest to the importance and influence of T. W. Anderson's scholarship and research (see attached bibli- ography). Born on June 5, 1918, in Minneapolis, Minnesota, Anderson studied at North Park College and received his B.S. from Northwestern University in 1939 and his Ph.D. from Princeton University in 1945. He participated actively in the research program of the Cowles Commission for Research in Eco- nomics at the University of Chicago in 1945-1946, and from there went on to Columbia University as an instructor in mathematical statistics in 1946, rising through the ranks to full professor by 1956. He moved to his present position at Stanford as Professor of Statistics and Economics in 1967. Anderson has been recognized in many ways during his distinguished career. He serves on the editorial boards of many international scientific journals. He is a member of the International Statistical Institute, a fellow of the Institute of Mathemat- ical Statistics (President, 1962-1963), a fellow of the American Statistical Asso- ciation (Vice-President, 1971-1973), a fellow of the Econometric Society; and he was elected to the American Academy of Arts and Sciences in 1974 and to the National Academy of Sciences in 1976. Anderson's early work pioneered a number of "non-null" distributions, for which he derived explicit forms of the noncentral Wishart distribution (over- lapping some independent results of M. A. Girshick). These results are fun- damental to the development of essential statistical theory for characteristic roots and vectors of covariance matrices and related quantities (for tests, esti- mates, asymptotic distributions, and confidence regions). Anderson's work xiii XIV Biographical Note has found immediate applications in econometrics in terms of the Limited Information Maximum Likelihood estimate for every single equation in a sys- tem. This method and its subsequent simplification (by Theil and Basmann) have played an important role in estimating simultaneous equation models in economics. Anderson has recently published many results concerning the ex- act and approximate distributions of these estimators and their subsequent modifications. Anderson has also concentrated on developing inference procedures for covariance matrices that are linear combinations of known matrices. He dis- covered that the probability of a convex symmetric set decreases as the center of the set departs along a ray from the center for a symmetric unimodal density; this result supports a broad hierarchy of basic inequalities for multi- variate distributions. His work on time series analysis extended and applied the basic Neyman- Pearson theory to tests of independence and order of dependence for time series. Recent papers present methods for general autoregressive moving aver- age models. Anderson's work has also had a substantial influence on quantitative psy- chology and sociology. He wrote (with Herman Rubin) the first comprehen- sive treatment of statistical inference in factor analysis, introducing new con- cepts and methods. Later papers dealt incisively with problems in ''scaling" and "measurement." In evaluating T. W. Anderson's many outstanding and versatile contribu- tions to statistical and econometric theory, we would be remiss in not high- lighting his masterful text, An Introduction to Multivariate Statistical Analysis (an updated revised edition is in the works), which is among the most frequently cited works in the statistical literature and which has been widely applied throughout the sciences. Samuel Karlin Takeshi Amemiya Leo Goodman May 1, 1983 Bibliography of Theodore W. Anderson Books 1. "An Introduction to Multivariate Statistical Analysis," 374 pp. Wiley, New York (1958). [Russian translation: Vvedenie v Mnogomernyi Statisticeskii Analiz, 500 pp. Gosudarstvennoe Izdatel'stvo Fiziko-matematiceskoi Literatury, Moscow (1963).] 2. "The Statistical Analysis of Time Series," 704 pp. Wiley, New York (1971). [Russian trans- lation: Statisticeskii Analiz Vremennykh Rjadov, 755 pp. Izdatelstvo MIR, Moscow (1976).] 3. "A Bibliography of Multivariate Statistical Analysis (with S. Das Gupta and G. P. H. Styan) 642 pp. Oliver & Boyd, Edinburgh, and Halsted Press (1972). [Reprinted, Robert E. Krieger, Huntington, New York (1977).] 4. "Introductory Statistical Analysis" (with Stanley L. Sclove), 499 pp. Houghton Mifflin, Boston (1974). 5. "An Introduction to the Statistical Analysis of Data" (with Stanley L. Sclove), 704 pp. Houghton Mifflin, Boston (1978). Papers 1. Some significance tests for normal bivariate distributions (with D. S. Villars). Annals of Mathematical Statistics 14, 141-148 (1943). 2. On card matching. Annals of Mathematical Statistics 14, 426-435 (1943). 3. Some extensions of the Wishart distribution (with M. A. Girshick). Annals of Mathematical Statistics 15, 345-357 (1944). [Correction 35, 923-924 (1964).] 4. The non-central Wishart distribution and certain problems of multivariate statistics. Annals of Mathematical Statistics 17, 409-431 (1946). (Correction 35, 923-924 (1964).] 5. A note on a maximum-likelihood estimate. Econometrica 15, 241-244 (1947). 6. On the theory of testing serial correlation. Skandinavisk Aktuarietidskrifi 31, 88-116 (1948). 7. The asymptotic distributions of the roots of certain determinantal equations. Journal of the Royal Statistical Society, Ser. B 10, 132-139 (1948). 8. Estimation of the parameters of a single equation in a complete system of stochastic equa- tions (with Herman Rubin). Annals of Mathematical Statistics 20, 46-63 (1949). [Reprinted in "Readings in Econometric Theory" (J. Malcolm Dowling and Fred R. Glahe, eds.), Colora- do Associated University Press, pp. 358-375 (1970).] 9. Distribution of the circular serial correlation coefficient for residuals from a fitted Fourier series (with R. L. Anderson). Annals of Mathematical Statistics 21, 59-81 (1950). 10. Estimation of the parameters of a single equation by the limited-information maximum- likelihood method. In "Statistical Inference in Dynamic Economic Models" (Tjalling C. Koopmans, ed.), pp. 311-322. Wiley, New York (1950). xv XVI Bibliography 11. The asymptotic properties of estimates of the parameters of a single equation in a complete system of stochastic equations (with Herman Rubin). Annals of Mathematical Statistics 21, 570-582 (1950). [Reprinted in "Readings in Econometric Theory" (J. Malcolm Dowling and Fred R. Glahe, eds.). Colorado Associated University Press, pp. 376-388 (1970).] 12. Classification by multivariate analysis. Psychometrika 16, 31-50 (1951). 13. The asymptotic distribution of certain characteristic roots and vectors. In "Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability" (Jerzy Ney- man, ed.), pp. 105-130. Univ. California Press, Berkeley, California (1951). 14. Estimating linear restrictions on regression coefficients for multivariate normal distributions. Annals of Mathematical Statistics 22, 327-351 (1951). [Correction, Annals of Statistics 8, 1400 (1980.)] 15. Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes (with D. A. Darling). Annals of Mathematical Statistics 23, 193-212 (1952). 16. Probability models for analyzing time changes in attitudes. In "Mathematical Thinking in the Social Sciences" (Paul F. Lazarsfeld, ed.), pp. 17-66. The Free Press, Glencoe, Illinois (1954). 17. On estimation of parameters in latent structure analysis. Psychometrika 19, 1-10 (1954). 18. A test of goodness of fit (with D. A. Darling). Journal of the American Statistical Association 49, 765-769 (1954). 19. Some statistical problems in relating experimental data to predicting performance of a pro- duction process. Journal of the American Statistical Association 50, 163-177 (1955). 20. The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proceedings of the American Mathematical Society 6, 170-176 (1955). 21. Department of Mathematical Statistics. In "History of the Faculty of Political Science." pp. 250-255. Columbia Univ. Press, New York (1955). 22. Some recent results in latent structure analysis. In "Proceedings of the Invitational Confer- ence on Testing Problems, October 30, 1954," pp. 49-53. Educational Testing Service, Princeton, New Jersey (1955). 23. Statistical inference in factor analysis (with Herman Rubin). In "Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability" (Jerzy Neyman, ed.), Vol. 5, pp. 111-150. Univ. California Press, Berkeley, California (1956). 24. Statistical inference in Markov chains (with Leo A. Goodman). Annals of Mathematical Statis- tics 28, 89-110 (1957). [Reprinted in "Readings in Mathematical Psychology" (R. Duncan Luce, Robert R. Bush, and Eugene Galanter, eds.), Vol. 1, pp. 241-262. Wiley, New York (1963).] 25. Maximum likelihood estimates for a multivariate normal distribution when some observa- tions are missing. Journal of the American Statistical Association 52, 200-203, (1957). 26. On asymptotic distributions of estimates of parameters of stochastic difference equations. Annals of Mathematical Statistics 30, 676-687 (1959). 27. Some scaling models and estimation procedures in the latent class model. In "Probability and Statistics: The Harald Cramer Volume" (Ulf Grenander, ed.), pp. 9-38. Almqvist and Wiksell, Stockholm (1959). 28. A modification of the sequential probability ratio test to reduce the sample size. Annals of Mathematical Statistics 31, 165-197 (1960). 29. Harold Hotelling's research in statistics. The American Statistician 14, 17-21 (1960). 30. A limitation of the optimum property of the sequential probability ratio test (with Milton Friedman). In "Contributions to Probability and Statistics; Essays in Honor of Harold Hotelling" (Ingram Olkin, Sudhish G. Ghurye, Wassily Hoeffding, William G. Madow, and Henry B. Mann, eds.), pp. 57-69. Stanford Univ. Press, Stanford, California (1960). 31. Some stochastic process models for intelligence test scores. In "Mathematical Methods in the Social Sciences" (Kenneth J. Arrow, Samuel Karlin, and Patrick Suppes, eds.), pp. 205-220. Stanford Univ. Press, Stanford, California (1960). 32. The choice of the degree of a polynomial regression as a multiple decision problem. Annals of Mathematical Statistics 33, 255-265 (1962).