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**Generalized** **Least** **Squares** **Generalized** **Least** **Squares** Takeaki Kariya and Hiroshi Kurata 2004 John Wiley & Sons, Ltd ISBN: 0-470-86697-7 (PPC) WILEY SERIES IN PROBABILITY AND STATISTICS Established by WALTER A SHEWHART and SAMUEL S WILKS Editors: David J Balding, Peter Bloomfield, Noel A C Cressie, Nicholas I Fisher, Iain M Johnstone, J B Kadane, Geert Molenberghs, Louise M Ryan, David W Scott, Adrian F M Smith, Jozef L Teugels; Editors Emeriti: Vic Barnett, J Stuart Hunter, David G Kendall A complete list of the titles in this series appears at the end of this volume **Generalized** **Least** **Squares** Takeaki Kariya Kyoto University and Meiji University, Japan Hiroshi Kurata University of Tokyo, Japan Copyright 2004 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wileyeurope.com or www.wiley.com 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, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to permreq@wiley.co.uk, or faxed to (+44) 1243 770620 This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the Publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional should be sought Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Library of Congress Cataloging-in-Publication Data Kariya, Takeaki **Generalized** **least** **squares** / Takeaki Kariya, Hiroshi Kurata p cm – (Wiley series in probability and statistics) Includes bibliographical references and index ISBN 0-470-86697-7 (alk paper) **Least** **squares** I Kurata, Hiroshi, 1967-II Title III Series QA275.K32 2004 511 42—dc22 2004047963 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-470-86697-7 (PPC) Produced from LaTeX files supplied by the author and processed by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by TJ International, Padstow, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at **least** two trees are planted for each one used for paper production To my late GLS co-worker Yasuyuki Toyooka and to my wife Shizuko —Takeaki Kariya To Akiko, Tomoatsu and the memory of my fathers —Hiroshi Kurata Contents Preface xi Preliminaries 1.1 Overview 1.2 Multivariate Normal and Wishart Distributions 1.3 Elliptically Symmetric Distributions 1.4 Group Invariance 1.5 Problems 1 16 21 **Generalized** **Least** **Squares** Estimators 2.1 Overview 2.2 General Linear Regression Model 2.3 **Generalized** **Least** **Squares** Estimators 2.4 Finiteness of Moments and Typical GLSEs 2.5 Empirical Example: CO2 Emission Data 2.6 Empirical Example: Bond Price Data 2.7 Problems 25 25 26 33 40 49 55 63 67 67 68 73 82 90 95 Nonlinear Versions of the Gauss–Markov Theorem 3.1 Overview 3.2 **Generalized** **Least** **Squares** Predictors 3.3 A Nonlinear Version of the Gauss–Markov Theorem in Prediction 3.4 A Nonlinear Version of the Gauss–Markov Theorem in Estimation 3.5 An Application to GLSEs with Iterated Residuals 3.6 Problems SUR and Heteroscedastic Models 4.1 Overview 4.2 GLSEs with a Simple Covariance Structure 4.3 Upper Bound for the Covariance Matrix of a GLSE 4.4 Upper Bound Problem for the UZE in an SUR Model 4.5 Upper Bound Problems for a GLSE in a Heteroscedastic Model vii 97 97 102 108 117 126 viii CONTENTS 4.6 4.7 Empirical Example: CO2 Emission Data 134 Problems 140 Serial Correlation Model 5.1 Overview 5.2 Upper Bound for the Risk Matrix of a GLSE 5.3 Upper Bound Problem for a GLSE in the Anderson Model 5.4 Upper Bound Problem for a GLSE in a Two-equation Heteroscedastic Model 5.5 Empirical Example: Automobile Data 5.6 Problems Normal Approximation 6.1 Overview 6.2 Uniform Bounds for Normal Approximations to the Probability Density Functions 6.3 Uniform Bounds for Normal Approximations to the Cumulative Distribution Functions 6.4 Problems 143 143 145 153 158 165 170 171 171 176 182 193 Extension of Gauss–Markov Theorem 7.1 Overview 7.2 An Equivalence Relation on S(n) 7.3 A Maximal Extension of the Gauss–Markov Theorem 7.4 Nonlinear Versions of the Gauss–Markov Theorem 7.5 Problems 195 195 198 203 208 212 Some Further Extensions 8.1 Overview 8.2 Concentration Inequalities for the Gauss–Markov Estimator 8.3 Efficiency of GLSEs under Elliptical Symmetry 8.4 Degeneracy of the Distributions of GLSEs 8.5 Problems 213 213 214 223 233 241 Growth Curve Model and GLSEs 9.1 Overview 9.2 Condition for the Identical Equality between the GME and the OLSE 9.3 GLSEs and Nonlinear Version of the Gauss–Markov Theorem 9.4 Analysis Based on a Canonical Form 9.5 Efficiency of GLSEs 9.6 Problems 244 244 249 250 255 262 271 CONTENTS ix A Appendix 274 A.1 Asymptotic Equivalence of the Estimators of θ in the AR(1) Error Model and Anderson Model 274 Bibliography 281 Index 287 Preface Regression analysis has been one of the most widely employed and most important statistical methods in applications and has been continually made more sophisticated from various points of view over the last four decades Among a number of branches of regression analysis, the method of **generalized** **least** **squares** estimation based on the well-known Gauss–Markov theory has been a principal subject, and is still playing an essential role in many theoretical and practical aspects of statistical inference in a general linear regression model A general linear regression model is typically of a certain covariance structure for the error term, and the examples are not only univariate linear regression models such as serial correlation models, heteroscedastic models and equi-correlated models but also multivariate models such as seemingly unrelated regression (SUR) models, multivariate analysis of variance (MANOVA) models, growth curve models, and so on When the problem of estimating the regression coefficients in such a model is considered and when the covariance matrix of the error term is known, as an efficient estimation procedure, we rely on the Gauss–Markov theorem that the Gauss–Markov estimator (GME) is the best linear unbiased estimator In practice, however, the covariance matrix of the error term is usually unknown and hence the GME is not feasible In such cases, a **generalized** **least** **squares** estimator (GLSE), which is defined as the GME with the unknown covariance matrix replaced by an appropriate estimator, is widely used owing to its theoretical and practical virtue This book attempts to provide a self-contained treatment of the unified theory of the GLSEs with a focus on their finite sample properties We have made the content and exposition easy to understand for first-year graduate students in statistics, mathematics, econometrics, biometrics and other related fields One of the key features of the book is a concise and mathematically rigorous description of the material via the lower and upper bounds approach, which enables us to evaluate the finite sample efficiency in a general manner In general, the efficiency of a GLSE is measured by relative magnitude of its risk (or covariance) matrix to that of the GME However, since the GLSE is in general a nonlinear function of observations, it is often very difficult to evaluate the risk matrix in an explicit form Besides, even if it is derived, it is often impractical to use such a result because of its complication To overcome this difficulty, our book adopts as a main tool the lower and upper bounds approach, xi xii PREFACE which approaches the problem by deriving a sharp lower bound and an effective upper bound for the risk matrix of a GLSE: for this purpose, we begin by showing that the risk matrix of a GLSE is bounded below by the covariance matrix of the GME (Nonlinear Version of the Gauss–Markov Theorem); on the basis of this result, we also derive an effective upper bound for the risk matrix of a GLSE relative to the covariance matrix of the GME (Upper Bound Problems) This approach has several important advantages: the upper bound provides information on the finite sample efficiency of a GLSE; it has a much simpler form than the risk matrix itself and hence serves as a tractable efficiency measure; furthermore, in some cases, we can obtain the optimal GLSE that has the minimum upper bound among an appropriate class of GLSEs This book systematically develops the theory with various examples The book can be divided into three parts, corresponding respectively to Chapters and 2, Chapters to 6, and Chapters to The first part (Chapters and 2) provides the basics for general linear regression models and GLSEs In particular, we first give a fairly general definition of a GLSE, and establish its fundamental properties including conditions for unbiasedness and finiteness of second moments The second part (Chapters 3–6), the main part of this book, is devoted to the detailed description of the lower and upper bounds approach stated above and its applications to serial correlation models, heteroscedastic models and SUR models First, in Chapter 3, a nonlinear version of the Gauss–Markov theorem is established under fairly mild conditions on the distribution of the error term Next, in Chapters and 5, we derive several types of effective upper bounds for the risk matrix of a GLSE Further, in Chapter 6, a uniform bound for the normal approximation to the distribution of a GLSE is obtained The last part (Chapters 7–9) provides further developments (including mathematical extensions) of the results in the second part Chapter is devoted to making a further extension of the Gauss–Markov theorem, which is a maximal extension in a sense and leads to a further generalization of the nonlinear Gauss–Markov theorem proved in Chapter In the last two chapters, some complementary topics are discussed These include concentration inequalities, efficiency under elliptical symmetry, degeneracy of the distribution of a GLSE, and estimation of growth curves This book is not intended to be exhaustive, and there are many topics that are not even mentioned Instead, we have done our best to give a systematic and unified presentation We believe that reading this book leads to quite a solid understanding of this attractive subject, and hope that it will stimulate further research on the problems that remain The authors are indebted to many people who have helped us with this work Among others, I, Takeaki Kariya, am first of all grateful to Professor Morris L Eaton, who was my PhD thesis advisor and helped us get in touch with the publishers I am also grateful 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Taylor WE 1978 The heteroscedastic linear model: exact finite sample results Econometrica 46, 663–675 Tong YL 1990 The Multivariate Normal Distribution Springer-Verlag Toyooka Y 1987 An iterated version of the Gauss-Markov theorem in **generalized** **least** **squares** estimation Journal of Japan Statistical Society 17, 129–136 Toyooka Y and Kariya T 1986 An approach to upper bound problems for risks of the **generalized** **least** **squares** estimators Annals of Statistics 14, 679–690 286 BIBLIOGRAPHY Toyooka Y and Kariya T 1995 A note on sufficient condition for the existence of 2nd moments of **generalized** **least** **squares** estimator in a linear model Mathematica Japonica 42, 509–510 Usami Y and Toyooka Y 1997a On the degeneracy of the distribution of a GLSE in a regression with a circularly distributed error Mathematica Japonica 45, 423–431 Usami Y and Toyooka Y 1997b Errata of Kariya and Toyooka (1992), Bounds for normal approximations to the distributions of **generalized** **least** **squares** predictors and estimators Journal of Statistical Planning and Inference 58, 399–405 von Rosen D 1991 The growth curve model: a review Communications in Statistics: Theory and Methods 20, 2791–2822 Wang SG and Shao J 1992 Constrained Kantorovich inequalities and relative efficiency of **least** **squares** Journal of Multivariate Analysis 42, 284–298 Wang SG and Ip WC 1999 A matrix version of the Wielandt inequality and its applications to statistics Linear Algebra and its Applications 296, 171–181 Wu L and Perlman MD 2000 Lattice conditional independence model for seemingly unrelated regressions Communications in Statistics: Simulation and Computation 29, 361– 384 Zellner A 1962 An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias Journal of American Statistical Association 57, 348–368 Zellner A 1963 Estimators for seemingly unrelated regressions: some finite sample results Journal of American Statistical Association 58, 977–992 Zyskind G 1967 On canonical forms, non-negative covariance matrices and best and simple **least** **squares** estimators in linear models Annals of Mathematical Statistics 38, 1092– 1109 Zyskind G 1969 Parametric augmentations and error structures under which certain simple **least** **squares** and analysis of variance procedures are also best Journal of American Statistical Association 64, 1353–1368 Index Anderson model, 29, 40, 53, 182, 187, 234, 237 Anderson’s theorem, 217 AR(1) error model, 28, 79 Automobile data, 165 conditional distribution, 13 marginal distribution, 13 Equi-correlated model, 29 Equivalence theorem, 215 Equivariant estimator, 124, 132 location-equivariant estimator, 37, 197, 220 Bartlet decomposition, Best linear unbiased estimator (BLUE), 34 Best linear unbiased predictor (BLUP), 70 Bond price data, 55 Gamma — function, Multivariate — function, Gauss–Markov estimator (GME), 34, 70, 98, 144, 172, 196, 214 Gauss–Markov predictor (GMP), 71, 172 Gauss–Markov theorem, 34, 196 — in prediction, 70 maximal extension of —, 206 nonlinear version of — in estimation, 83, 209, 260 nonlinear version of — in prediction, 76 General linear regression model, 26, 34, 41, 82, 97, 171, 195, 214, 244 General multivariate analysis of variance (GMANOVA) model, 245 **Generalized** **least** **squares** estimator (GLSE), 35, 72, 91, 98, 144, 173, 220, 245 Cauchy–Schwarz inequality, 41, 43 Chi-square (χ ) distribution, Cholesky decomposition, Circularly distributed error model, 234 CO2 emission data, 135, 181 Concentration inequality, 215 Concentration probability, 215 Conditional covariance matrix, 15 Convex set, 215 Covariance matrix, Degeneracy, 233 Elasticity, 49, 136 Elliptically symmetric distribution — with finite moments, 11 definition, 11 characterization, 87 **Generalized** **Least** **Squares** Takeaki Kariya and Hiroshi Kurata 2004 John Wiley & Sons, Ltd ISBN: 0-470-86697-7 (PPC) 287 288 **Generalized** **least** **squares** predictor (GLSP), 72, 173 Group — of nonsingular lower-triangular matrices, 123 — of nonsingular lower-triangular matrices with positive diagonal elements, 16 — of nonsingular matrices, 16 — of orthogonal matrices, 16 definition, 16 group action, 17 group invariance, 16 subgroup, 17 transitivity, 17 Growth curve model, 244 Heteroscedastic model, 30, 43, 84, 93, 105, 127, 158, 165 Hypergeometric function, 161 Indicator function, 14 Invariance property — of GLSE, 43, 125 — of loss function, 115, 125 Invariant function, 18 Inversion formula, 176 Kantorovich inequality, 111 Kronecker product, 32 Linear unbiased estimator, 34 Linear unbiased predictor, 70 Location-equivariant estimator, 37, 197, 220 Log-concavity, 218 Loss function, 114, 125, 132, 150, 160 Maximal invariant, 18 Maximum likelihood estimator (MLE), 39 Mean vector, INDEX Moore–Penrose **generalized** inverse, 66 Most concentrated, 215 Multivariate linear regression model, 21 Multivariate normal distribution characteristic function, 21 characterization, conditional distribution, marginal distribution, moment generating function, 21 probability density function, Nonlinear version of Gauss–Markov theorem — in estimation, 83, 209, 260 — in prediction, 76 Normal approximation, 171 Orbit, 17 Ordinary **least** **squares** estimator (OLSE), 35, 104, 145, 194, 197 Ordinary **least** **squares** (OLS) residual, 20, 245, 250 Rao’s covariance structure, 152, 200, 249 Restricted GLSE, 45 Restricted Zellner estimator (RZE), 48 Risk function, 125, 133 Risk matrix, 70, 98, 144, 196, 245 Seemingly unrelated regression (SUR) model, 31, 47, 85, 94, 101, 107, 118, 135, 180, 187, 224 Serial correlation model, 143 Simple covariance structure, 102 Spectral decomposition, Spherically symmetric distribution — with finite moments, 10 definition, INDEX Symmetric convex set, 215 Symmetric inverse property, 115, 150, 160 Symmetric set, 215 Symmetry about the origin, 217 Uniform bound — for cdf, 186 — for pdf, 176 Uniform distribution on the unit sphere, 4, 10 Unimodal, 217 289 Unrestricted GLSE, 46, 106, 129, 160, 168 Unrestricted Zellner estimator (UZE), 48, 107, 118, 225 Upper bound for risk (covariance) matrix, 109, 144 Upper bound problem, 102 Wishart distribution marginal distribution, probability density function, **Generalized** **Least** **Squares** Takeaki Kariya and Hiroshi Kurata 2004 John Wiley & Sons, Ltd ISBN: 0-470-86697-7 (PPC) WILEY SERIES IN PROBABILITY AND STATISTICS ESTABLISHED BY WALTER A SHEWHART AND SAMUEL S WILKS Editors: David J Balding, Noel A C Cressie, Nicholas I Fisher, Iain M Johnstone, J B Kadane, Geert Molenberghs, Louise M Ryan, David W Scott, Adrian F M Smith, Jozef L Teugels Editors Emeriti: Vic Barnett, J Stuart Hunter David G Kendall The Wiley Series in Probability and Statistics is well established and authoritative It covers many topics of current research interest in both pure and applied statistics and probability theory Written by leading statisticians and 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Takeaki Generalized least squares / Takeaki Kariya, Hiroshi Kurata p cm – (Wiley series in probability and statistics) Includes bibliographical references and index ISBN 0-470-86697-7 (alk paper) Least. .. Kendall A complete list of the titles in this series appears at the end of this volume Generalized Least Squares Takeaki Kariya Kyoto University and Meiji University, Japan Hiroshi Kurata University

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