Generalized Least Squares

by Kariya, Takeaki; Kurata, Hiroshi

• ISBN13: 9780470866979
• ISBN10: 0470866977
• Format: Hardcover
• Copyright: 2004-07-23
• Publisher: WILEY

Summary

"Generalised Least Squares adopts a concise and mathematically rigorous approach. It will provide an up-to-date self-contained introduction to the unified theory of generalized least squares estimations, adopting a concise and mathematically rigorous approach. The book covers in depth the 'lower and upper bounds approach', pioneered by the first author, which is widely regarded as a very powerful and useful tool for generalized least squares estimation, helping the reader develop their understanding of the theory. The book also contains exercises at the end of each chapter and applications to statistics, econometrics, and biometrics, enabling use for self-study or as a course text.

Author Biography

Takeaki Kariya is the author of Generalized Least Squares, published by Wiley.

Hiroshi Kurata is the author of Generalized Least Squares, published by Wiley.

Table of Contents

Preface	xi
Preliminaries	1	(24)
Overview	1	(1)
Multivariate Normal and Wishart Distributions	1	(7)
Elliptically Symmetric Distributions	8	(8)
Group Invariance	16	(5)
Problems	21	(4)
Generalized Least Squares Estimators	25	(42)
Overview	25	(1)
General Linear Regression Model	26	(7)
Generalized Least Squares Estimators	33	(7)
Finiteness of Moments and Typical GLSEs	40	(9)
Empirical Example: CO2 Emission Data	49	(6)
Empirical Example: Bond Price Data	55	(8)
Problems	63	(4)
Nonlinear Versions of the Gauss--Markov Theorem	67	(30)
Overview	67	(1)
Generalized Least Squares Predictors	68	(5)
A Nonlinear Version of the Gauss--Markov Theorem in Prediction	73	(9)
A Nonlinear Version of the Gauss--Markov Theorem in Estimation	82	(8)
An Application to GLSEs with Iterated Residuals	90	(5)
Problems	95	(2)
SUR and Heteroscedastic Models	97	(46)
Overview	97	(5)
GLSEs with a Simple Covariance Structure	102	(6)
Upper Bound for the Covariance Matrix of a GLSE	108	(9)
Upper Bound Problem for the UZE in an SUR Model	117	(9)
Upper Bound Problems for a GLSE in a Heteroscedastic Model	126	(8)
Empirical Example: CO2 Emission Data	134	(6)
Problems	140	(3)
Serial Correlation Model	143	(28)
Overview	143	(2)
Upper Bound for the Risk Matrix of a GLSE	145	(8)
Upper Bound Problem for a GLSE in the Anderson Model	153	(5)
Upper Bound Problem for a GLSE in a Two-equation Heteroscedastic Model	158	(7)
Empirical Example: Automobile Data	165	(5)
Problems	170	(1)
Normal Approximation	171	(24)
Overview	171	(5)
Uniform Bounds for Normal Approximations to the Probability Density Functions	176	(6)
Uniform Bounds for Normal Approximations to the Cumulative Distribution Functions	182	(11)
Problems	193	(2)
Extension of Gauss--Markov Theorem	195	(18)
Overview	195	(3)
An Equivalence Relation on S(n)	198	(5)
A Maximal Extension of the Gauss--Markov Theorem	203	(5)
Nonlinear Versions of the Gauss--Markov Theorem	208	(4)
Problems	212	(1)
Some Further Extensions	213	(31)
Overview	213	(1)
Concentration Inequalities for the Gauss--Markov Estimator	214	(9)
Efficiency of GLSEs under Elliptical Symmetry	223	(10)
Degeneracy of the Distributions of GLSEs	233	(8)
Problems	241	(3)
Growth Curve Model and GLSEs	244	(30)
Overview	244	(5)
Condition for the Identical Equality between the GME and the OLSE	249	(1)
GLSEs and Nonlinear Version of the Gauss-Markov Theorem	250	(5)
Analysis Based on a Canonical Form	255	(7)
Efficiency of GLSEs	262	(9)
Problems	271	(3)
A Appendix	274	(7)
A.1 Asymptotic Equivalence of the Estimators of θ in the AR(1) Error Model and Anderson Model	274	(7)
Bibliography	281	(6)
Index	287