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Generalized Least Squares

ISBN: 978-0-470-86697-9
312 pages
July 2004
Generalized Least Squares (0470866977) cover image
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.

 

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Preface.

1 Preliminaries.

1.1 Overview.

1.2 Multivariate Normal and Wishart Distributions.

1.3 Elliptically Symmetric Distributions.

1.4 Group Invariance.

1.5 Problems.

2 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.

3 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.

4 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.

4.6 Empirical Example: CO2 Emission Data.

4.7 Problems.

5 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.

6 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.

7 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.

8 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.

9 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.

A. Appendix.

A.1 Asymptotic Equivalence of the Estimators of θ in the AR(1) Error Model and Anderson Model.

Bibliography.

Index.

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"…provides a systematic discussion about statistical methods and theory for handling linear regression analysis when such conventional model assumptions are violated." (Technometrics, November 2005)

"...an accessible introduction to GLSE...an excellent source of reference, can be used as a course text, and will help to stimulate further research into this flourishing topic..." (Mathematical Reviews, 2005)

"...provides an up-to-date, self-contained introduction to the unified theory ..." (Zentralblatt Math, Vol. 1057, No. 8. 2005)

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