Contents

Preface xi

1 Preliminaries 1

1.1 Overview 1

1.2 Multivariate Normal and Wishart Distributions 1

1.3 Elliptically Symmetric Distributions 8

1.4 Group Invariance 16

1.5 Problems 21

2 Generalized Least Squares Estimators 25

2.1 Overview 25

2.2 General Linear RegressionModel 26

2.3 Generalized Least Squares Estimators 33

2.4 Finiteness of Moments and Typical GLSEs 40

2.5 Empirical Example: CO2 Emission Data 49

2.6 Empirical Example: Bond Price Data 55

2.7 Problems 63

3 Nonlinear Versions of the Gauss–Markov Theorem 67

3.1 Overview 67

3.2 Generalized Least Squares Predictors 68

3.3 A Nonlinear Version of the Gauss–Markov Theorem

in Prediction 73

3.4 A Nonlinear Version of the Gauss–Markov Theorem

in Estimation 82

3.5 An Application to GLSEs with Iterated Residuals 90

3.6 Problems 95

4 SUR and Heteroscedastic Models 97

4.1 Overview 97

4.2 GLSEs with a Simple Covariance Structure 102

4.3 Upper Bound for the Covariance Matrix of a GLSE 108

4.4 Upper Bound Problem for the UZE in an SUR Model 117

4.6 Empirical Example: CO2 Emission Data 134

4.7 Problems 140

5 Serial Correlation Model 143

5.1 Overview 143

5.2 Upper Bound for the Risk Matrix of a GLSE 145

5.3 Upper Bound Problem for a GLSE in the Anderson Model 153

5.4 Upper Bound Problem for a GLSE in a Two-equation

HeteroscedasticModel 158

5.5 Empirical Example: Automobile Data 165

5.6 Problems 170

6 Normal Approximation 171

6.1 Overview 171

6.2 Uniform Bounds for Normal Approximations

to the Probability Density Functions 176

6.3 Uniform Bounds for Normal Approximations

to the Cumulative Distribution Functions 182

6.4 Problems 193

7 Extension of Gauss–Markov Theorem 195

7.1 Overview 195

7.2 An Equivalence Relation on S(n) 198

7.3 A Maximal Extension of the Gauss–Markov Theorem 203

7.4 Nonlinear Versions of the Gauss–Markov Theorem 208

7.5 Problems 212

8 Some Further Extensions 213

8.1 Overview 213

8.2 Concentration Inequalities for the Gauss–Markov Estimator 214

8.3 Efficiency of GLSEs under Elliptical Symmetry 223

8.4 Degeneracy of the Distributions of GLSEs 233

8.5 Problems 241

9 Growth Curve Model and GLSEs 244

9.1 Overview 244

9.2 Condition for the Identical Equality between the GME

and the OLSE 249

9.3 GLSEs and Nonlinear Version of the Gauss–Markov Theorem 250

9.4 Analysis Based on a Canonical Form 255

9.5 Efficiency of GLSEs 262

A Appendix 274

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

Model and AndersonModel 274

Bibliography 281