Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd Edition

Chapter 13The linear regression model

1 INTRODUCTION

In this chapter, we consider the general linear regression model

(1)

where y is an n × 1 vector of observable random variables, X is a nonstochastic n × k matrix (n ≥ k) of observations of the regressors, and ɛ is an n × 1 vector of (not observable) random disturbances with

(2)

where V is a known positive semidefinite n × n matrix and σ² is unknown. The k × 1 vector β of regression coefficients is supposed to be a fixed but unknown point in the parameter space ℬ. The problem is that of estimating (linear combinations of) β on the basis of the vector of observations y.

To save space, we shall denote the linear regression model by the triplet

(3)

We shall make varying assumptions about the rank of X and the rank of V.

We assume that the parameter space ℬ is either the k‐dimensional Euclidean space

(4)

or a nonempty affine subspace of ℝ^k, having the representation

(5)

where the matrix R and the vector r are nonstochastic. Of course, by putting ...