Generalized Least Squares Theory

One of the important assumptions of the standard regression model (Model 1) is the assumption that the covariance matrix of the error terms is a scalar times the identity matrix. In this chapter we shall relax this assumption. In Section

6.1 the case in which the covariance matrix is known will be considered. It is pedagogically useful to consider this case before we consider, in Section 6.2, a more realistic case of an unknown covariance matrix. Then in Sections 6.3 through 6.7 various ways of specifying the covariance matrix will be con­sidered.

6.1 The Case of a Known Covariance Matrix

6.1.1 Model 6

The linear regression model we shall consider in this chapter, called Model 6, is defined by

y = X0+u, (6.1.1)

where X is a TX К matrix of known constants with rank K(^T), fi is a K-vector of unknown parameters, and u is a Г-vector of random variables with Ea = 0 and Гии’ = X, a known T X Tpositive-definite covariance matrix. (In Section 6.1.5 we shall consider briefly the case of a singular covariance ma­trix.) We shall write the /,jth element of 2 as au.

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