TABLE OF CONTENTS In the previous development for ordinary least-squares linear regression, it was assumed that the measurement errors had zero mean, and were uncorrelated with equal variance. In many practical cases, the assumptions of uncorrelated measurement errors and homogeneous variances are not valid. Thus, it is necessary to make the least-squares model more general by assuming that
The N N noise covariance matrix V is nonsingular and positive definite. The least-squares estimator for this modified model is obtained by minimizing
The minimizing parameter vector is computed from
which is called the generalized least-squares estimator. Using calculations similar to those shown earlier for ordinary least squares, it can be shown that 
GLS is asymptotically unbiased, E( GLS)= ?, with covariance matrix given by
Furthermore, under the assumptions (5.103), it can be shown that GLS is the best linear estimator of ? (see Ref. 3).
If the measurement errors for the dependent variable are uncorrelated, but with different variances, then V is a diagonal matrix with unequal elements on the diagonal. Introducing W = V ?1, the elements of W are weights for each equation in the regression problem, and the parameter estimation procedure is called weighted least squares. The expressions for the parameter estimates and covariance matrix in this case are the same as for the generalized least squares, with V ?1 now being a diagonal matrix [cf. Eqs. (5.105) and (5.106), respectively]. This approach...
