A Singular Covariance Matrix
If the covariance matrix X is singular, we obviously cannot define GLS by
(6.1.3) . Suppose that the rank of X is S < T. Then, by Theorem 3 of Appendix 1, we can find an orthogonal matrix H = (H,, H2), where H, is T X S and H2 is TX (T — S), such that HJXH, = A, a diagonal matrix consisting of the S positive characteristic roots of X, H’,XH2 = 0, and H2XH2 = 0. The premultiplication of (6.1.1) by H’ yields two vector equations:
Щу = ЩХ0. (6.1.12)
Note that there is no error term in (6.1.12) because АШ2ии’Н2 = H2XH2 = 0 and therefore H2u is identically equal to a zero vector. Then the best linear unbiased estimator of fi is GLS applied to (6.1.11) subject to linear constraints
(6.1.12) .1 Or, equivalently, it is LS applied to
A-,/2H’,y = A-1/2H’,X0 + A-,/2H’,u (6.1.13)
subject to the same constraints. Thus it can be calculated by appropriately redefining the symbols X, Q, and c in formula (1.4.5).