The Gauss–Markov Theorem for Regression Models with Possibly Singular Covariances

In this note we extend the Gauss–Markov theorem to the case of singular covariances and give an explicit formula for the BLUE of an estimable parameter. The formula reduces to the usual one when the covariance is nonsingular. An easy by-product of this result reestablishes known conditions for coincidence of the BLUE and naive least squares estimate.