A sufficient condition on the covariance matrix for F tests in linear models to be valid

Abstract

Consider the linear model for the random vector Y distributed in N (Xβ, C ); X is not necessarily of full rank and C is a nonnegative-definite matrix. This note gives a sufficient condition on C for the usual F test of a null hypothesis H β = 0 to be valid. A discussion of necessity of the condition is included.