Effect of adding regressors on the equality of the BLUEs under two linear models

In this paper we consider the estimation of regression coefficients in two partitioned linear models, shortly denoted as M12={y,X1β1+X2β2,V}M12={y,X1β1+X2β2,V}, and M̲12={y,X1β1+X2β2,V̲}, which differ only in their covariance matrices. We call M12M12 and M̲12 full models, and correspondingly, Mi={y,Xiβi,V}Mi={y,Xiβi,V} and M̲i={y,Xiβi,V̲} small models. We give a necessary and sufficient condition for the equality between the best linear unbiased estimators (BLUEs) of X1β1X1β1 under M12M12 and M̲12. In particular, we consider the equality of the BLUEs under the full models assuming that they are equal under the small models.