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 M 12 = { y , X 1 β 1 + X 2 β 2 , V } , and M ̲ 12 = { y , X 1 β 1 + X 2 β 2 , V ̲ } , which differ only in their covariance matrices. We call M 12 and M ̲ 12 full models, and correspondingly, M i = { y , X i β i , V } and M ̲ i = { y , X i β i , V ̲ } small models. We give a necessary and sufficient condition for the equality between the best linear unbiased estimators (BLUEs) of X 1 β 1 under M 12 and M ̲ 12 . In particular, we consider the equality of the BLUEs under the full models assuming that they are equal under the small models.