Properties of BLUEs in full versus small linear models

In this article we consider the partitioned linear model M12={y, X1β1+X2β2, V} and the corresponding small model M1={y, X1β1, V}. We focus on comparing the best linear unbiased estimators, BLUEs, of X1β1 under M12 and M1. In other words, we are interested in the effect of adding regressors on the BLUEs. Particular attention is paid on the consistency of the model, that is, whether the realized value of the response vector y belongs to the column space of (X1:V) or (X1:X2:V).