Equivalence of predictors under real and over-parameterized linear models

Assume that a real linear model y=Xβ+ε${\bf y}= {\bf X}\pmb {\beta }+ \pmb {\varepsilon }$ is over-parameterized as y=Xβ+Zγ+ε${\bf y}= {\bf X}\pmb {\beta }+ {\bf Z}\pmb {\gamma }+ \pmb {\varepsilon }$ by adding some new regressors Zγ${\bf Z}\pmb {\gamma }$. In such a case, results of statistical inferences of the unknown parameters β$\pmb {\beta }$ and ε$\pmb {\varepsilon }$ under the two models are not necessarily the same. This paper aims at characterizing relationships between the best linear unbiased predictors (BLUPs) of the joint vector ϕ=Kβ+Jε$\pmb {\phi }= {\bf K}\pmb {\beta }+ {\bf J}\pmb {\varepsilon }$ of the unknown parameters in the two models. In particular, we derive necessary and sufficient conditions for the BLUPs of ϕ$\pmb {\phi }$ to be equivalent under the real model and its over-parameterized counterpart.