Summary: Assume that a real linear model \(\mathbf{y}=\mathbf{X\beta}+\mathbf{\varepsilon}\) is over-parameterized as \(\mathbf{y}=\mathbf{X\beta}+\mathbf{Z\gamma}+\mathbf{\varepsilon}\) by adding some new regressors \(\mathbf{Z\gamma}\). In such a case, results of statistical inferences of the unknown parameters \(\mathbf{\beta}\) and \(\mathbf{\varepsilon}\) 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 \(\mathbf{\phi}=\mathbf{K\beta}+\mathbf{J\varepsilon}\) of the unknown parameters in the two models. In particular, we derive necessary and sufficient conditions for the BLUPs of \(\mathbf{\phi}\) to be equivalent under the real model and its over-parameterized counterpart.