On equivalence of predictors/estimators under a multivariate general linear model with augmentation

Assume that a true multivariate general linear model for an observed random matrix is over-parameterized by adding some new regressors due to model uncertainty. Then predictors and estimators of parameter spaces in the true and over-parameterized models are not necessarily the same. In this article, we study the comparison problem of predictors/estimators of parameter spaces under the two models. In particular, we derive necessary and sufficient conditions for the best linear unbiased predictors/best linear unbiased estimators of the parameter spaces to be equivalent under the two models.