Linear prediction sufficiency in the misspecified linear model

We consider the general linear model y=Xβ+ε supplemented with the new (future) unobservable random vector y*, coming from y*=X*β+ε*, where the expectation of y* is X*β and the covariance matrix of y* is known as well as the cross-covariance matrix between y* and y. We denote the supplemented model as M*. The misspecified supplemented model is denoted as M¯*, and the misspecification concerns the covariance part of the setup. Suppose that Fy is linearly sufficient for estimable parametric function X*β under M*. We give necessary and sufficient conditions that Fy continues to be linearly sufficient for X*β under the model M¯*. The corresponding properties regarding the linear prediction sufficiency with respect to ε* and y* are also studied.