Abstract In this article we consider the general linear model {y, X ß, V} where y is the observable random vector with expectation X ß and covariance matrix V. Our interest is on predicting the unobservable random vector y* which comes from y* = X* ß + ƹ* where the expectation of y* isX* ß and the covariance matrix of y* is known aswell as the cross-covariance matrix between y* and y. The random vector y* can be considered as a kind of unknown future value.We introduce upper bounds for the Euclidean distances between the BLUPs, the best linear unbiased predictors, when the prediction is based on the original model and when it is based on the transformed model { Fy, FX ß, FVF’}. We also show how the upper bounds are related to the concept of linear sufficiency, and we apply our results into the mixed linear model.