How data or error covariance can change and still retain BLUEs as well as their covariance or the sum of squares of errors

Misspecification of the error covariance in linear models usually leads to incorrect inference and conclusions. We consider two linear models, $A$ and $B$, with the same design matrix but different error covariance matrices. The conditions under which every representation of the best linear unbiased estimator (BLUE) of any estimable parametric vector under $A$ remains BLUE under $B$ have been well known since C.R. Rao's paper in 1971: Unified theory of linear estimation, Sankhyā Ser. A, Vol. 33, pp. 371–394. However, there are no previously published results on retaining the weighted sum of squares of errors (SSE) for non-full-rank design or error covariance matrices, and the question of when the covariance matrix of the BLUEs is also retained has been partially explored only recently. For change in any specified error covariance matrix, we provide necessary and sufficient conditions (nasc) for both BLUEs and their covariance matrix to remain unaltered and to retain this property for all submodels. We also consider nasc for SSE to be unchanged. We decompose SSE under error covariance changes, and derive nasc under which error covariance change leaves hypothesis tests for fixed-effect deletion under normality unaltered. We also show that simultaneous retention of BLUEs and both their covariance and SSE is not possible. We outline the effects of weak and strong error covariance singularity. We provide applications (via data cloning) to maintaining data confidentiality in Official Statistics without using Confidentialised Unit Record Files (CURFs), to certain types of experimental design and to estimation of fixed parameters for linear models for single nucleotide polymorphisms (SNPs) in genetics.