Some Decompositions of OLSEs and BLUEs Under a Partitioned Linear Model
We consider in this paper a partitioned linear model<formula format="inline"><simplemath>{y, X<sub>1</sub>β<sub>1</sub>+X<sub>2</sub>β<sub>2</su b>, σ-super-2σ}</simplemath></formula>and two corresponding small models<formula format="inline"><simplemath>{y, X<sub>1</sub>β<sub>1</sub>, σ-super-2σ}</simplemath></formula>and<formula format="inline"><simplemath>{y, X<sub>2</sub>β<sub>2</sub>, σ-super-2σ}</simplemath></formula>. We derive necessary and sufficient conditions for (i) the ordinary least squares estimator under the full model to be the sum of the ordinary least squares estimators under the two small models; (ii) the best linear unbiased estimator under the full model to be the sum of the best linear unbiased estimators under the two small models; (iii) the best linear unbiased estimator under the full model to be the sum of the ordinary least squares estimators under the two small models. The proofs of the main results in this paper also demonstrate how to use the matrix rank method for characterizing various equalities of estimators under general linear models. Copyright 2007 The Authors. Journal compilation (c) 2007 International Statistical Institute.
