On a partitioned linear model and some associated reduced models

In this paper we consider a partitioned linear model (Y, X1β1 + X2β2, σ2Ω) where Ω is positive definite but X = (X1 : X2) can be rank-deficient. We also consider some associated reduced models involving only β2 (and not β1) and study how far the inferences on the estimable linear functions of β2 in the full model coincide with those in the reduced models. The results of Aigner and Balestra (1988) and Nurhonen and Puntanen (1992) follow as special cases. We also point out a flaw in the paper by Nurhonen and Puntanen (1992) and correct it suitably. Finally we apply the results proved to obtain explicitly the statistic for Turkeyʹs test of nonadditivity in two-way classified data with a block intraclass correlation structure.