Estimation in singular linear models with stepwise inclusion of linear restrictions

In this paper, we consider the general linear model ℳ={y,Xβ,Σ}, without any rank assumptions to the model matrix X and covariance matrix Σ, and its two restricted models ℳr1={y,Xβ|A1β=r1,Σ} and ℳr12={y,Xβ|Aβ=r,Σ}, where r=(r1′,r2′)′ and A=(A1′,A2′)′. We give the necessary and sufficient conditions for the BLUEs to equal under ℳ and ℳr1, as well as under ℳr1 and ℳr12. We also derive that the BLUEs under ℳr1 are superior to the BLUEs under ℳ, and that the BLUEs under ℳr12 are superior to the BLUEs under ℳr1 in the sense of the covariance matrix.