Suppose $Y \sim N(X\beta, \Sigma)$. Ordinary LS $\hat{\beta_{OLS}} = (X'X)^{-1}X'y$ and weighted LS $\hat{\beta_{WLS} }= (X'WX)^{-1}X'Wy$, where $W = \Sigma^{-1}$. It is well known that $Var(\hat{\beta_{OLS}}) \ge Var(\hat{\beta_{WLS}})$. This implies for the predicted value $Var(X\hat{\beta_{OLS}}) \ge Var(X\hat{\beta_{WLS}})$. Can we conclude that for the residual $Var(Y-X\hat{\beta_{OLS}}) \le Var(Y-X\hat{\beta_{WLS}})$ ?

I think this does not hold since the variance decomposition does not hold for the OLS under $Y \sim N(X\beta, \Sigma)$. Does anybody have any thought on this ?


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