Solution to Review Question by Qiang Gao, updated at May 15, 2017
Chapter 1 Finite-Sample Properties of OLS Section 6 Generalized Least Squares (GLS) ...
Review Question 1.6.4 (Sampling error of GLS) Show: $$ \hat{ \boldsymbol{ \beta } }_{\mathrm{GLS}} - \boldsymbol{ \beta } = ( \mathbf{X}' \mathbf{V}^{-1} \mathbf{X} )^{-1} \mathbf{X}' \mathbf{V}^{-1} \boldsymbol{ \varepsilon } $$.
$$
\begin{align}
\hat{ \boldsymbol{ \beta } }_{\mathrm{GLS}}
& =
( \mathbf{X}' \mathbf{V}^{-1} \mathbf{X} )^{-1} \mathbf{X}' \mathbf{V}^{-1} \mathbf{y}
\tag{1.6.5}
\ & =
( \mathbf{X}' \mathbf{V}^{-1} \mathbf{X} )^{-1} \mathbf{X}' \mathbf{V}^{-1} ( \mathbf{X} \boldsymbol{ \beta } + \boldsymbol{ \varepsilon } )
\ & =
( \mathbf{X}' \mathbf{V}^{-1} \mathbf{X} )^{-1} \mathbf{X}' \mathbf{V}^{-1} \mathbf{X} \boldsymbol{ \beta } + ( \mathbf{X}' \mathbf{V}^{-1} \mathbf{X} )^{-1} \mathbf{X}' \mathbf{V}^{-1} \boldsymbol{ \varepsilon }
\ & =
\boldsymbol{ \beta } + ( \mathbf{X}' \mathbf{V}^{-1} \mathbf{X} )^{-1} \mathbf{X}' \mathbf{V}^{-1} \boldsymbol{ \varepsilon }.
\end{align}
$$
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