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 1.6.2 (Generalized $$ SSR $$) Show that $$ \hat{ \boldsymbol{ \beta } }_{ \mathrm{GLS} } $$ minimizes $$ ( \mathbf{y} - \mathbf{X} \tilde{ \boldsymbol{ \beta } } )' \mathbf{V}^{-1} ( \mathbf{y} - \mathbf{X} \tilde{ \boldsymbol{ \beta } } ) $$.
Note that for symmetric $$ \mathbf{V} $$, its inverse $$ \mathbf{V}^{-1} $$ is also symmetric.
The objective function is
$$
\begin{align}
& ( \mathbf{y} - \mathbf{X} \tilde{ \boldsymbol{ \beta } } )' \mathbf{V}^{-1} ( \mathbf{y} - \mathbf{X} \tilde{ \boldsymbol{ \beta } } )
\ = &
( \mathbf{y} - \mathbf{X} \tilde{ \boldsymbol{ \beta } } )' ( \mathbf{V}^{-1} \mathbf{y} - \mathbf{V}^{-1} \mathbf{X} \tilde{ \boldsymbol{ \beta } } )
\ = &
\mathbf{y}' \mathbf{V}^{-1} \mathbf{y} - 2 \cdot \mathbf{y}' \mathbf{V}^{-1} \mathbf{X} \tilde{ \boldsymbol{ \beta } } + \tilde{ \boldsymbol{ \beta } }' \mathbf{X}' \mathbf{V}^{-1} \mathbf{X} \tilde{ \boldsymbol{ \beta } }.
\end{align}
$$
Taking derivative with respect to $$ \tilde{ \boldsymbol{ \beta } } $$ leads to the first-order condition
$$
2 \cdot \mathbf{X}' \mathbf{V}^{-1} \mathbf{y} + 2 \cdot \mathbf{X}' \mathbf{V}^{-1} \mathbf{X} \tilde{ \boldsymbol{ \beta } } = \mathbf{0},
$$
and it reduces to
$$
\mathbf{X}' \mathbf{V}^{-1} \mathbf{X} \tilde{ \boldsymbol{ \beta } } = \mathbf{X}' \mathbf{V}^{-1} \mathbf{y},
$$
which solves that
$$
\tilde{ \boldsymbol{ \beta } }_{ \mathrm{GLS} } = ( \mathbf{X}' \mathbf{V}^{-1} \mathbf{X} )^{-1} \mathbf{X}' \mathbf{V}^{-1} \mathbf{y}.
$$
Copyright ©2017 by Qiang Gao