9.3 Incorrect Variance Matrix

If the first column of $\mathbf{X}$ is $\mathbf{1}_{n}$ and

use (9.8) to show that

Hint: Consider $\mathbf{P 1}_{n}$.

When $\operatorname {Var}\left[\varepsilon \right] = \sigma^{2} \mathbf{V}$, the appropriate estimate of $\mathbf{\beta }$ is the generalized least squares estimate $\beta * = \left(\mathbf{X}^{T} \mathbf{V}^{-1} \mathbf{X}\right)^{-1} \mathbf{X}^{T} \mathbf{V}^{-1} \mathbf{Y}$. If $\mathcal{C}\left(\mathbf{V}^{-1} \mathbf{X}\right) = \mathcal{C}(\mathbf{X})$, show that $\mathbf{\beta }^{*}$ and $\widehat{\mathbf{\beta }}$ are identical. Hint: $\mathbf{V}^{-1} \mathbf{X} = \mathbf{X W}$, where $\mathbf{W}$ is nonsingular.