10.4 Nonconstant Variance and Serial Correlation: 10.4.1 Detecting Nonconstant Variance, 10.4.2 Estimating Variance Functions, 10.4.3 Transforming to Equalize Variances, 10.4.4 Serial Correlation and the Durbin-Watson Test

Let $\mathbf{G} = \mathbf{\Sigma }^{-1} \mathbf{X}\left(\mathbf{X}^{T} \mathbf{\Sigma }^{-1} \mathbf{X}\right)^{-1}$, and let $\mathbf{Q}$ be an orthonormal basis for $\mathcal{C}(\mathbf{X})^{\perp }$, so that $\mathbf{Q} \mathbf{Q}^{T} = \mathbf{I}_{n}-\mathbf{X}\left(\mathbf{X}^{T} \mathbf{X}\right)^{-1} \mathbf{X}^{T}$. Verify that $\mathbf{G}^{T} \mathbf{\Sigma } \mathbf{Q} = \mathbf{0}$ and that $\mathbf{G}^{T} \mathbf{\Sigma } \mathbf{G} = \left(\mathbf{X}^{T} \mathbf{\Sigma }^{-1} \mathbf{X}\right)^{-1}$.

Let $\mathbf{M} = \left(\mathbf{\Sigma }^{1 / 2} \mathbf{Q}, \mathbf{\Sigma }^{1 / 2} \mathbf{G}\right)$. Show that $\mathbf{M}$ is of full rank, so that $\mathbf{M}\left(\mathbf{M}^{T} \mathbf{M}\right)^{-1} \mathbf{M}^{T} = \mathbf{I}_{n}$.

Use (b) to prove (10.38).

Use the fact that $[\operatorname {det}\left(\mathbf{M}\right)]^{2} = \operatorname {det}\left(\mathbf{M}^{T} \mathbf{M}\right)$ to prove (10.39).

Suppose that $\mathbf{Y} = \left(Y_{1}, \ldots , Y_{n}\right)^{T}$, where the $Y_{i}$ 's are independently and identically distributed as $N\left(\mu , \sigma^{2}\right)$. Find the restricted likelihood for $\mathbf{Q}^{T} \mathbf{Y}$.

Show that for this example the REML estimate of $\sigma^{2}$ is the usual unbiased estimate $S^{2} = (n-1)^{-1} \sum_{i = 1}^{n}\left(Y_{i}-\bar{Y}\right)^{2}$. Thus in this case the REML estimate is unbiased but the MLE is not.