Past exam of the options I course of the University of Oxford / 2012 / O / O2 / b

The design columns are $1$, $x$ and $z$. Their pairwise inner products are zero by the three supplied sums. Put $S_{xx}={\sum }_i{x}_i^2$ and $S_{zz}={\sum }_i{z}_i^2$, positive under the full-rank assumption. ThenApplying part (a),The off-diagonal covariances vanish because the design columns are orthogonal. With Gaussian errors, the coefficient estimators are also jointly normal and therefore independent.

The future mean response is $E_{\ast }=\alpha +\beta x_{\ast }+\gamma z_{\ast }$. By invariance of maximum likelihood estimation, its estimator is the fitted meanIts sampling variance is the quadratic form of the coefficient covariance matrix with $c=(1,x_{\ast },z_{\ast }{)}^T$:This estimator is unbiased because each fitted coefficient is unbiased. The future error ${ϵ}_{\ast }$ does not enter the estimator of the mean, so its independent variance is not part of this expression.