 Solution

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.