Past exam of the options I course of the University of Oxford / 2004 / O / O2

For nondegenerate design let $S_{xx}={\sum }_i{x}_i^2>0$. Maximizing the Gaussian likelihood over the mean parameters is equivalent to minimizing ${\sum }_i(Y_i-\alpha -\beta x_i{)}^2$. Since ${\sum }_i{x}_i=0$, the normal equations separate:Its error is the linear combination $\hat{\beta }-\beta ={\sum }_i{x}_i{ϵ}_i/S_{xx}$. Independence and normality therefore giveIf every design point is zero, the slope is not identifiable and no such slope estimator can be inferred; the standard regression answer assumes the stated nondegeneracy.

For $n>2$, the usual unbiased estimator of variance isThe fitted mean lies in the two-dimensional span of the constant design vector and $x$. Orthogonal projection of the Gaussian error onto its complement gives $n-2$ independent standardized normal coordinates. HenceThis residual projection is independent of the fitted-coefficient projection. If maximum likelihood for variance is intended instead, its estimator is ${\hat{\sigma }}_{\mathrm{ML}}^2=\mathrm{RSS}/n$, with $n{\hat{\sigma }}_{\mathrm{ML}}^2/{\sigma }^2\sim {\chi }_{n-2}^2$; the denominator differs, not the residual degrees of freedom.

Independence of the slope estimate and residual variance, together with their normal and chi-square laws, gives the Student pivotal quantityThus an exact Gaussian-model two-sided 95% confidence interval isUsing a normal quantile without estimating the variance uncertainty would not give this exact finite-sample coverage.