Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / O / O5 / d

Put $C=(X^{T}X{)}^{-1}$ and $\nu =n-p$. The normal law of the estimator gives $({\hat{\theta }}_i-{\theta }_i)/(\sigma \sqrt{C_{ii}})\sim N(0,1)$. The fitted and residual Gaussian projections are orthogonal and hence independent, so this quantity is independent of $\nu s^2/{\sigma }^2\sim {\chi }_{\nu }^2$. Their ratio therefore has Student's t-distribution:If $t_{\nu ,1-\alpha /2}$ is its upper $1-\alpha /2$ quantile, rearranging its central probability event givesThis interval has exact coverage $1-\alpha$ under the given normal linear model, rather than merely large-sample coverage.