Past exam of the options II course of the University of Oxford / 2009 / O / O4 / Solution

Minimizing the sum of squared residuals gives the normal equations for linear least squares, $X^{T}X\hat{\theta }=X^{T}Y$. Full column rank makes $X^{T}X$ positive definite and invertible, so the unique minimizer isBecause $\mathbb{E}Y=X\theta$ and $\mathrm{Cov}(Y)={\sigma }^2{I}_n$, one has $AX=I_p$ andFor the alternative linear estimator the corresponding general formulas areNo normality assumption on the errors is needed for these claims.