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

The normal-error log-likelihood, for fixed variance, is a constant minus $\Vert Y-X\theta {\Vert }^2/(2{\sigma }^2)$. Differentiating the squared residual gives the normal equations $X^{T}X\hat{\theta }=X^{T}Y$. Since $X$ has full column rank, $X^{T}X$ is positive definite and the squared residual is strictly convex in $\theta$. Hence the unique minimizer, and thus the maximum-likelihood estimator, isProfiling over the unknown positive variance does not change the residual-minimizing value of $\theta$.