Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 5J / b / Solution

For the normal linear model, the Gaussian likelihood givesMaximization over $\beta$ gives the ordinary least squares estimator $\hat{\beta }$, and maximization over the variance givesAt this value the quadratic term in $-2\ell$ is $n$. There are $p$ regression coefficients and one variance parameter, so substitution into the Akaike information criterion yields