 Residual estimate of Gaussian noise variance

For a correctly specified full-rank fixed-design Gaussian linear regression with $p<n$, dividing the residual sum of squares by the residual degrees of freedom $n-p$ gives an unbiased estimator of the noise variance. The projection removes the whole deterministic mean, so the numerator's expectation is $(n-p){\sigma }^2$. A misspecified subset model need not have this property.