 Joint distribution of least-squares and variance estimators

With a full-column-rank $n\times p$ design and isotropic normal errors, ordinary least squares gives $\hat{\beta }\sim N_p(\beta ,{\sigma }^2(X^{T}X{)}^{-1})$, independent of the residual squared length. Orthogonal design and residual projections of the multivariate normal distribution are independent, and $\mathrm{RSS}/{\sigma }^2\sim {\chi }_{n-p}^2$. The maximum likelihood variance estimate is $\mathrm{RSS}/n$, not the unbiased estimate $\mathrm{RSS}/(n-p)$.