(Sufficient Statistics) Show that $(\hat{\beta}, s^2)$ are sufficient statistics for $(\beta_0, \sigma^2)$ conditional on $X$, in the conditional normal model $y | X \sim \mathcal{N}(X\beta_0, \sigma^2 I_N)$, for $X$ full-column rank. That is, show that the distribution of $y$ conditional on $X$, $\hat{\beta}$, and $s^2$ does not depend on the population parameters $\beta_0$ and $\sigma^2$.