I have in my notes that $\displaystyle \frac{\hat u^T \hat u}{T-k} = \sigma ^2 $. I also have that $\displaystyle \frac{\hat u^T \hat u}{ \sigma ^2} \sim \chi^2_{T-k} $. I know how to derive the first part, but how the heck do we know the second thing is true? Does it come from the derivation of the first equation or... what? Help!