[Math] Show $\hat{\beta}$ and $s^2$ are independent

Question

I have the model:

$y=X{\beta}+{\epsilon}$

I know $\hat{\beta}=(X'X)^{-1}X'y$ and that it is an unbiased estimator of ${\beta}$ and that $s^2=\hat{\epsilon}'\hat{\epsilon}/(n-k)$ and is an unbiased estimator of the variance.

How do I show that $\hat{\beta}$ and $s^2$ are independent?

Answer 1

Since $s^2:=\hat\epsilon^T\hat\epsilon/(n-k)$ is a function of the residual vector $\hat\epsilon:=Y-X\hat\beta$, this result follows from the independence of $\hat\beta$ and $Y-X\hat\beta$.