Consider a multiple regression model $\displaystyle \displaystyle y = X \beta + u$ where $\displaystyle \displaystyle u \sim N (0,\sigma^2 I_n)$

where $\displaystyle \displaystyle \beta$ is a $\displaystyle \displaystyle K \times 1$ vector, $\displaystyle \displaystyle X$ is a non-stochastic full rank $\displaystyle \displaystyle n \times K$ matrix and $\displaystyle \displaystyle \sigma^2 > 0$ in an unknown parameter.

We have $\displaystyle \displaystyle \hat{\sigma}^2 = \frac{1}{n}\hat{u}'\hat{u}$ and $\displaystyle \displaystyle s^2 = \frac{1}{n-K}\hat{u}'\hat{u}$ where $\displaystyle \displaystyle \hat{u}=y-X\hat{\beta}$

(a) Calculate the MSE of $\displaystyle \displaystyle s^2$ and $\displaystyle \displaystyle \hat{\sigma}^2$

(b) For each $\displaystyle \displaystyle K \in \{ 1,...,7 \}$ determine the valye of $\displaystyle \displaystyle n$ for which $\displaystyle \displaystyle MSE(\hat{\sigma}^2) \le MSE(s^2)$

Now I can do part of (a). I can calculate the MSE of $\displaystyle \displaystyle s^2$. But I don't know how to do it for $\displaystyle \displaystyle \hat{\sigma}^2$ nor do I know how to do (b). Any hints or tips?