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

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

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

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

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


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

Thanks