A hideous Linear Regression/confidence set question

[Phillips101]

Take the linear model Y=X*beta+e, where e~Nn(0, sigma^2 * I), and it has MLE beta.hat

First, find the distribution of (beta.hat-beta)' * X'*X * (beta.hat-beta), where t' is t transpose. I think I've done this. I think it's a sigma^2 chi-squared (n-p) distribution.

Next, Hence find a (1-a)-level confidence set for beta based on a root with an F distribution. I can't do this to save my life. I'm aware that an F distribution is the ratio of two chi-squareds, but where the hell I'm going to get another chi squared from I have no idea. Also, we're dealing in -vectors- and I don't know how,what,why any confidence set is going to be or even look like, and I've no idea how to even try to get one.

-Any- help would be appreciated. Thanks

 

[statdad]

Notice that

$$\frac{\hat{\beta}' X'X \hat{\beta}}{\sigma^2}$$

has a $$\Chi^2$$ distribution. however, the variance is unknown, so you need to estimate it (with another expression from the regression). What would you use for the estimate, and what is its distribution?

 

[Phillips101]

Use the MLE sigma2.hat=(1/n)*||Y-Xbeta.hat||^2 ? This is distributed as a chi-squared n-1 variable if I remember correctly...

 

[Phillips101]

If that's correct, then the thing you posted is distributed as an F distribution, which is what I need? And would swapping beta.hat for beta.hat-beta make any difference to this?