(Multiple regression model with p's predictor variables.)
Derive the distribution of
$$\frac{(b-\beta)X'X(b-\beta)}{MSE\cdot p}$$
As far as I know,
$b\sim N(\beta,\sigma^2 (X'X)^{-1})$
$b-\beta \sim N(0,\sigma^2 (X'X)^{-1})$
$\frac{b-\beta}{\sigma} \sim N(0,(X'X)^{-1})$
$\frac{X(b-\beta)}{\sigma } \sim N(0,X(X'X)^{-1}X')$
$\frac{(b-\beta)'X'X(b-\beta)}{\sigma^2}\sim \chi^2(rank(X'X);\delta=0)$ **
Since $X'X$ is full rank, $rank(X'X)=p$ **
Thus, $\frac{(b-\beta)'X'X(b-\beta)}{\sigma^2}\sim \chi^2(p)$
Since we don't know about the parameter $\sigma^2$, plug in MSE in place of that.
It becomes approximately $\frac{(b-\beta)'X'X(b-\beta)}{MSE}\sim \chi^2(p)$ **
And finally dividing p, we get $\frac{(b-\beta)'X'X(b-\beta)}{MSE\cdot p}\sim \chi^2(1)$ **
The mark "**" means I'm not sure because of my lack of mathematical justification. Also I'm not sure whether this solution is correct.(the solution is from me, of course.) Is there anybody who can help me get the answer and improve my statistical skills?