We consider $Y=X\beta+\epsilon$, $\epsilon \sim N(0,\sigma^2 I)$.
Let H$_0$ is $\beta_0=\dots\beta_k=0$ vs
exist $i\in{0,1,\dots,k}: \beta_i\neq 0$
How to proof that if H$_0$ is true then $F=\frac{n-1-k}{k}\frac{\sum_{i=1}^n(\hat{Y}_i-\overline Y)^2}{\sum_{i=1}^n(\hat{Y}_i-Y_i)^2}\sim F_{(k,n-k-1)}$. I can only proof, that $\frac{1}{\sigma^2}\sum_{i=1}^n(\hat{Y}_i-Y_i)^2\sim \chi^2_{n-k-1}$. So exactly, we should show, that
$\frac{1}{\sigma^2}\sum_{i=1}^n(\hat{Y}_i-\overline Y)^2\sim\chi^2_{k}$. But I don't know, how to do it. Can anyone give me some advice? i will be very gratefull.
Note that $\hat{y}_i$ is a linear combination of $\hat{\beta}$, so it is normal and from the OLS derivation you know that $\frac{1}{n}\sum_{i=1}^n\hat{y}_i = \frac{1}{n}\sum_{i=1}^ny_i$, thus $\frac{1}{\sigma^2}\sum_{i=1}^n(\hat{y}_i - \bar{y}_n)^2$ is compatible with the very definition of the $\chi^2$ distribution; $Z_{1},...,Z_{n}$ are $N(0,1)$, then $\sum Z_i^2\sim \chi^2{(n)}$.