Say we want to test the overall model adequacy in a multiple linear regression model:
$$H_0: B_1 = B_2 = ... = 0 $$ $$H_1: B_j \neq 0 \text{ for at least one j}$$
the random errors $\epsilon$ are statistically independent and normally distributed with mean zero and constant variance $\sigma^2$.
If the variance is known, can we still do an F test by replacing $MS_{Res}$ with the known $\sigma^2$. If so do we still use the same degrees of freedom $(n-k-1)$? In other words is our test model:
$$F_0 = (SS_{Reg}/k)/\sigma^2$$
and does that follow a an F distribution with k and n-k-1 degrees?
