Test for overall significance of regression when the variance of errors is known 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?

The F-distribution is defined as the distribution of
$$ F=\frac{S_1/\nu_1}{S_2/\nu_2}$$
where $S_1, S_2$ are independent chi-squared random variables with $\nu_1, \nu_2$ degrees of freedom.
The error estimate is represented by $S_2$, and when that is known, in some sense it have $\nu_2=\infty$ degrees of freedom. But the limit (in distribution or in probability) of $\frac{S_2}{\nu_2}$  when $\nu_2 \to \infty$ is 1, and then the F ratio defined above reduces to $\frac{S_1}{\nu_1}$, which has a chisquared distribution.
So the usual F test reduces to a chisquared test.
