# Why is this statistic F-distributed?

A book I'm reading claims that the statistic:

$$\frac{(RSS_0 - RSS_1) / (p_1 - p_0)}{RSS_1 / (N - p_1 - 1)}$$ has an F distribution. Why is this? I know that an F distribution is something like $$\frac{\chi^2_p / p}{\chi^2_q / q}$$, where the two chi-square distributions are independent, but I fail to see why $$RSS_0 - RSS_1$$ is chi-squared and also why $$RSS_0 - RSS_1$$ and $$RSS_1$$ are independent.

For some context, $$RSS_1$$ is the $$RSS$$ of a least squares model with $$p_1 + 1$$ parameters, and $$RSS_0$$ is a smaller model with $$p_1 - p_0$$ of the parameters in the first model set to 0.

• One of the main assumptions of the OLS model is that the error term is normally distributed. As you might know the square of a normal random variable is distributed as a Chi square. The sum of Chi squares is once again Chi square – RScrlli Mar 27 at 21:36
• @RScrlli the sum of chi squares is chi square, but I don't think the difference is. – serendipity Mar 27 at 21:59
• In this case, the difference can be expressed as a sum of squares. – whuber Mar 27 at 22:26
• @serendipity Difference of (independent) chi-square variables is definitely not another chi-square. But this is not the setup here. – StubbornAtom Mar 28 at 10:30
• And I think your result follows from a theorem on quadratic forms related to Fisher-Cochran theorem. – StubbornAtom Mar 28 at 11:16