# What is the probability distribution and variance of the OLS estimate $s^2$ of the error variance $\sigma^2$ in linear regression?

Consider the standard linear regression model $$y = X \beta + \varepsilon,$$ where the error $$\varepsilon$$ has fixed variance $$\sigma^2$$. We can make an unbiased estimate of the error variance in a linear regression model using the OLS estimator $$s^2 = \frac{1}{n-p}\sum_{i=1}^n(y_i-\hat y_i)^2$$ where $$p$$ is the number of regressors.

I am wondering what is the variance of the estimate $$s^2$$ and what is it's probability distribution? If we need to make assumptions about $$\varepsilon$$ before we can say anything about the distribution of $$s^2$$, then assume $$\varepsilon$$ is normally distributed.

I'm interested in the case of stochastic regressors (as opposed to fixed), if that makes a difference.

• en.wikipedia.org/wiki/… – user2974951 Nov 6 '20 at 11:44
• I don't see the variance or probability distribution of $s^2$ listed there. – Bertus101 Nov 6 '20 at 12:45
• Hints: (1) You must make a strong distributional assumption, as you suggest. (2) Because $\hat y,$ being a linear combination of the $y_i,$ is Normal when $\varepsilon$ is Normal, the distribution of a suitable multiple of $s^2$ is a chi-square distribution. The rest is just calculation. – whuber Nov 6 '20 at 12:54
• Ok, I'll try it out thanks. – Bertus101 Nov 6 '20 at 13:41
• Something else is that as $n$ increases, it asymptotically converges in distribution to the normal distribution by a central limit theorem argument (under certain regularity conditions). This is consistent with @whuber's comment for the special case where $\epsilon$ is normal: the $\chi^2$ distribution with $k = n-p$ degrees of freedom approaches the normal distribution for large $k$. – Matthew Gunn Nov 6 '20 at 14:22