How do you derive the conditional variance for $s^2$, the OLS estimator of $\sigma^2$?

Question

I just need a little bit of a push in the right direction. I'm working my way through Hayashi's Econometrics and hit a snag in section 1.4. Review question 7 asks:

Show that, under Assumptions 1.1-1.5,

$Var(s^2|X)=\frac{2\sigma^4}{n-K}$

Hint: If a random variable is distributed as $\chi^2(m)$, then its mean is $m$ and variance $2m$.

I figure this needs to be broken into two parts – the first showing that $s^2$ follows a $\chi^2$ distribution, and the second part showing that the mean is the expression above sans the 2.

The book gives a couple of hints about the kinds of things that follow $\chi^2$ distributions. Here's a footnote on page 41:

Fact: Let x be an $m$ dimensional random vector. If $x~N(\mu,\Sigma)$ with $\Sigma$ nonsingular, then $(x-\mu)'\Sigma^{-1}(x-\mu)\sim\chi^2(m)$.

This doesn't do me much good though. Secondly, there's this bit on page 37:

Fact: If $x\sim N(0,I_n)$ and $A$ is idempotent, then $x'Ax$ has a chi-squared distribution with degrees of freedom equal to the rank of A.

But $\varepsilon$ (measurement error) doesn't follow the standard normal distribution – its variance is $\sigma^2$, so this isn't much use to me either. I'm just starting out and not really sure how to tackle this. Could someone give me a hand?

Answer 1

Browsing around in the online Google version of the book it seems to me that Assumption 1.5 is the normality assumption. In that case the proof of Proposition 1.3 says that $q|X \sim \chi^2(n-K)$ where $q = (n-K)s^2/\sigma^2$. Thus $$\begin{array}{rcl} \text{Var}(s^2|X) & = & \text{Var}(\sigma^2 q/(n-K)|X) \\ & = & \frac{\sigma^4}{(n-K)^2} \text{Var}(q|X) \\ & = & \frac{\sigma^4}{(n-K)^2} 2(n-K) \\ & = & \frac{2\sigma^4}{n-K} \end{array}$$ where we used the hint for the third equality.