Standard Error of Noise Variance in Least Squares Suppose I'm doing ordinary least squares with homoskedastic errors, so something like:
$$
y = X \beta + \epsilon
$$
where $\epsilon \sim \mathcal{N}(0,\sigma^2)$.
I know how to estimate the expected value of $\sigma^2$, but I'm having trouble identifying how to compute a standard error on that estimate. Something analogous to what's shown on page 47 (pdf page 58) of this book.

$\DeclareMathOperator{\Var}{Var}$
Suppose the design matrix is $N \times (p + 1)$, I am assuming you want to calculate the variance of $\hat{\sigma}^2 = \frac{1}{N - p - 1}\epsilon'(I - H)\epsilon$, which is an unbiased estimator of $\sigma^2$, where $I$ is the order $N + 1$ identity matrix and $H = X(X'X)^{-1}X'$.  In this thread, it has been shown that
\begin{align}
(N - p - 1)\hat{\sigma}^2/\sigma^2 \sim \chi_{N - p - 1}^2,
\end{align}
by which and the variance of a $\chi_k^2$ r.v. is $2k$ follows that
\begin{align}
\Var(\hat{\sigma}^2) = \frac{\sigma^4}{(N - p - 1)^2} \times 2(N - p - 1) = 
\frac{2\sigma^4}{N - p - 1}.
\end{align}
Estimate the $\sigma^4$ in the numerator by $\hat{\sigma}^4$, it follows that the standard error of $\hat{\sigma}^2$ is $\frac{\sqrt{2}\hat{\sigma}^2}{\sqrt{N - p - 1}}$.
