I was wondering whether $\frac{\chi^2_n}{n}$ is stochastically increasing in $n$.
My main problem: Suppose $\hspace{5pt}\frac{(n-p)\hat{\sigma}^2}{\sigma^2} \sim \chi^2_{n-p}$. Then the expected length of C.I. increases if one over-parametrizes the linear model or, in other words,
if $n-p=k \uparrow$ $\Rightarrow$ $k[\frac{1}{u_k}-\frac{1}{v_k}] \downarrow$ where $P(u_k<\chi^2_k<v_k)=1-\alpha$
Of course, I was unable to show it. So I considered the simple case: $v_k=\infty$, in that case the above reduces to
$$k \uparrow \hspace{5pt}\Rightarrow \frac{k}{\chi^2_{k;\alpha}} \downarrow \hspace{5pt}\Rightarrow \frac{\chi^2_{k;\alpha}}{k} \uparrow$$
If $\frac{\chi^2_n}{n}$ is Stochastically increasing in $n$ then the above claim will be true, however it may not be mandatory.
Let $X_n\sim \chi_n^2$. Asymptotically
$$P\left\{\frac{X_n}{n}\le x\right\}\to 1\{x\ge 1\}$$
as $n\to \infty$ because
$$\varphi_{X_n/n}(t)=\left(1-\frac{2it}{n}\right)^{-n/2}\to e^{it}$$
which is the c.f. of $\delta_1$. Thus $\left[\frac{u_n}{n},\frac{v_n}{n}\right]$ shrinks as $n\uparrow$. You may also note that since $\frac{X_n}{n}\sim \Gamma\left(\frac{n}{2},\frac{2}{n}\right)$
$$P\left\{\left|\frac{X_n}{n}-1\right|\le \frac{2}{\alpha n}\right\}\ge1-\alpha$$
so that $\left(\frac{v_n}{n}-\frac{u_n}{n}\right)=O(n^{-1})$ (for large $n$, $X_n/n$ is roughly symmetric around 1).