= Solution
Model 2 is obtained from model 3 by setting $\beta_0=0$. Model 3 is contained in model 1 by choosing the symmetric block parameters
$$
\beta_{k\ell}^{(1)}=
\begin{cases}
\beta_k+\beta_\ell,&k\ne\ell,\\
2\beta_k+\beta_0,&k=\ell.
\end{cases}
$$
Hence
$$
\boxed{\text{model 2}\subset\text{model 3}\subset\text{model 1}.}
$$
Assuming every college-pair cell is observed and the models are identifiable, model 1 has $C(C+1)/2$ parameters and model 3 has $C+1$. The <likelihood-ratio test statistic> for model 3 against model 1 is
$$
D=2\{\ell_1(\widehat\beta_1)-\ell_3(\widehat\beta_3)\}.
$$
Under model 3 and the regularity conditions of <Wilks theorem>,
$$
\boxed{D\ \xrightarrow{d}\ \chi^2_{\,C(C+1)/2-(C+1)}
=\chi^2_{(C-2)(C+1)/2}.}
$$
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