I have the following dataset: at time $t_1$, a person faces the following ordered choice set $y \in [0,1]$, and at time $t_2$, he faces the ordered choice set $y \in [0,1,2]$.

Assume the utility is $y^*=X\beta + \epsilon$. I know that at time t, the underlying process can be modeled like this

$y=\begin{cases} 0, y^* \leq \mu_0 \\1, \mu_0 < y^* \end{cases}$

At time t2, the underlying process is this

$y=\begin{cases} 0, y^* \leq \mu_0 \\1, \mu_0 < y^* \leq \mu_1, \\2, \mu_1 < y^* \leq \mu_2\end{cases} $

The two are ordered logit. My question is how to estimate the parameters in observations in $t_1$ and observations in $t_2$ in one go (ensure $\mu_0$ and $\beta$ are the same across two periods)?

Any suggestions or recommended readings are welcomed! Thanks!


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