# Ordered logit with different choice sets at different times

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!