Consider the Chamberlain (1985) fixed effects conditional logit model with a lagged dependent variable given in (11.16). Show that for \(T=3, \operatorname{Pr}\left[A / y_{i l}+y_{i 2}=1, \mu_{i}\right]\) and therefore \(\operatorname{Pr}\left[B / y_{i 1}+y_{i 2}=1, \mu_{i}\right]\) do not depend on \(\mu_{i}\). Note that \(A\) and \(B\) are defined in (11.17) and (11.18), respectively.

\[\begin{align*}
& \operatorname{Pr}\left[y_{i 0}=1 / \mu_{i}\right]=p_{0}\left(\mu_{i}\right) \\
& \operatorname{Pr}\left[y_{i t}=1 / \mu_{i}, y_{i 0}, y_{i 1}, \ldots, y_{i, t-1}\right]=\frac{e^{\gamma y_{i, t-1}+\mu_{i}}}{1+e^{\gamma y_{i, t-1}+\mu_{i}}} \quad t=1, \ldots, T \tag{11.16}
\end{align*}\]

\[\begin{align*}
A & =\left\{y_{i 0}=d_{0}, y_{i 1}=0, y_{i 2}=1, y_{i 3}=d_{3}\right\}  \tag{11.17}\\
B & =\left\{y_{i 0}=d_{0}, y_{i 1}=1, y_{i 2}=0, y_{i 3}=d_{3}\right\} \tag{11.18}
\end{align*}\]