Consider the Honore and Kyriazidou (2000b) fixed effects logit model given in (11.19).

(a) Show that for \(T=3, \operatorname{Pr}\left[A / x_{i}^{\prime}, \mu_{i}, A \cup B\right]\) and \(\operatorname{Pr}\left[B / x_{i}^{\prime}, \mu_{i}, A \cup B\right]\) both depend on \(\mu_{i}\). This means that the conditional likelihood approach will not eliminate the fixed effect \(\mu_{i}\).

(b) If \(x_{i 2}^{\prime}=x_{i 3}^{\prime}\), show that \(\operatorname{Pr}\left[A / x_{i}^{\prime}, \mu_{i}, A \cup B, x_{i 2}^{\prime}=x_{i 3}^{\prime}\right]\) and \(\operatorname{Pr}\left[B / x_{i}^{\prime}, \mu_{i}\right.\), \(A \cup B\), \(\left.x_{i 2}^{\prime}=x_{i 3}^{\prime}\right]\) do not depend on \(\mu_{i}\).

\[\begin{align*}
& \operatorname{Pr}\left[y_{i 0}=1 / x_{i}^{\prime}, \mu_{i}\right]=p_{0}\left(x_{i}^{\prime}, \mu_{i}\right)  \tag{11.19}\\
& \operatorname{Pr}\left[y_{i t}=1 / x_{i}^{\prime}, \mu_{i}, y_{i 0}, \ldots, y_{i, t-1}\right]=\frac{e^{x_{i t}^{\prime} \beta+\gamma y_{i, t-1}+\mu_{i}}}{1+e^{x_{i t}^{\prime} \beta+\gamma y_{i, t-1}+\mu_{i}}} \quad t=1, \ldots, T \end{align*}\]