We considered the fixed effects logit model with \(T=2\).

(a) In this problem, we look at \(T=3\) and we ask the reader to compute the conditional probabilities that would get rid of the individual effects by conditioning on \(\sum_{t=1}^{3} y_{i t}\). Note that this sum can now be \(0,1,2\), or 3 . First show that terms in the conditional likelihood function, which are conditioned upon \(\sum_{t=1}^{3} y_{i t}=0\) or 3 add nothing to the likelihood. Then focus on terms that condition on \(\sum_{t=1}^{3} y_{i t}=1\) or 2.

(b) Show that for \(T=10\), one has to condition on the sum being \(1,2, \ldots, 9\). One can see that the number of probability computations are increasing. To convince yourself, write down the probabilities conditioning on \(\sum_{t=1}^{10} y_{i t}=\) 1 .