# panel data logit model

## Consider the following panel data logit model

Y =1{X′β+a−u >0} it it i it

for i = 1,2,··· ,N and t = 1,2. uit is i.i.d over (i,t) with a logistic distribution. The regressors X are strictly exogenous with respect to the transitory shock uit. However, the individual effects ai can be correlated with the regressors. Define Yi ≡ {Yi1,Yi2},Xi ≡ {Xi1, Xi2} and Si ≡ Yi1 + Yi2.

a. Obtain the closed-form CCP of Yit given both Xit and ai for each t = 1, 2.

b. Obtain the closed-form CCP of Yi given Xi, ai and Si for Si = 0 and Si = 2. Show this does not depend on ai.

c. Obtain the closed-form CCP of Yi given Xi, ai and Si for Si = 1. Show this does not depend on ai.

d. Can you construct a maximum-likelihood estimator using the CCP for which Si = 0 and Si = 2? How about using the CCP for which Si = 1?