498 Discrete Choice Modeling
population values of both coefficients are 1.0. The results, which are consistent
with other studies, e.g., Katz (2001), suggest the persistence of the “smallTbias”
out to fairly largeT.These problems, particularly the last, have made the full fixed effects approach
unattractive. The specification, however, remains an attractive alternative to the
random effects approach considered next. Two approaches have been taken to
work around the incidental parameters problem in the fixed effects model. A vari-
ety of semiparametric models have been suggested, such as Honore and Kyriazidou
(2000a, 2000b) and Honore (2002).^15 In a few cases, including the binomial logit
(but not the probit), it is possible to condition the fixed effects out of the model.
The operation is similar to the group mean deviations transformation in the lin-
ear regression model. For the binary logit model (omitting the time invariant
variables), we have:
Prob(dit=jit|xit)=
exp[jit(αi+x′itβ)]
1 +exp(αi+x′itβ)wherejitis the observed value.This is the term that enters the unconditional log-likelihood function. However,
conditioning onTt=i 1 jit=Si, we have the joint probability:
Prob(di 1 =ji 1 ,di 2 =ji 2 ,...|xit,tT=i 1 dit=Si)=exp(tT=i 1 jitx′itβ)
∑
tdit=Siexp(Ti
t= 1 ditx′
itβ).(See Rasch, 1960; Andersen, 1970; Chamberlain, 1980.) The denominator of
the conditional probability is the summation over the different realizations of
(di 1 ,...,di,Ti)that can sum toSi. Note that, in this formulation, ifSi=0orTi,
there is only one way for the realizations to sum toSi, and the one term in the
denominator equals the observed result in the numerator. The probability equals
one and, as noted in point 3 above, this group falls out of the estimator. The
conditional log-likelihood is the sum of the logs of the joint probabilities. The
log-likelihood is free of the fixed effects, so the estimator has the usual proper-
ties, including consistency. This estimator was used by Cecchetti (1986) and Willis
(2006) to analyze magazine price changes.
The conditional estimator is consistent, so it bypasses the incidental parameter
problem. However, it does have a major shortcoming. By avoiding the estimation of
the fixed effects, we have precluded computation of the partial effects or estimates
of the probabilities for the outcomes. So, like the robust semiparametric estimators,
this approach limits the analyst to simple inference aboutβitself. One approach
that might provide some headway out of this constraint is to compute second-step
estimates ofαi. Since we have in hand a consistent estimator ofβ, we treat that as
known, and return to the unconditional log-likelihood function. For individuali,
the contribution to the log-likelihood is:
lnLi=∑Ti
t= 1 lnF[qit(αi+x′
itβ)].