William Greene 503random effects model that can be estimated, as in the static case, by Hermite
quadrature of maximum simulated likelihood (MSL).
Much of the contemporary literature has focused on methods of avoiding the
strong parametric assumptions of the probit and logit models. Manski (1987) and
Honore and Kyriadizou (2000a) show that Manski’s (1986) maximum score esti-
mator can be applied to the differences of unequal pairs of observations in a
two-period panel with fixed effects. An extension of lagged effects to a paramet-
ric model is Chamberlain (1980), Jones and Landwehr (1988) and Magnac (1997),
who added state dependence to Chamberlain’s fixed effects logit estimator. Unfor-
tunately, once the identification issues are settled, the model is only operational if
there are no other exogenous variables in it, which limits its usefulness for practi-
cal application. Lewbel (2000) has extended his fixed effects estimator to dynamic
models as well. In this framework, the narrow assumptions about the independent
variables once again limit its practical applicability. Honore and Kyriazidou (2000b)
have combined the logic of the conditional logit model and Manski’s maximum
score estimator. They specify:
Prob(di 0 = 1 |Xi,zi,αi)=F 0 (Xi,zi,αi), whereXi=(xi 1 ,xi 2 ,...,xiT),
Prob(dit= 1 |Xi,zi,αi,di 0 ,di 1 ,...,di,t− 1 )=F(x′itβ+z′iγ+αi+λdi,t− 1 )t=1,...,T.The analysis assumes a single regressor and focuses on the case ofT =3. The
resulting estimator resembles Chamberlain’s but relies on observations for which
xit=xi,t− 1 , which rules out direct time effects as well as, for practical purposes,
any continuous variable. The restriction to a single regressor limits the generality
of the technique as well. The need for observations with equal values ofxitis a
considerable restriction, and the authors propose a kernel density estimator for
the difference,xit−xi,t− 1 , instead, which does relax that restriction a bit. The
end result is an estimator which converges (they conjecture) but to a non-normal
distribution and at a rate slower thann−^1 /^3.
Semiparametric estimators for dynamic models at this point in the development
are still primarily of theoretical interest. Models that extend the parametric for-
mulations to include state dependence have a much longer history, including
Heckman (1978, 1981a, 1981b), Heckman and Macurdy (1981), Jakubson (1988),
Keane (1993) and Beck, Epstein and Jackman (2001), to name just a few.^19
11.3.6.5 Parameter heterogeneity: random parameters and latent class models
Among the central features of panel data treatments of individual data is the
opportunity to model individual heterogeneity, both observed and unobserved.
The preceding discussion develops a set of models in which latent heterogeneity is
embodied in the additive effect,αi. We can extend the model to allow heterogeneity
in the other model parameters as well,. The resulting specification is:
d∗it=w′itθi+αi+εi, dit= 1 (dit∗> 0 ). (11.10)