532 Discrete Choice Modeling
finesses the incidental parameters problem – the estimator ofβin this model is
consistent. This is not the case for the FENB2 form, where:
Qi=
θ
θ+tT=i 1 λit.For estimation purposes, we have a negative binomial distribution forYi=tyit
with mean%i=tλit.
Like the fixed effects model, introducing random effects into the negative bi-
nomial model adds some additional complexity. We do note, since the negative
binomial model derives from the Poisson model by adding latent heterogeneity to
the conditional mean, that adding a random effect to the negative binomial model
might well amount to introducing the heterogeneity a second time. However, one
might prefer to interpret the negative binomial as the density foryitin its own
right, and treat the common effects in the familiar fashion. Hausman, Hall and
Griliches’ (1984) random effects negative binomial model is a hierarchical model
that is constructed as follows. The heterogeneity is assumed to enterλitadditively
with a gamma distribution with mean 1,(θi,θi). Then,θi/(1+θi)is assumed to have
a beta distribution with parametersaandb. The resulting unconditional density
after the heterogeneity is integrated out is:
p(yi 1 ,yi 2 ,...,yiTi)=(a+b)(
a+tT=i 1 λit)
(
b+Tt=i 1 yit)(a)(b)(
a+tT=i 1 λit+b+Tt=i 1 yit).As before, the relationship between the heterogeneity and the conditional mean
function is unclear, since the random effect impacts upon the parameter of the
skedastic function. An alternative approach that maintains the essential flavor of
the Poisson model (and other random effects models) is to augment the NB2 form
with the random effect:
Prob(Y=yit|xit,εi)=(θ+yit)
(yit+ 1 )(θ)
r
yit
it(^1 −rit)θ,λit=exp(xit′β+εi),
rit=λit/(θ+λit).We then estimate the parameters by forming the conditional (onεi)log-likelihood
and integratingεiout either by quadrature or simulation. The parameters are sim-
pler to interpret by this construction. Estimates of the two forms of the random
effects model are presented below for comparison.
11.6.4 Application
The study by Ripahn, Wambach and Million (2003) that provided the data we
have used in numerous earlier examples analyzed the two count variables DocVis
and HospVis. The authors were interested in the joint determination of these two