536 Discrete Choice Modeling
health care system and a process that generates the count when they do. Nonethe-
less, there is little doubt that both are improvements on the Poisson regression.
The average predicted probability of the zero outcome is 0.04826, so the Poisson
model predictsnPˆ 0 =1, 319 zero observations. The frequency in the sample is
10,135. The counterparts for the ZIP model are 0.36340 and 9,930. The Poisson
model is not nested in the ZIP model – setting the ZIP coefficients to zero forces
the regime probability to 0.5, not to 1.0. Thus the models cannot be compared
by log-likelihoods. The Vuong statistic strongly supports the zero inflation model,
being+47.05. Similar results are obtained for the hurdle model with the same
specification.
The German health care panel data set contains 7,293 individuals with group
sizes ranging from 1 to 7 and Table 11.9 presents the fixed and random effects
estimates of the equation for DocVis. The pooled estimates are also shown for
comparison. Overall, the panel data treatments bring large changes in the esti-
mates compared to the pooled estimates. There is also a considerable amount of
variation across the specifications. With respect to the parameter of interest,Public,
we find that the size of the coefficient falls substantially with all panel data treat-
ments. Whether using the pooled, fixed or random effects specifications, the test
statistics (Wald, LR) all reject the Poisson model in favor of the negative binomial.
Similarly, either common effects specification is preferred to the pooled estimator.
There is no simple basis for choosing between the fixed and random effects mod-
els, and we have further blurred the distinction by suggesting two formulations for
each of them. We do note that the two random effects estimators are producing
similar results, which one might hope for, but the two fixed effects estimators are
producing very different estimates. The NB1 estimates include two coefficients, on
IncomeandEducation, that are positive, but negative in every other case.
Moreover, the coefficient onPublic, which is large and significant throughout
the table, becomes small and less significant with the fixed effects estimators.
11.7 Multinomial unordered choices
We now extend the random utility, discrete choice model of sections 11.2–11.4 to a
setting in which the individual chooses among multiple alternatives (see Hensher,
Rose and Greene, 2005). The random utility model is:
Uit,j=x′it,jβ+z′itγ+εit,j, j=1,...,Jit, t=1,...,Ti,where, as before, we consider individualiin choice situationt, choosing among a
possibly variable number of choices,Jit, and a possibly individual specific number
of choice situations. For the present, for convenience, we assumeTi=1 – a single-
choice situation. This will be generalized later. The extension to variable choice set
sizes,Jit, turns out to be essentially a minor modification of the mathematics, so it
will also prove convenient to assumeJitis fixed atJ. The random utility model is
thus:
Ui,j=x′i,jβ+z′iγ+εi,j, j=1,...,J, i=1,...,n.