Palgrave Handbook of Econometrics: Applied Econometrics

504 Discrete Choice Modeling

The specification is completed by the assumptions about the process that generates
the individual specific parameters. Note that, in this formulation, the “effect”αiis
now merely an individual specific constant term. It is thus convenient to absorb it
into the rest of the parameter vector,θi, and assume thatwitcontains a constant.
A random parameters model (or mixed model or hierarchical model), in which
parameters are continuously distributed across individuals, can be written:

θi=θ 0 + zi+ui,

whereuiis a set of uncorrelated random variables with zero means (means are
absorbed inθ 0 )and unit variances (non-unit variances are contained in the param-
eter matrix). The random effects model examined earlier emerges if = 0 and
the only random component inθiis the constant term, in which casewould
have a single nonzero diagonal element equal toσu. For the more general case, we
have a random parameters formulation in which:

E[θi|zi]=θ 0 + zi

Var[θi|zi]=′.

A random parameters model of this sort can be estimated by Hermite quadrature
(see Rabe-Hesketh, Skrondal and Pickles, 2005) or by MSL (see Train, 2003; Greene,
2008a, Ch. 17, 23). The simulated log-likelihood function for this model will be:

lnLS=

∑n

i= 1 ln

1

R

∑R

r= 1

∏Ti

t= 1 #

(

qit(w′it(θ 0 +zi+uir))

)

.

Partial effects in this model can be computed by averaging the partial effects at the
population conditional means of the parameters,E[θi|zi]=θ 0 + zi.

11.3.7 Application

Riphahn, Wambach and Million (2003) were interested in counts of physician and
hospital visits. In this application, they were particularly interested in the impact
that the presence of private insurance had on utilization counts, i.e., whether the
data contain evidence of moral hazard. The sample is an unbalanced panel of
7,293 households. The number of households varies over seven periods (1,525;
1,079; 825; 926; 1,051; 1,000; 887) with a total number of 27,326 observations.
The variables in the data file are listed in Table 11.2. (Only a few of these were used
in the applications.)
The model to be examined here (not the specification used in the original
study) is:

Prob(Doctorit= 1 |xit)=F(β 1 +β 2 Ageit+β 3 Incomeit+β 4 Kidsit

+β 5 Educationit+β 6 Marriedit).