530 Discrete Choice Modeling
whereγ=α/( 1 −ρ^2 )^1 /^2 andτ=ρ/( 1 −ρ^2 )^1 /^2. There is a minor extension of
this model that might be interesting for the health care application examined in
this study. The count variables and all the covariates in both equations would be
observed for all observations. Thus, to use the full sample of data, the appropriate
log-likelihood would be:
f(yi,zi|xi,zi)=∫∞−∞f(yi|xi,εi)]#(
( 2 si− 1 )[z′iγi+τεi])
φ(εi)dεi.11.6.2.4 Bivariate Poisson model
The application from which our examples are drawn was a study of the two count
variables, DocVis (visits to the doctor) and HospVis (visits to the hospital). Riphahn,
Wambach and Million (2003) were interested in a bivariate count model for the
two outcomes. One approach to formulating a two-equation Poisson model is to
treat the correlation as arising from the intervention of a latent common Poisson
process. The model is:
y 1 =y 1 ∗+U
y 2 =y∗ 2 +U,wherey∗ 1 ,y∗ 2 andUare three independent Poisson processes. This model is anal-
ogous to the seemingly unrelated regressions model (see King, 1989). The major
shortcoming of this approach is that it forces the two variables to be positively cor-
related. For the application considered here, it is at least possible that the preventive
motivation for physician visits could result in a negative correlation between physi-
cian and inpatient hospital visits. The approach proposed by Riphahn, Wambach
and A. Million (2003) adapted for a random effects panel data model, isyit,j∼
Poisson (λit,j), where:
λit,j=exp(x′it,jβ+ui,j+εit,j), j=1, 2.and where the unique heterogeneity, (εit,1,εit,2), has a bivariate normal distri-
bution with correlationρ, and the random effects, which are constant through
time, have independent normal distributions. Thus the correlation between the
conditional means is that induced by the two log-normal variables, exp(εit,1)and
exp(εit,2). The implied correlation between yit,1and yit,2was not derived. This
would be weaker thanρ, since both variables have additional variation around the
correlated conditional mean functions.
In order to formulate the log-likelihood function, the random components must
be integrated out. There are no closed forms for the integrals based on the nor-
mal distribution – the problem is similar to that in the sample selection model.
The authors used a quadrature procedure to approximate the integrals. The log-
likelihood could also be maximized by using simulation. Separate models were
fitted to men and women in the sample. The pooling hypothesis was rejected for
all specifications considered.