598 Panel Data Methods
estimated using Gauss–Hermite quadrature. An application to the 1997 reform of
co-payments for prescription drugs in Germany uses data on quarterly doctor vis-
its in the GSOEP. This confirms Deb and Trivedi’s (2002) result that finite mixture
models outperform Negbin hurdle models, but the results show that the normal
hurdle model fits better than both of these specifications.
The innovation in Van Ourti (2004) is to include a Gaussian random effect in
the two-part model. The time invariant individual effect appears as a common
factor in both parts of the model, with the factor loading in the first equation
normalized to one for identification. The individual effect is then integrated out
with the resulting integral evaluated by Gauss–Hermite quadrature. In an empirical
application to GP and specialist visits and to nights in hospital in the Panel Study
of Belgian Households (PSBH), the panel version of the two-part model (2PM-PA)
is compared to a one part-model with a Gaussian individual effect (1PM-PA) and to
pooled versions of the one-part and two-part models (1PM-PO, 2PM-PO). On the
basis of the log-likehoods and the Akaike information criterion (AIC) and Bayesian
information criterion (BIC), the 2PM-PA specification is preferred to the simpler
specifications.
While Van Ourti (2004) extends the Gaussian random effects model to the two-
part specification for count data, Munkin and Trivedi (1999) and Riphahnet al.
(2003) do the same for a bivariate count data model, dealing with the case where
there are two dependent variables both measured as integer counts. Munkin and
Trivedi (1999) propose a model that is designed for cross-section data and that is
applied to the number of emergency room visits and of hospitalizations in data
from the US National Medical Expenditure Survey, 1987–88 (as used by Deb and
Trivedi, 1997). They construct the bivariate model by specifying the marginal dis-
tributions of the counts and then adding a correlated heterogeneity term to give a
Poisson-log-normal mixture. This has conditional mean functions:
λi 1 =exp(x′i 1 β 1 +εi 1 )
λi 2 =exp(x′i 2 β 2 +εi 2 )
(εi 1 ,εi 2 )∼N[(0, 0),(σ 12 ,ρσ 1 σ 2 ,σ 22 )].(12.33)The log-likelihood function for the model involves a two-dimensional integral and
is estimated by MSL with antithetics and a correction for first order simulation bias.
Using a bivariate model does not lead to substantial changes in the estimated effects
of the regressors, but the overall fit of the model does improve.
The model proposed by Riphahnet al.(2003) is designed for panel data and they
apply it to separate measures of doctor visits and hospital inpatient visits from 12
waves of the GSOEP for 1984–95, focusing on West Germans aged between 25 and
- Their specification extends the single equation model of Geilet al.(1997) by
adding two Gaussian error components to each equation; one, a time invariant
individual effect (uij); the other, a time-varying idiosyncratic error term (εitj):
λit 1 =exp(x′it 1 β 1 +ui 1 +εit 1 )
λit 2 =exp(x′it 2 β 2 +ui 2 +εit 2 ).(12.34)