Palgrave Handbook of Econometrics: Applied Econometrics

600 Panel Data Methods

by Bago d’Uva (2006). Her model uses a panel of individuals across time: individuals
iare observedTitimes. Letyitrepresent the number of doctor visits in yeart. The

joint density ofyi=

(

yi 1 ,...,yiTi

)

is given by:

g

(

yi|xi;πi 1 ,...,πiC;θ 1 ,...,θC

)

=

∑C

j= 1

πij

∏Ti

t= 1

fj

(

yit|xit;θj

)

, (12.36)

wherexiis a vector of covariates, including a constant, theθjare vectors of parame-

ters, and 0<πij<1 and

∑C
j= 1 πij=1. Conditional on the class that the individual
belongs to, the number of visits in periodt,yit, is assumed to be determined by a
hurdle model. The underlying distribution for the two stages of the hurdle model
is the Negbin. Formally, for each componentj=1,...C, it is assumed that the
probability of zero visits and the probability of observingyitvisits, given thatyitis
positive, are given by the following expressions:

fj

(

0 |xit;βj 1

)

=P

[

yit= 0 |xit,βj 1

]

=

(

λ^1 j1,−itk+ 1

)−λkj1,it

fj

(

yit|yit>0,xit;βj 2

)

=



(

yit+

λkj2,it

αj

)(

αjλ^1 j2,−itk+ 1

)−λ

k

j2,it

αj

(

1 +

λkj2,−it^1

αj

)−yit



(

λkj2,it

αj

)

(yit+ (^1) )
⎡
⎢⎢
⎣^1 −
(
αjλ^1 j2,−itk+ 1
)−λ
k
j2,it
αj
⎤
⎥⎥
⎦
, (12.37)
whereλj1,it=exp
(
x′itβj 1
)
,λj2,it=exp
(
x′itβj 2
)
,αjare overdispersion parameters
andkis an arbitrary constant.
As in the standard hurdle model,βj 1 can be different fromβj 2 , reflecting the fact
that the determinants of care are allowed to have different effects on the probability
of seeking care and on the conditional number of visits. On the other hand, having[
βj 1 ,βj 2
]
=
[
βl 1 ,βl 2
]
forj=lreflects the differences between the latent classes.
Various special cases are nested within the general model. It can be assumed that
all the slopes are the same, varying only the constant terms,βj1,0andβj2,0, and
the overdispersion parametersαj. This represents a case where there is unobserved
individual heterogeneity, but not in the responses to the covariates. The most
flexible version allowsαjand all elements ofβj 1 andβj 2 to vary across classes. The
finite mixture hurdle model also accommodates a mixture of sub-populations for
which health care use is determined by a Negbin model (the two decision processes
are indistinguishable) and sub-populations for which utilization is determined by
a hurdle model. This is obtained by settingβj 1 =βj 2 for some classes. Setting them
equal for all of the classes gives a panel data version of Deb and Trivedi’s (1997)
latent class Negbin model. Bago d’Uva (2006) applies the latent class hurdle model
to panel data from the RAND Health Insurance Experiment and finds a higher price
effect on health care utilization for the latent class of “low users.” This is mostly
attributable to the impact of price on the probability of seeing a doctor rather than
the conditional number of visits.