Palgrave Handbook of Econometrics: Applied Econometrics

590 Panel Data Methods

“normal” and “complicated” pregnancies leads to evidence that early prenatal care
is effective: on average, bringing the onset of prenatal care forward by one week
increases birth weights by 30–60 g in normal pregnancies. Using a finite mixture
model to capture the bimodality of the distribution of birth weights counteracts
evidence from the standard 2SLS approach that the effects of prenatal care are
weak or nonexistent. Estimates of the mixture model use observational data from
the 1988 US National Maternal and Infant Health Survey, and the empirical find-
ings are augmented by simulation results that show that the conventional findings
could be attributable to the existence of a relatively small proportion (10–15%) of
“complicated” pregnancies in the population.
To specify a finite mixture model, consider a vector of outcomesyithat are
observed for individuali: these may be repeated observations in a panel data model
or related outcomes in a multiple equation model and they are linked by common
unobservable heterogeneity. Then assume that each individual belongs to one of
a set of latent classesj=1,...,C, and that individuals are heterogeneous across
classes. Conditional on the observed covariates, there is homogeneity within a
given classj. Given the class that individualibelongs to, the outcomes have a
joint densityfj(yi|xi;θj)where theθjare vectors of parameters that are specific to
each class. The probability of belonging to classjisπij, where 0<πij<1 and
∑C
j= 1 πij=1. Unconditionally on the latent class the individual belongs to, the
joint density ofyiis given by:

f

(

yi|xi;πi 1 ,...,πiC;θ 1 ,...,θC

)

=

∑C

j= 1

πijfj

(

yi|xi;θj

)

. (12.26)

The discrete distribution of the heterogeneity hasCmass points and theπs need
to be estimated along with theθjs.
In many empirical applications of finite mixture models the class membership
probabilities are treated as fixed parametersπij=πj,j=1,...,C, (e.g., Atellaet al.,
2004; Bago d’Uva, 2006; Deb, 2001; Deb and Holmes, 2000; Deb and Trivedi, 1997,
2002; Jiménez-Martinet al., 2002). A more general approach is to parameterize
the heterogeneity as a function of individual characteristics. To implement this
approach in the case of the latent class model, class membership can be modeled
as a multinomial logit (as in, e.g., Clarket al.,2005; Etilé, 2006):

πij=

exp

(

zi′γj

)

∑C

k= 1

exp

(

zi′γk

),j=1,...,C, (12.27)

with the normalizationγC=0. This approach specifies the determinants of class
membership. In a panel data context, this parameterization provides a way to
account for the possibility that the observed regressors may be correlated with
the individual heterogeneity. Lettingzi=x ̄ibe the average over the observed
panel of the observations on the covariates is in line with what has been done in
recent studies to allow for the correlation between covariates and random effects,