Palgrave Handbook of Econometrics: Applied Econometrics

William Greene 527

other functional forms proposed for count data models, including the generalized
Poisson, gamma, and Polya–Aeppli forms described in Winkelmann (2003) and
Greene (2007a, Ch. 24).
The heteroskedasticity in the count models is induced by the relationship
between the variance and the mean. The single parameterθpicks up an implicit
overall scaling, so it does not contribute to this aspect of the model. As in the linear
model, microeconomic data are likely to induce heterogeneity in both the mean
and variance of the response variable. A specification that allows independent
variation of both will be of some virtue. The result:

Var[yi|xi]=λi( 1 +( 1 /θ)λPi−^1 ),

suggests that a natural platform for separately modeling heteroskedasticity will be
the dispersion parameter,θ, which we now parameterize as:

θi=θexp(z′iδ).

Operationally, this is a relatively minor extension of the model. But it is likely to
introduce a quite substantial increase in the flexibility of the specification. Indeed,
a heterogeneous NBP model is likely to be sufficiently parameterized to accommo-
date the behavior of most datasets. (Of course, the specialized models discussed
below, e.g., the zero inflation models, may yet be more appropriate for a given
situation.)

11.6.2 Extended models for counts: two-part, zero inflation, sample
selection, bivariate

“Non-Poissonness” arises from a variety of sources in addition to the latent
heterogeneity modeled in the previous section. A variety oftwo-part modelshave
been proposed to accommodate elements of the decision process.

11.6.2.1 Hurdle model

The hurdle model (Mullahy, 1986; Gurmu, 1997) consists of a participation
equation and a conditional Poisson or negative binomial model. The structural
equations are:

Prob(yi> 0 |zi) =a binary choice mechanism, such as probit or logit

Prob(yi=j|yi>0,xi) =truncated Poisson or negative binomial.

(See Shaw, 1988.) For a logit participation equation and a Poisson count,
the probabilities for the observed data that enter the log-likelihood function
would be:

Prob(yi= 0 |zi)=

1

1 +exp(z′iα)

Prob(yi=j|xi,zi)=Prob(yi> 0 |zi)×Prob(yi=j|yi>0,xi)

=

exp(z′iα)

1 +exp(z′iα)

exp(−λi)λ

j

i

j![ 1 −exp(−λi)]

.