Palgrave Handbook of Econometrics: Applied Econometrics

528 Discrete Choice Modeling

This model might apply for on-site counts of the use of certain facilities such as
recreational sites. The expectation in the hurdle model is easily found using the
rules of probability:

E[yi|xi,zi]=

exp(z′iα)

1 +exp(z′iα)E[yi|yi>0,xi,zi]

=

exp(z′iα)

1 +exp(z′iα)

λi

[ 1 −exp(−λi)].

As usual, the intricacy of the function mandates some caution in interpreting the
model coefficients. In particular:

δi(xi) =∂E[y∂i|xxii,zi]

=

{

exp(z′iα)

1 +exp(z′iα)

λi

[ 1 −exp(−λi)]

(

1 −[ 1 λ−iexpexp((−−λλii))]

)}

β.

The complication of the partial effects is compounded ifzicontains any of the
variables that also appear inxi. The other part of the partial effect is:

δi(zi)=

{

exp(z′iα)

[ 1 +exp(z′iα)]^2

λi

[ 1 −exp(−λi)]

}

α.

11.6.2.2 Zero inflation models

A related formulation is thezero inflation model, which is a type oflatent class model.
The model accommodates a situation in which the zero outcome can arise in either
of two mechanisms. In one regime, the outcome is always zero; in the other, the
outcome is generated by the Poisson or negative binomial process that might also
produce a zero. The example suggested in Lambert’s (1992) pioneering application
is a manufacturing process that produces a number of defective parts,yi, equal to
zero if the process is under control, or equal to a Poisson outcome if the process is
not under control. The applicable distribution is:

Prob(yi= 0 |xi,zi)=Prob(regime 0|zi)+Prob(regime 1|zi)

Prob(yi= 0 |regime1,xi)

=F(ri|zi)+[ 1 −F(ri|zi)]Prob(yi= 0 |xi)

Prob(yi=j|yi>0,xi,zi)=[ 1 −F(ri|zi)]Prob(yi=j|xi).

The density governing the count process may be the Poisson or negative binomial
model. The regime process is typically specified as a logit model, though the probit
model is often used as well. Finally, two forms are used for the regime model,
the standard probit or logit model with covariate vector,zi, and the zero inflated
poisson, ZIP(τ) form, which takes the form (for the logit–Poisson model):

Prob(yi= 0 |xi)=%(τx′iβ)+[ 1 −%(τx′iβ)]exp(−λi)

Prob(yi=j|yi>0,xi)=[ 1 −%(τx′iβ)]exp(λi)λji/j!,