Palgrave Handbook of Econometrics: Applied Econometrics

William Greene 529

whereλi = exp(x′iβ)andτ is a single new, free parameter to be estimated.
(Researchers usually find that theτform of the model is more restrictive than
desired.) The conditional mean function is:

E[yi|xi,zi]=[ 1 −F(ri|zi)]λi.

11.6.2.3 Sample selection

We consider an extension of the classic model of sample selection (Heckman, 1979)
to the models for count outcomes. In the context of the applications considered
here, for example, we might consider a sample based on only those individuals
who have health insurance. The generic model will take the form:

s∗i=z′iα+ui, ui∼N[0, 1],

si= 1 (s∗i> 0 )(probit selection equation)

λi|εi=exp(β′xi+σεi),εi∼N[0, 1] (index function with heterogeneity)^26

yi|xi,εi∼Poisson(yi|xi,εi)(Poisson model for outcome)

[ui,εi]∼N[(0, 1),(1,ρ,1)]

yi,xiare observed only whensi=1.

The count model is the heterogeneity model suggested earlier with log-normal
rather than log-gamma heterogeneity. The conventional approach of fitting the
probit selection equation, computing an inverse Mills ratio, and adding it as an
extra regressor in the Poisson model, is inappropriate here (see Greene, 1995, 1997,
2006). A formal approach for this model is developed in Terza (1994, 1998) and
Greene (1995, 2006, 2007b). Formal results are collected in Greene (2006). The
generic result for the count model (which can be adapted to the negative binomial
or other models) is:

f(yi,si|xi,zi)=

∫∞

−∞

[( 1 −si)+sif(yi|xi,εi)]

#

(

( 2 si− 1 )[z′iαi+ρεi]/

√

1 −ρ^2

)

φ(εi)dεi,

with:

f(yi|xi,εi)=

exp(−λi|xi,εi)(λi|xi,εi)yi

(yi+ 1 )

,λi|xi,εi=exp(β′xi+σεi).

The integral does not exist in closed form, but the model can be fitted by
approximating the integrals with Hermite quadrature:

lnLQ=

∑N

i= 1

log

[

1

√

π

∑H

h= 1

ωh

[

( 1 −si)+sif(yi|xi,vh)

]

#

[

( 2 si− 1 )

(

z′iγi+τvh

)]]

,

or simulation, for which the simulated log-likelihood is:

lnLS=

∑N

i= 1 log

1

R

∑R

r= 1 [(^1 −si)+sif(yi|xi,σεir)]#[(^2 si−^1 )

(

z′iγi+τεir

)

],