512 Discrete Choice Modeling
Interpretation of the components of this model is particularly complicated.
Typically, interest will center on the second equation. In Greene (1998),
the second equation concerned the presence of a gender economics course
in a college curriculum, while the first equation specified the presence of a
women’s studies program on the campus. In Kassouf and Hoffman (2006), the
authors were interested in the occurrence of work-related injuries while the first
conditioning equation specified the use or non-use of protective equipment.
Fabbri, Monfardini and Radice (2004) analyzed the choice of Cesarean delivery
conditioned on hospital type (public or private). In Greene, Rhine and Toussaint-
Comeau (2003), the main equation concerned use of a check cashing facility while
the conditioning event in the first equation was whether or not the individual
participated in the banking system. In all of these cases, the margin of interest is
the impact of the variables in the model on the probability thatdi,2equals one.
Becausedi,1appears in the second equation, there is (potentially) a direct effect (in
wi,2)and an indirect effect transmitted todi,2through the impact of the variable
in question on the probability thatdi,1equals one. Details on these computations
appear in Greene (2008a) and Kassouf and Hoffmann (2006).
11.4.3 Sample selection in a bivariate probit model
Another bivariate probit model that is related to the recursive model of the
preceding section is thebivariate probit with sample selection. The structural
equations are
di∗,1=wi′,1θ 1 +εi,1, di,1=1ifdi∗,1>0, 0 otherwise,di∗,2=w′i,2θ 2 +εi,2, di,2=1ifd∗i,2>0, 0 otherwise, and ifdi,1=1,di,2,wi,2are unobserved whendi,1=0,
(εi,1εi,2)∼N 2 [(0, 0),(1, 1),ρ].The first equation is a “selection equation.” Presence in the sample of observa-
tions for the second equation is determined by the first. Like the recursive model,
this framework has been used in a variety of applications. The first was a study
of the choice of deductibles in insurance coverage by Wynand and van Praag
(1981). Boyes, Hoffman and Low (1989) and Greene (1992) studied loan default in
which the application is the selection rule. More recently, McQuestion (2000) has
used the model to analyze health status (selection) and health behavior, and Lee,
Lee and Eastwood (2003) have studied consumer adoption of computer banking
technology.
Estimation of this sample selection model is done by maximum likelihood in
one step.^20 The log-likelihood is:
lnL=∑
di,a= 0
ln#(−w′i,1θ 1 )+∑n
i=1,di,1= 1
ln# 2[
w′i,1θ 1 ,qi,2w′i,2θ 2 ,qi,2ρ]
.As before, estimation and inference in this model follows the standard procedures.