510 Discrete Choice Modeling
The 5% critical value from the chi-squared distribution with 6 degrees of freedom
is 12.592, so this hypothesis is rejected as well.
11.4 Bivariate and multivariate binary choice
The health care data contain two binary variables, DOCTOR and HOSPITAL, which
one would expect to be at least correlated if not jointly determined. The extension
of the binary choice model to more than one choice is relatively uncomplicated,
but does bring new statistical issues as well as new practical complications. We
consider several two equation specifications first, as these are the leading cases,
then consider the extension to an arbitrary number of binary choices.
11.4.1 Bivariate binary choice
A two-equation binary choice model would take the form of a seemingly unrelated
regressions model:
di∗,1=wi′,1θ 1 +εi,1, di,1=1ifd∗i,1>0,di∗,2=wi′,2θ 2 +εi,2, di,2=1ifd∗i,2>0,where “1” and “2” distinguish the equations (and are distinct from the periods in
a panel data case). The bivariate binary choice model arises when the two distur-
bances are correlated. There is no convenient approach for this model based on
the logistic model, so we assume bivariate normality at the outset. The bivariate
probit model has:
F(εi,1,εi,2)=N 2 [(0, 0),(1, 1),ρ], − 1 <ρ<1.The probability associated with the joint eventdi,1=di,2=1 is then:
Prob(di,1=1,di,2= 1 |wi,1,wi,2)=# 2[
w′i,1θ 1 ,w′i,2θ 2 ,ρ]
,where# 2 [t 1 ,t 2 ,ρ]denotes the bivariate normal c.d.f. The log-likelihood function
is the joint density for the observed outcomes. By extending the formulation of
the univariate probit model in the preceding section, we obtain:
lnL=∑n
i= 1 ln#^2[(
qi,1w′i,1θ 1)
,(
qi,2w′i,2θ 2)
,(
qi,1qi,2ρ)]
.The bivariate normal integral does not exist in closed form, and must be approxi-
mated, typically with Hermite quadrature.
The model is otherwise conventional and the standard conditions for MLEs
are obtained. Interpretation of the model does bring some complications, how-
ever. First,θ 1 andθ 2 are not the slopes of any recognizable conditional mean
function and neither are the derivatives of the possibly interesting Prob(di,1=
1,di,2= 1 |wi,1,wi,2). Both of these are complicated functions of all the model
parameters and both data vectors (see Greene, 2008a, sec. 23.8.3; Christofides,
Stengos and Swidinsky, 1997; Christofides, Hardin and Stengos, 2000). Since
this is a two-equation model, it is unclear what quantity should be analyzed
when interpreting the coefficients in relation to partial effects. One possibility