William Greene 511

is the joint probability, Prob(di,1=1,di,2= 1 )=# 2

[

wi′,1θ 1 ,w′i,2θ 2 ,ρ

]

, that is

analyzed by Christofides, Stengos and Swidinsky (1997). Greene (1996, 2008a)
considers, instead, the conditional mean function E[di,1|di,2= 1,wi,1,wi,2]=

2

[

w′i,1θ 1 ,w′i,2θ 2 ,ρ

]
/#[w′i,2θ 2 ]. In either case, the raw coefficients bear little
resemblance to the partial effects.
For hypothesis testing about the coefficients, the standard results for Wald, LM
and LR tests apply. The LM test is likely to be cumbersome because the derivatives
of the log-likelihood function are complicated. The other two are straightforward.
A hypothesis of interest is that the correlation is zero. For testing:

H 0 :ρ=0,

all three likelihood-based procedures are straightforward, as the application below
demonstrates. The LM statistic derived by Kiefer (1982) is:

λLM=

{

∑n

i= 1

[

qi,1qi,2

φ(w′i,1θ 1 )φ(w′i,2θ 2 )

#(w′i,1θ 1 )#(w′i,2θ 2 )

]} 2

∑n

i= 1

⎧

⎨

⎩

[

φ(w′i,1θ 1 )φ(w′i,2θ 2 )

] 2

#(w′i,1θ 1 )#(−w′i,1θ 1 )#(w′i,2θ 2 )#(−w′i,2θ 2 )

⎫

⎬

⎭

,

where the two coefficient vectors are the MLEs from the univariate probit models
estimated separately.

11.4.2 Recursive simultaneous equations

Section 11.3.5 considered a type of simultaneous equations model in which an
endogenous regressor appears on the right-hand side of a probit model. Two other
simultaneous equations specifications have attracted interest. Amemiya (1985)
demonstrates that a fully simultaneous bivariate probit model,

di∗,1=w′i,1θ 1 +γ 1 di,2+εi,1, di,1=1ifd∗i,1>0,

di∗,2=w′i,2θ 2 +γ 2 di,1+εi,2, di,2=1ifdi∗,2>0,

is internally inconsistent and unidentified. However, a recursive model:

d∗i,1=w′i,1θ 1 +εi,1, di,1=1ifd∗i,1>0,

d∗i,2=w′i,2θ 2 +γ 2 di,1+εi,2, di,2=1ifd∗i,2>0,

(εi,1εi,2)∼N 2 [(0, 0),(1, 1),ρ],

is a straightforward extension of the bivariate model. For estimation of this model,
we have the counterintuitive result that it can be fitted as an ordinary bivariate
probit model with the additional right-hand-side variable in the second equation,
ignoring the simultaneity. The recent literature provides a variety of applications
of this model, including Greene (1998), Fabbri, Monfardini and Radice (2004),
Kassouf and Hoffman (2006), White and Wolaver (2003), Gandelman (2005) and
Greene, Rhine and Toussaint-Comeau (2006).