William Greene 51311.4.4 Multivariate binary choice and the panel probit model
In principle, the bivariate probit model can be extended to an arbitrary number of
equations, as:
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...
di∗,M=w′i,MθM+εi,M,di,M=1ifdi∗,M>0,
⎛
⎜
⎜⎜
⎝εi,1
εi,2
...
εi,M⎞
⎟
⎟⎟
⎠
∼N⎡
⎢
⎢⎢
⎣⎛
⎜
⎜⎜
⎝0
0
...
0⎞
⎟
⎟⎟
⎠
,⎛
⎜
⎜⎜
⎝1 ρ 12 ... ρ 1 M
ρ 12 1 ... ρ 2 M
... ... ... ...
ρ 1 M ρ 2 M ... 1⎞
⎟
⎟⎟
⎠⎤
⎥
⎥⎥
⎦
= N 2 [ 0 ,R].The obstacle to the use of this model is its computational burden. The log-
likelihood is computed as follows. Let:
Qi=diag(qi,1,qi,2,...,qi,M)bi=(w′i,1θ 1 ,w′i,2θ 2 ,...,w′i,MθM)′
ci=Qibi
Di=QiRQi.Then:
lnL=∑n
i= 1
ln#M[ci,Di].Evaluation of theM-variate normal c.d.f. cannot be done analytically or with
quadrature. It is done with simulation, typically using the GHK (Geweke, Haji-
vassilou and Keane) simulator.
This form of the model also generalizes the random effects probit model exam-
ined earlier. We can relax the assumption of equal cross-period correlations by
writing:
dit∗=w′itθ+εit, dit=1ifd∗it>0, 0 otherwise,
(εi 1 ,...,εiT)∼N[ 0 ,R].This is precisely the model immediately above with the constraint that the coeffi-
cients in the equations are all the same. In this form, it is conventionally labeled
thepanel probit model.^21 Bertschek and Lechner (1998) devised a GMM estimator
to circumvent the computational burden of this model. Greene (2004a) examined
the same model, and considered alternative computational procedures as well as
some variations of the model specification.