492 Discrete Choice Modeling
11.3.3.1 Gibbs sampler for the binomial probit model
- ComputeW′Wonce at the outset and obtainLsuch thatLL′=(W′W)−^1.
- Startθat any value such as 0.
- Obtain drawsUi,rfrom the standard uniform distribution. Greene (2008a,
p. 575, result (17–1)) shows how to transform a draw fromU[0,1] to a draw
from the truncated normal with underlying meanμand standard deviationσ.
For this application,μ=w′iθandσ=1, so the draws fromp(d∗|θ,d,W) are
obtained as:
di∗,r(r)=w′iθr− 1 +Φ−^1
[
1 −( 1 −Ui,r)Φ(w′iθr− 1 )]
ifdi= 1
di∗,r(r)=w′iθr− 1 +Φ−^1[
Ui,rΦ(−w′iθr− 1 )]
ifdi=0.This step is used to draw thenobservations ond∗i,r(r).- To draw an observation from the multivariate normal population of
p(θ|d∗,d,W], we need to draw from the normal population with meanq∗r− 1
and variance (W′W)−^1. For this application, we use the results at step 3 to com-
puteq∗=(W′W)−^1 W′d∗(r). We obtain a vector,v,ofKdraws from theN[0,1]
population, and then computeθ(r)=q∗+Lv.
The iteration cycles between steps 3 and 4. This should be repeated several thou-
sand times, discarding the burn-in draws, and then the estimator ofθis the sample
mean of the retained draws. The posterior variance is computed with the variance
of the retained draws. Posterior estimates ofdi∗would typically not be useful.
This application of the Gibbs sampler demonstrates, in an uncomplicated case,
how the algorithm can provide an alternative to actually maximizing the log-
likelihood. The similarity of the method to the EM algorithm (Dempster, Laird
and Rubin, 1977) is not coincidental. Both procedures use an estimate of the
unobserved, censored data, and both estimateθby using OLS using the predicted
data.
11.3.4 Semiparametric models
The fully parametric probit and logit models remain by far the mainstays of empir-
ical research on binary choice. Fully nonparametric discrete choice models are
fairly exotic and have made only limited inroads in the literature, most of which is
theoretical (e.g., Matzkin, 1993). The middle ground is occupied by a few semipara-
metric models that have been proposed to relax the detailed assumptions of the
probit and logit specifications. The single index model of Klein and Spady (1993)
has been used in several applications, including Gerfin (1996), Horowitz (1993)
and Fernandez and Rodriguez-Poo (1997), and provides the theoretical platform
for a number of extensions.^10
The single index formulation departs from a regression formulation:
E[di|wi]=E[di|w′iθ].
Then:
Prob(di= 1 |wi)=F(w′iθ|wi)=G(w′iθ),