58 K. Li and N.R. Prabhala
6.2. Bayesian methods for selection models
To illustrate the implementation of the Bayesian approach to selection models, consider
the switching regression model of Section3.1. For notational convenience, rewrite this
model as the system of equations
I= (^1) Ziγ+ηi> 0 , (28)
YE,i=XE,iβE+E,i, (29)
YNE,i=XNE,iβNE+NE,i, (30)
where 1{.}denotes the indicator function and the other notation follows that in Sec-
tion3.1. As before, assume that the errors are trivariate normal with the probit error
variance in equation(28)normalized to unity.
The critical unobservability issue, as discussed in Section4, is that if a firm self-
selectsE, we observe the outcomeYE,i. However, we do not observe the counterfactual
YNE,ithat would have occurred had firmichosenNEinstead ofE. FollowingTanner
and Wong (1987), a Bayesian estimation approach generates counterfactuals by aug-
menting the observed data with simulated observations of the unobservables through a
“data augmentation” step. When augmented data are generated in a manner consistent
with the structure of the model, the distribution of the augmented data converges to
the distribution of the observed data. The likelihood of both the observed and the aug-
mented data can be used as a proxy for the likelihood of the observed data. Conditional
on the observed and augmented data and given a prior on parametersγ,βand the error
covariances, approximate posteriors for the model parameters can be obtained by using
standard simulation methods. The additional uncertainty introduced by simulating un-
observed data can then be integrated out (Gelfand and Smith, 1990) to obtain posteriors
conditional on only the observed data.
Explicitly modeling the unobserved counterfactuals offers advantages in the con-
text of selection models. The counterfactuals that are critical in estimating treatment
effects are merely the augmented data that are anyway employed in Bayesian esti-
mation. The augmented data also reveal deficiencies in the model that are not iden-
tified by simple tests for the existence of selectivity bias. In addition, one can ob-
tain exact small sample distributions of parameter estimates that are particularly use-
ful when sample sizes are small to moderate, such as self-selection involving rela-
tively infrequent events. Finally, we can impose parameter constraints without com-
promising estimation. In later sections, we review empirical applications that employ
the Bayesian approach towards estimating counterfactuals (Li and McNally, 2004;
Scruggs, 2006). We also illustrate an application to a matching problem (Sørensen,
2005 ) in which the tractability of the conditional distributions given subsets of para-
meters leads to computationally feasible estimators in a problem where conventional
maximum likelihood estimators are relatively intractable.