  Sample selection models represent a common methodology for correcting bias
induced by data missing not at random. It is well known that these models are
not empirically identifiable without exclusion restrictions. In other words,
some variables predictive of missingness do not affect the outcome model of
interest. The drive to establish this requirement often leads to the inclusion
of irrelevant variables in the model. A recent proposal uses adaptive LASSO to
circumvent this problem, but its performance depends on the so-called
covariance assumption, which can be violated in small to moderate samples.
Additionally, there are no tools yet for post-selection inference for this
model. To address these challenges, we propose two families of spike-and-slab
priors to conduct Bayesian variable selection in sample selection models. These
prior structures allow for constructing a Gibbs sampler with tractable
conditionals, which is scalable to the dimensions of practical interest. We
illustrate the performance of the proposed methodology through a simulation
study and present a comparison against adaptive LASSO and stepwise selection.
We also provide two applications using publicly available real data. An
implementation and code to reproduce the results in this paper can be found at
https://github.com/adam-iqbal/selection-spike-slab
