Bayesian variable selection in sample selection models using spike-and-slab priors

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