Bayesian Variable Selection Under Sample Selection and Model Misspecification

Sample selection bias arises when missingness in the outcome of interest correlates with the outcome itself, leading to non-randomly selected samples. A common approach to correct bias from sample selection is to use sample selection models that jointly model the selection mechanism and the outcome of interest. Formulating these models typically rely on exclusion restrictions (variables that are predictors of selection but not appearing in the outcome equation) to ensure identifiability of the parameters. However, the choice of exclusion restrictions often depends on heuristics or expert judgment, potentially leading to the inclusion of irrelevant variables or the omission of important ones. Additionally, distributional misspecification and omitted variable bias are frequent challenges in this framework. To formally address these issues, we propose a Bayesian variable selection (BVS) methodology that incorporates both local priors (LPs) and non-local priors (NLPs), enabling the identification of variables with predictive power for the outcome and selection processes. We develop computational tools to conduct BVS in sample selection models based on a Laplace approximation of the marginal likelihood, and characterize the resulting Bayes factor rates under model misspecification. We establish model selection consistency for both classes of priors, showing that the proposed methodology correctly identifies active variables for both the selection process and outcome process asymptotically. The priors are calibrated to account for the possibility of distributional misspecification and omitted variable bias. We present a simulation study and real-data applications to explore the finite-sample effects of model misspecification on BVS. We compare the performance of the proposed methodology against BVS based on spike-and-slab (SS) priors and the Adaptive LASSO (ALASSO), an adaptive weighting of the least absolute shrinkage and selection operator (LASSO).