In this paper, we establish sharp bounds on the joint distribution of potential outcomes and the distribution of treatment effects in parametric switching regime models with normal mean-variance mixture errors and in the semi-parametric switching regime models of Heckman (1990). Our results for parametric switching regime models with normal mean-variance mixture errors extend some existing results for the Gaussian switching regime model and our results for semi-parametric switching regime models supplement the point identification results of Heckman (1990). Compared with the corresponding sharp bounds when selection is random, we observe that self-selection tightens the bounds on the joint distribution of the potential outcomes and the distribution of treatment effects. These bounds depend on the identified model parameters only and can be easily estimated once the identified model parameters are estimated. The important issue of inference is briefly discussed.

This article studies competition under information heterogeneity in selection markets and examines the impact of public information regulations aimed at reducing information asymmetries between competing firms. We develop a novel model and introduce new empirical strategies to analyse imperfect competition in markets where firms have heterogeneous information about consumers, vary in cost structures, and offer differentiated products. Using data from the Italian auto insurance market, we find substantial differences in the precision of risk ratings across insurers, and those with less accurate risk-rating algorithms tend to have more efficient cost structures. We assess the equilibrium effects of giving firms equal access to aggregated risk information from a centralized bureau. This policy significantly reduces prices by increasing competition, leading to a 15.7% boost in consumer surplus, almost reaching the efficiency benchmark where firms have full knowledge of consumers’ true risk. Aggregating information through the bureau favours low-risk consumers and reduces average costs by 12 euros per contract through more efficient insurer–insuree matching.