Treatment effects in sample selection models and their nonparametric estimation

In a sample-selection model with the ‘selection’ variable Q and the ‘outcome’ variable Y ∗ , Y ∗ is observed only when Q = 1 . For a treatment D affecting both Q and Y ∗ , three effects are of interest: ‘participation’ (i.e., the selection) effect of D on Q , ‘visible performance’ (i.e., the observed outcome) effect of D on Y ≡ Q Y ∗ , and ‘invisible performance’ (i.e., the latent outcome) effect of D on Y ∗ . This paper shows the conditions under which the three effects are identified, respectively, by the three corresponding mean differences of Q , Y , and Y | Q = 1 (i.e., Y ∗ | Q = 1 ) across the control ( D = 0 ) and treatment ( D = 1 ) groups. Our nonparametric estimators for those effects adopt a two-sample framework and have several advantages over the usual matching methods. First, there is no need to select the number of matched observations. Second, the asymptotic distribution is easily obtained. Third, over-sampling the control/treatment group is allowed. Fourth, there is a built-in mechanism that takes into account the ‘non-overlapping support problem’, which the usual matching deals with by choosing a ‘caliper’. Fifth, a sensitivity analysis to gauge the presence of unobserved confounders is available. A simulation study is conducted to compare the proposed methods with matching methods, and a real data illustration is provided.