“Treatment Effects in Sample Selection Models and their Nonparametric Estimation” by Myoung-jae Lee
In a sample-selection model with the ‘selection' variable Q and the ‘outcome' variable Y*, Y* is observed only when Q = 1; i.e., Y ≡ QY* is observed along with Q. For a treatment D that affects both Q and Y*, three treatment effects are of interest: ‘participation' (i.e., selection) effect of D on Q, ‘visible performance' (i.e., observed outcome) effect of D on Y , and ‘invisible performance' (i.e., latent outcome) effect of D on Y*. The visible performance effect combines the participation and invisible performance effects, and can be decomposed into the effect on ‘the always-participants', the effect on ‘the compliers' and so on. This paper shows under what conditions the three effects are identified, respectively, with 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. With identification issue settled, nonparametric estimators for the effects are proposed. The estimators under two-sample framework take the form of two-sample U-statistics of order (1,1), and have three advantages. First, there is no need to decide the number of matched observations as in the usual matching methods. Second, there is a built-in protection against the problem of ‘non-overlapping supports' across the control and treatment groups. Third, over-sampling of the control/treatment group is allowed. A real-data illustration of job-training effects on female work-hours is provided.