Conformalized Lee Inference: Distribution-Free Prediction Sets for Individual Treatment Effect under Monotone Sample Selection

Treatment can change which outcomes are observed, so treated-selected and selected-control units need not represent the same latent population. This paper proposes conformalized Lee inference for counterfactual prediction under randomized treatment and monotone sample selection. The procedure uses treated-selected observations to fit and calibrate an arbitrary prediction rule and replaces the usual $(1-\alpha)$ score quantile with the adjusted $(1-\alpha\pi)$ quantile, where $\pi$ is the identified share of always-selected units among treated-selected units. The resulting prediction set has finite-sample, distribution-free marginal coverage over the sharp Lee ambiguity set. For a selected-control unit, subtracting the observed untreated outcome yields a marginal prediction set for the realized individual treatment effect. The adjusted population cutoff is minimax optimal over the reduced-information identification region. Simulations show that ordinary split-conformal prediction can under-cover under distribution shifts induced by selection, whereas the adjusted procedures restore coverage. Empirical analysis uses the National Job Corps Study data to illustrate prediction sets for individual wage effects of assignment to program access.