Sensitivity Analysis for the Average Treatment Effect under Discrete Unobserved Confounders

We model unobserved confounding through an unknown finite number of latent types. This assumption induces finite-mixture representations of the treated and control outcome distributions. Using the identified mixture components, we characterize the sharp identified set for the number of latent types and derive the sharp identified set for the average treatment effect (ATE) corresponding to each admissible value, thereby providing a natural framework for sensitivity analysis. We further obtain a cutoff beyond which the identified set for the ATE coincides with a version of the Manski bounds, whereas below the cutoff it is strictly smaller. This cutoff grows only linearly with the numbers of mixture components in the treated and control groups, although the maximum admissible number of latent types grows quadratically. We also provide estimation and inference procedures with asymptotic guarantees and illustrate our methodology using LaLonde's data.