This dissertation contains two chapters. The first chapter studies causal parameters that depend on a moment of the joint distribution of potential outcomes. Such parameters are especially relevant in policy evaluation settings, where noncompliance is common and accommodated through the model of Imbens & Angrist (1994). The sharp identified set for these parameters is an interval with endpoints characterized by the value of optimal transport problems. Sample analogue estimators are proposed based on the dual problem of optimal transport. These estimators are root-n consistent and converge in distribution under mild assumptions. Inference procedures based on the bootstrap are straightforward and computationally convenient. The ideas and estimators are demonstrated in an application revisiting the National Supported Work Demonstration job training program. Estimates suggest that workers who would see below average earnings without treatment tend to see above average benefits from treatment.

The second chapter proposes a methodology for studying the robustness of results drawn from incomplete datasets. Selection is measured as the squared Hellinger divergence between the distributions of complete and incomplete observations, which has a natural interpretation. The breakdown point is defined as the minimal amount of selection needed to overturn a given result. Reporting point estimates and lower confidence intervals of the breakdown point is a simple, concise way to communicate a result’s robustness. An estimator of the breakdown point of results drawn from GMM models is proposed and shown root-n consistent and asymptotically normal under mild assumptions. Lower confidence intervals of the breakdown point are simple to construct. The chapter concludes with a simulation study illustrating the good finite sample performance of the procedure.