  Many causal parameters 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). This paper shows that 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. I
find suggestive evidence that workers who would see below average earnings
without treatment tend to see above average benefits from treatment.
