  We consider the estimation of average treatment effects in observational
studies and propose a new framework of robust causal inference with unobserved
confounders. Our approach is based on distributionally robust optimization and
proceeds in two steps. We first specify the maximal degree to which the
distribution of unobserved potential outcomes may deviate from that of observed
outcomes. We then derive sharp bounds on the average treatment effects under
this assumption. Our framework encompasses the popular marginal sensitivity
model as a special case, and we demonstrate how the proposed methodology can
address a primary challenge of the marginal sensitivity model that it produces
uninformative results when unobserved confounders substantially affect
treatment and outcome. Specifically, we develop an alternative sensitivity
model, called the distributional sensitivity model, under the assumption that
heterogeneity of treatment effect due to unobserved variables is relatively
small. Unlike the marginal sensitivity model, the distributional sensitivity
model allows for potential lack of overlap and often produces informative
bounds even when unobserved variables substantially affect both treatment and
outcome. Finally, we show how to extend the distributional sensitivity model to
difference-in-differences designs and settings with instrumental variables.
Through simulation and empirical studies, we demonstrate the applicability of
the proposed methodology.
