What to Observe When Assuming Selection on Observables

Political scientists regularly rely on a selection-on-observables assumption to identify causal effects of interest. Once a causal effect has been identified in this way, a wide variety of estimators can, in principle, be used to consistently estimate the effect of interest. While these estimators are all justified by appeals to the same causal identification assumptions, they often differ greatly in how they make use of the data at hand. For instance, methods based on regression rely on an explicit model of the outcome variable but do not explicitly model the treatment assignment process, whereas methods based on propensity scores explicitly model the treatment assignment process but do not explicitly model the outcome variable. Understanding the tradeoffs between estimation methods is complicated by these seemingly fundamental differences. In this paper we seek to rectify this problem. We do so by clarifying how most estimators of causal effects that are justified by an appeal to a selection-on-observables assumption are all special cases of a general weighting estimator. We then explain how this commonality provides for diagnostics that allow for meaningful comparisons across estimation methods—even when the methods are seemingly very different. We illustrate these ideas with two applied examples.