The Identification Power of Combining Experimental and Observational Data for Distributional Treatment Effect Parameters

This study investigates the identification power gained by combining experimental data, in which treatment is randomized, with observational data, in which treatment is self-selected, for distributional treatment effect (DTE) parameters. While experimental data identify average treatment effects, many DTE parameters, such as the distribution of individual treatment effects, are only partially identified. We examine whether and how combining these two data sources tightens the identified set for such parameters. For broad classes of DTE parameters, we derive nonparametric sharp bounds under the combined data and clarify the mechanism through which data combination improves identification relative to using experimental data alone. Our analysis highlights that self-selection in observational data is a key source of identification power. We establish necessary and sufficient conditions under which the combined data shrink the identified set, showing that such shrinkage generally occurs unless selection-on-observables holds in the observational data. We also propose a linear programming approach to compute sharp bounds that can incorporate additional structural restrictions, such as positive dependence between potential outcomes and the generalized Roy selection model. An empirical application using data on negative campaign advertisements in the 2008 U.S. presidential election illustrates the practical relevance of the proposed approach.