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

This paper 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 the two data sources tightens the identified set for these parameters. For broad classes of DTE parameters, we derive sharp bounds under the combined data and clarify the mechanism through which the data combination tightens the identified set relative to using experimental data alone. Our analysis highlights that self-selection in the observational data is a key source of identification power. We also characterize 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 the sharp bounds, which can accommodate additional structural restrictions such as mutual stochastic monotonicity of potential outcomes and the generalized Roy model. An empirical application using data on negative campaign advertisements in a U.S. presidential election illustrates the practical relevance of the approach.