Designing Experimental Selection Correction with Multiple Secondary Outcomes

Researchers increasingly have access to observational data containing primary and multiple secondary outcomes, together with randomized experiments in which only secondary outcomes are measured. Experimental selection correction (ESC) leverages experimental evidence to correct selection bias in observational estimation of the primary average treatment effect (ATE). Existing ESC methods, however, typically treat the set of measured secondary outcomes and the treatment-assignment rule as fixed, resulting in potential efficiency loss. We address this limitation by developing a novel experimental design framework that jointly optimizes secondary-outcome selection and treatment assignment under resource constraints, accounting for the tradeoff between the informativeness and measurement costs of secondary outcomes. We characterize the oracle design that minimizes the semiparametric efficiency bound for estimation of the ATE. To learn the design, we propose EfficientESC, a two-stage procedure that uses observational data and a randomized pilot to determine the design for the main experiment. We show that the learned design converges to the oracle design, and the proposed ATE estimator is asymptotically normal and attains the minimized semiparametric efficiency bound. Extensive simulations demonstrate substantial efficiency gains over competing methods in finite samples.