Efficient combination of observational and experimental datasets under general restrictions on outcome mean functions

A researcher collecting data from a randomized controlled trial (RCT) often has access to an auxiliary observational dataset that may be confounded or otherwise biased for estimating causal effects. Common modeling assumptions impose restrictions on the outcome mean function - the conditional expectation of the outcome of interest given observed covariates - in the two datasets. Running examples from the literature include settings where the observational dataset is subject to outcome-mediated selection bias or to confounding bias taking an assumed parametric form. We propose a succinct framework to derive the efficient influence function for any identifiable pathwise differentiable estimand under a general class of restrictions on the outcome mean function. This uncovers surprising results that with homoskedastic outcomes and a constant propensity score in the RCT, even strong parametric assumptions cannot improve the semiparametric lower bound for estimating various average treatment effects. We then leverage double machine learning to construct a one-step estimator that achieves the semiparametric efficiency bound even in cases when the outcome mean function and other nuisance parameters are estimated nonparametrically. The goal is to empower a researcher with custom, previously unstudied modeling restrictions on the outcome mean function to systematically construct causal estimators that maximially leverage their assumptions for variance reduction. We demonstrate the finite sample precision gains of our estimator over existing approaches in extensions of various numerical studies and data examples from the literature.