  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.
