  We consider the problem of estimating the effects of a binary treatment on a
continuous outcome of interest from observational data in the absence of
confounding by unmeasured factors. We provide a new estimator of the population
average treatment effect (ATE) based on the difference of novel double-robust
(DR) estimators of the treatment-specific outcome means. We compare our new
estimator with previously estimators both theoretically and via simulation.
DR-difference estimators may have poor finite sample behavior when the
estimated propensity scores in the treated and untreated do not overlap. We
therefore propose an alternative approach, which can be used even in this
unfavorable setting, based on locally efficient double-robust estimation of a
semiparametric regression model for the modification on an additive scale of
the magnitude of the treatment effect by the baseline covariates $X$. In
contrast with existing methods, our approach simultaneously provides estimates
of: i) the average treatment effect in the total study population, ii) the
average treatment effect in the random subset of the population with
overlapping estimated propensity scores, and iii) the treatment effect at each
level of the baseline covariates $X$.
  When the covariate vector $X$ is high dimensional, one cannot be certain,
owing to lack of power, that given models for the propensity score and for the
regression of the outcome on treatment and $X$ used in constructing our DR
estimators are nearly correct, even if they pass standard goodness of fit
tests. Therefore to select among candidate models, we propose a novel approach
to model selection that leverages the DR-nature of our treatment effect
estimator and that outperforms cross-validation in a small simulation study.
