  Semiparametric inference on average causal effects from observational data is
based on assumptions yielding identification of the effects. In practice,
several distinct identifying assumptions may be plausible; an analyst has to
make a delicate choice between these models. In this paper, we study three
identifying assumptions based on the potential outcome framework: the back-door
assumption, which uses pre-treatment covariates, the front-door assumption,
which uses mediators, and the two-door assumption using pre-treatment
covariates and mediators simultaneously. We provide the efficient influence
functions and the corresponding semiparametric efficiency bounds that hold
under these assumptions, and their combinations. We demonstrate that neither of
the identification models provides uniformly the most efficient estimation and
give conditions under which some bounds are lower than others. We show when
semiparametric estimating equation estimators based on influence functions
attain the bounds, and study the robustness of the estimators to
misspecification of the nuisance models. The theory is complemented with
simulation experiments on the finite sample behavior of the estimators. The
results obtained are relevant for an analyst facing a choice between several
plausible identifying assumptions and corresponding estimators. Our results
show that this choice implies a trade-off between efficiency and robustness to
misspecification of the nuisance models.
