  We argue that randomized controlled trials (RCTs) are special even among
settings where average treatment effects are identified by a nonparametric
unconfoundedness assumption. This claim follows from two results of Robins and
Ritov (1997): (1) with at least one continuous covariate control, no estimator
of the average treatment effect exists which is uniformly consistent without
further assumptions, (2) knowledge of the propensity score yields a uniformly
consistent estimator and honest confidence intervals that shrink at parametric
rates with increasing sample size, regardless of how complicated the propensity
score function is. We emphasize the latter point, and note that
successfully-conducted RCTs provide knowledge of the propensity score to the
researcher. We discuss modern developments in covariate adjustment for RCTs,
noting that statistical models and machine learning methods can be used to
improve efficiency while preserving finite sample unbiasedness. We conclude
that statistical inference has the potential to be fundamentally more difficult
in observational settings than it is in RCTs, even when all confounders are
measured.
