  Causal inference with observational data can be performed under an assumption
of no unobserved confounders (unconfoundedness assumption). There is, however,
seldom clear subject-matter or empirical evidence for such an assumption. We
therefore develop uncertainty intervals for average causal effects based on
outcome regression estimators and doubly robust estimators, which provide
inference taking into account both sampling variability and uncertainty due to
unobserved confounders. In contrast with sampling variation, uncertainty due
unobserved confounding does not decrease with increasing sample size. The
intervals introduced are obtained by deriving the bias of the estimators due to
unobserved confounders. We are thus also able to contrast the size of the bias
due to violation of the unconfoundedness assumption, with bias due to
misspecification of the models used to explain potential outcomes. This is
illustrated through numerical experiments where bias due to moderate unobserved
confounding dominates misspecification bias for typical situations in terms of
sample size and modeling assumptions. We also study the empirical coverage of
the uncertainty intervals introduced and apply the results to a study of the
effect of regular food intake on health. An R-package implementing the
inference proposed is available.
