  Various methods have recently been proposed to estimate causal effects with
confidence intervals that are uniformly valid over a set of data generating
processes when high-dimensional nuisance models are estimated by
post-model-selection or machine learning estimators. These methods typically
require that all the confounders are observed to ensure identification of the
effects. We contribute by showing how valid semiparametric inference can be
obtained in the presence of unobserved confounders and high-dimensional
nuisance models. We propose uncertainty intervals which allow for unobserved
confounding, and show that the resulting inference is valid when the amount of
unobserved confounding is small relative to the sample size; the latter is
formalized in terms of convergence rates. Simulation experiments illustrate the
finite sample properties of the proposed intervals and investigate an
alternative procedure that improves the empirical coverage of the intervals
when the amount of unobserved confounding is large. Finally, a case study on
the effect of smoking during pregnancy on birth weight is used to illustrate
the use of the methods introduced to perform a sensitivity analysis to
unobserved confounding.
