  Eliminating the effect of confounding in observational studies typically
involves fitting a model for an outcome adjusted for covariates. When, as
often, these covariates are high-dimensional, this necessitates the use of
sparse estimators such as the Lasso, or other regularisation approaches. Naiive
use of such estimators yields confidence intervals for the conditional
treatment effect parameter that are not uniformly valid. Moreover, as the
number of covariates grows with sample size, correctly specifying a model for
the outcome is non-trivial. In this work, we deal with both of these concerns
simultaneously, delivering confidence intervals for conditional treatment
effects that are uniformly valid, regardless of whether the outcome model is
correct. This is done by incorporating an additional model for the
treatment-selection mechanism. When both models are correctly specified, we can
weaken the standard conditions on model sparsity. Our procedure extends to
multivariate treatment effect parameters and complex longitudinal settings.
