Uniformly valid confidence intervals for conditional treatment effects in misspecified high-dimensional models

Abstract: 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.