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For observational studies, we study the sensitivity of causal inference whentreatment assignments may depend on unobserved confounders. We develop a lossminimization approach for estimating bounds on the conditional averagetreatment effect (CATE) when unobserved confounders have a bounded effect onthe odds ratio of treatment selection. Our approach is scalable and allowsflexible use of model classes in estimation, including nonparametric andblack-box machine learning methods. Based on these bounds for the CATE, wepropose a sensitivity analysis for the average treatment effect (ATE). Oursemi-parametric estimator extends/bounds the augmented inverse propensityweighted (AIPW) estimator for the ATE under bounded unobserved confounding. Byconstructing a Neyman orthogonal score, our estimator of the bound for the ATEis a regular root-$n$ estimator so long as the nuisance parameters areestimated at the $o_p(n^{-1/4})$ rate. We complement our methodology withoptimality results showing that our proposed bounds are tight in certain cases.We demonstrate our method on simulated and real data examples, and showaccurate coverage of our confidence intervals in practical finite sampleregimes with rich covariate information.

Bounds on the conditional and average treatment effect with unobserved confounding factors