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

arXiv: https://arxiv.org/abs/1808.09521 · HTML: https://mlsanity.com/paper/1808.09521/

## Key facts

- Cited by 39 papers in the map (data through 7 Oct 2026).
- First posted on arXiv 28 Aug 2018.

## Built on by (most cited first, 30 of 39)

- [Quantifying Ignorance in Individual-Level Causal-Effect Estimates under Hidden Confounding](https://mlsanity.com/paper/2103.04850/) (2021)
- [Diagnosing Model Performance Under Distribution Shift](https://mlsanity.com/paper/2303.02011/) (2023)
- [Doubly-Valid/Doubly-Sharp Sensitivity Analysis for Causal Inference with Unmeasured Confounding](https://mlsanity.com/paper/2112.11449/) (2021)
- [B-Learner: Quasi-Oracle Bounds on Heterogeneous Causal Effects Under Hidden Confounding](https://mlsanity.com/paper/2304.10577/) (2023)
- [Causal-BALD: Deep Bayesian Active Learning of Outcomes to Infer Treatment-Effects from Observational Data](https://mlsanity.com/paper/2111.02275/) (2021)
- [Long Story Short: Omitted Variable Bias in Causal Machine Learning](https://mlsanity.com/paper/2112.13398/) (2021)
- [What's the Harm? Sharp Bounds on the Fraction Negatively Affected by Treatment](https://mlsanity.com/paper/2205.10327/) (2022)
- [Model-Free and Model-Based Policy Evaluation when Causality is Uncertain](https://mlsanity.com/paper/2204.00956/) (2022)
- [Inference on Strongly Identified Functionals of Weakly Identified Functions](https://mlsanity.com/paper/2208.08291/) (2022)
- [Learning from a Biased Sample](https://mlsanity.com/paper/2209.01754/) (2022)
- [Partial Identification with Noisy Covariates: A Robust Optimization Approach](https://mlsanity.com/paper/2202.10665/) (2022)
- [Falsification before Extrapolation in Causal Effect Estimation](https://mlsanity.com/paper/2209.13708/) (2022)
- [Individualized Decision-Making Under Partial Identification: Three Perspectives, Two Optimality Results, and One Paradox](https://mlsanity.com/paper/2110.10961/) (2021)
- [Hidden yet quantifiable: A lower bound for confounding strength using randomized trials](https://mlsanity.com/paper/2312.03871/) (2023)
- [Partial Identification of Dose Responses with Hidden Confounders](https://mlsanity.com/paper/2204.11206/) (2022)
- [Robust Design and Evaluation of Predictive Algorithms under Unobserved Confounding](https://mlsanity.com/paper/2212.09844/) (2022)
- [Auditing Fairness under Unobserved Confounding](https://mlsanity.com/paper/2403.14713/) (2024)
- [Anytime-Valid Inference for Double/Debiased Machine Learning of Causal Parameters](https://mlsanity.com/paper/2408.09598/) (2024)
- [Minimax Regret Estimation for Generalizing Heterogeneous Treatment Effects with Multisite Data](https://mlsanity.com/paper/2412.11136/) (2024)
- [Distributionally Robust Policy Learning under Concept Drifts](https://mlsanity.com/paper/2412.14297/) (2024)
- [Efficient and Sharp Off-Policy Learning under Unobserved Confounding](https://mlsanity.com/paper/2502.13022/) (2025)
- [Minimax Optimal Estimation of Stability Under Distribution Shift](https://mlsanity.com/paper/2212.06338/) (2022)
- [Confounding-Robust Policy Improvement with Human-AI Teams](https://mlsanity.com/paper/2310.08824/) (2023)
- [StatEval: A Comprehensive Benchmark for Large Language Models in Statistics](https://mlsanity.com/paper/2510.09517/) (2025)
- [Distributionally robust risk evaluation with an isotonic constraint](https://mlsanity.com/paper/2407.06867/) (2024)
- [The Limits of Predicting Agents from Behaviour](https://mlsanity.com/paper/2506.02923/) (2025)
- [Distributionally Robust Causal Inference with Observational Data](https://mlsanity.com/paper/2210.08326/) (2022)
- [Causal Falsification of Digital Twins](https://mlsanity.com/paper/2301.07210/) (2023)
- [Operationalizing Counterfactual Metrics: Incentives, Ranking, and Information Asymmetry](https://mlsanity.com/paper/2305.14595/) (2023)
- [Information-Theoretic Causal Bounds under Unmeasured Confounding](https://mlsanity.com/paper/2601.17160/) (2026)

## Abstract (arXiv)

For observational studies, we study the sensitivity of causal inference when treatment assignments may depend on unobserved confounders. We develop a loss minimization approach for estimating bounds on the conditional average treatment effect (CATE) when unobserved confounders have a bounded effect on the odds ratio of treatment selection. Our approach is scalable and allows flexible use of model classes in estimation, including nonparametric and black-box machine learning methods. Based on these bounds for the CATE, we propose a sensitivity analysis for the average treatment effect (ATE). Our semi-parametric estimator extends/bounds the augmented inverse propensity weighted (AIPW) estimator for the ATE under bounded unobserved confounding. By constructing a Neyman orthogonal score, our estimator of the bound for the ATE is a regular root-n estimator so long as the nuisance parameters are estimated at the op(n-1/4) rate. We complement our methodology with optimality results showing that our proposed bounds are tight in certain cases. We demonstrate our method on simulated and real data examples, and show accurate coverage of our confidence intervals in practical finite sample regimes with rich covariate information.