Sharp Bounds for Continuous-Valued Treatment Effects with Unobserved Confounders

In causal inference, treatment effects are typically estimated under the ignorability, or unconfoundedness, assumption, which is often unrealistic in observational data. By relaxing this assumption and conducting a sensitivity analysis, we introduce novel bounds and derive confidence intervals for the Average Potential Outcome (APO)-a standard metric for evaluating continuous-valued treatment or exposure effects. We demonstrate that these bounds are sharp under a continuous sensitivity model, in the sense that they give the smallest possible interval under this model, and propose a doubly robust version of our estimators. In a comparative analysis with another method from the literature, using both simulated and real data sets, we show that our approach not only yields sharper bounds but also achieves good coverage of the true APO, with significantly reduced computation times.