When an exposure of interest is confounded by unmeasured factors, an instrumental variable (IV) can be used to identify and estimate certain causal contrasts. Identification of the marginal average treatment effect (ATE) from IVs relies on strong untestable structural assumptions. When one is unwilling to assert such structure, IVs can nonetheless be used to construct bounds on the ATE. Famously, Balke and Pearl (1997) proved tight bounds on the ATE for a binary outcome, in a randomized trial with noncompliance and no covariate information. We demonstrate how these bounds remain useful in observational settings with baseline confounders of the IV, as well as randomized trials with measured baseline covariates. The resulting bounds on the ATE are non-smooth functionals, and thus standard nonparametric efficiency theory is not immediately applicable. To remedy this, we propose (1) under a novel margin condition, influence function-based estimators of the bounds that can attain parametric convergence rates when the nuisance functions are modeled flexibly, and (2) estimators of smooth approximations of these bounds. We propose extensions to continuous outcomes, explore finite sample properties in simulations, and illustrate the proposed estimators in an observational study targeting the effect of higher education on wages.

翻译：当关注的暴露因素受未测量因素混杂时，工具变量可用于识别和估计某些因果对比。通过工具变量识别边际平均处理效应依赖于强且不可检验的结构性假设。若不愿断言此类结构，工具变量仍可用于构建平均处理效应的边界。经典研究（Balke and Pearl, 1997）证明了在不依从且无协变量信息的随机试验中，二值结果下平均处理效应的紧致边界。我们展示了这些边界如何适用于存在基线混杂因素（作为工具变量的混杂因素）的观察性研究，以及存在有测量基线协变量的随机试验。由此得到的平均处理效应边界是非光滑泛函，标准非参数效率理论无法直接适用。为解决此问题，我们提出：(1) 在新型边界条件下，基于影响函数的边界估计量，当 nuisance 函数灵活建模时可达到参数收敛速度；(2) 这些边界的光滑近似估计量。我们扩展至连续结果，通过模拟考察有限样本性质，并在一项关于高等教育对工资影响的目标效应的观察性研究中展示所提出的估计量。