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 typically relies on strong untestable structural assumptions. When one is unwilling to assert such structural assumptions, IVs can nonetheless be used to construct bounds on the ATE. Famously, Balke and Pearl (1997) employed linear programming techniques to prove 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 lower and upper bounds on the ATE are non-smooth functionals, and thus standard nonparametric efficiency theory is not immediately applicable. To remedy this, we propose (1) estimators of smooth approximations of these bounds, and (2) under a novel margin condition, influence function-based estimators of the ATE bounds that can attain parametric convergence rates when the nuisance functions are modeled flexibly. We propose extensions to continuous outcomes, and finally, illustrate the proposed estimators in a randomized experiment studying the effects of influenza vaccination encouragement on flu-related hospital visits.

翻译：当感兴趣的处理变量受到未测量因素的混杂影响时，工具变量可用于识别和估计特定因果对比。利用工具变量识别边际平均处理效应通常依赖于强不可检验的结构假设。若不愿假定此类结构假设，工具变量仍可用于构建平均处理效应的界。著名的Balke和Pearl（1997）通过线性规划技术，在无依从性且无协变量信息的随机试验中，证明了二元结局下平均处理效应的紧致界。我们论证了这些界在存在工具变量基线混杂因素的观察性研究以及测量基线协变量的随机试验中依然有效。由此得出的平均处理效应下界和上界是非光滑泛函，因此标准非参数效率理论无法直接适用。为解决这一问题，我们提出：(1)这些界的光滑近似估计量，以及(2)在新型边际条件下，基于影响函数的平均处理效应边界估计量，该估计量在灵活建模 nuisance 函数时可达到参数收敛速度。我们将方法扩展至连续结局，并最终在评估流感疫苗接种鼓励对流感相关住院影响的随机实验中展示所提议的估计量。