Instrumental variables (IVs) are widely used to estimate causal effects in the presence of unobserved confounding between exposure and outcome. An IV must affect the outcome exclusively through the exposure and be unconfounded with the outcome. We present a framework for relaxing either or both of these strong assumptions with tuneable and interpretable budget constraints. Our algorithm returns a feasible set of causal effects that can be identified exactly given relevant covariance parameters. The feasible set may be disconnected but is a finite union of convex subsets. We discuss conditions under which this set is sharp, i.e., contains all and only effects consistent with the background assumptions and the joint distribution of observable variables. Our method applies to a wide class of semiparametric models, and we demonstrate how its ability to select specific subsets of instruments confers an advantage over convex relaxations in both linear and nonlinear settings. We also adapt our algorithm to form confidence sets that are asymptotically valid under a common statistical assumption from the Mendelian randomization literature.

翻译：工具变量（IV）被广泛用于在暴露与结果间存在未观测混杂时估计因果效应。一个有效的工具变量必须仅通过暴露影响结果，且与结果无混杂。本文提出一个框架，通过可调节且可解释的预算约束来放宽这两个强假设中的任意一个或全部。我们的算法返回一个因果效应的可行集，该集合在给定相关协方差参数下可被精确识别。该可行集可能不连通，但为凸子集的有限并集。我们讨论了该集合成为尖锐集的条件，即其包含且仅包含与背景假设及可观测变量联合分布一致的所有效应。本方法适用于广泛的半参数模型，并通过数值实验证明其选择特定工具变量子集的能力在线性与非线性设定中均优于凸松弛方法。我们还调整算法以构建置信集，该置信集在孟德尔随机化文献中常见统计假设下具有渐近有效性。