We consider the efficient estimation of total causal effects in the presence of unmeasured confounding using conditional instrumental sets. Specifically, we consider the two-stage least squares estimator in the setting of a linear structural equation model with correlated errors that is compatible with a known acyclic directed mixed graph. To set the stage for our results, we characterize the class of linearly valid conditional instrumental sets that yield consistent two-stage least squares estimators for the target total effect and derive a new asymptotic variance formula for these estimators. Equipped with these results, we provide three graphical tools for selecting more efficient linearly valid conditional instrumental sets. First, a graphical criterion that for certain pairs of linearly valid conditional instrumental sets identifies which of the two corresponding estimators has the smaller asymptotic variance. Second, an algorithm that greedily adds covariates that reduce the asymptotic variance to a given linearly valid conditional instrumental set. Third, a linearly valid conditional instrumental set for which the corresponding estimator has the smallest asymptotic variance that can be ensured with a graphical criterion.

翻译：我们考虑了在存在未测量混杂因素的情况下，使用条件工具集对总因果效应进行高效估计的问题。具体而言，我们研究了线性结构方程模型（具有与已知无环有向混合图兼容的相关误差）中的两阶段最小二乘估计量。为奠定结果基础，我们刻画了能够为目标总效应生成一致两阶段最小二乘估计量的线性有效条件工具集类别，并推导了这些估计量的新渐近方差公式。基于这些结果，我们提供了三种用于选择更高效线性有效条件工具集的图形化工具：第一，一种图形化准则，可识别特定成对线性有效条件工具集中哪个对应的估计量具有更小的渐近方差；第二，一种贪心算法，通过逐步添加可降低渐近方差的协变量来优化给定的线性有效条件工具集；第三，一种线性有效条件工具集，其对应估计量具有可通过图形化准则保证的最小渐近方差。