Graphical tools for selecting accurate and valid conditional instrumental sets

We consider the accurate 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 fully characterise the class of conditional instrumental sets that result in a consistent two-stage least squares estimator for our target total effect. We refer to members of this class as valid conditional instrumental sets. Equipped with this definition, we provide three graphical tools for selecting accurate and valid conditional instrumental sets: First, a graphical criterion that for certain pairs of valid conditional instrumental sets identifies which of the two corresponding estimators has the smaller asymptotic variance. Second, a forward algorithm that greedily adds covariates that reduce the asymptotic variance to a valid conditional instrumental set. Third, a valid conditional instrumental set for which the corresponding estimator has the smallest asymptotic variance we can ensure with a graphical criterion.