  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.
