Covariate adjustment is a commonly used method for total causal effect estimation. In recent years, graphical criteria have been developed to identify all valid adjustment sets, that is, all covariate sets that can be used for this purpose.Different valid adjustment sets typically provide total causal effect estimates of varying accuracies. Restricting ourselves to causal linear models, we introduce a graphical criterion to compare the asymptotic variances provided by certain valid adjustment sets. We employ this result to develop two further graphical tools. First, we introduce a simple variance decreasing pruning procedure for any given valid adjustment set. Second, we give a graphical characterization of a valid adjustment set that provides the optimal asymptotic variance among all valid adjustment sets. Our results depend only on the graphical structure and not on the specific error variances or edge coefficients of the underlying causal linear model. They can be applied to directed acyclic graphs (DAGs), completed partially directed acyclic graphs (CPDAGs) and maximally oriented partially directed acyclic graphs (maximal PDAGs). We present simulations and a real data example to support our results and show their practical applicability.

Instrumental variable models allow us to identify a causal function between covariates X and a response Y , even in the presence of unobserved confounding. Most of the existing estimators assume that the error term in the response Y and the hidden confounders are uncorrelated with the instruments Z. This is often motivated by a graphical separation, an argument that also justifies independence. Posing an independence condition, however, leads to strictly stronger identifiability results. We connect to existing literature in econometrics and provide a practical method for exploiting independence that can be combined with any gradient-based learning procedure. We see that even in identifiable settings, taking into account higher moments may yield better finite sample results. Furthermore, we exploit the independence for distribution generalization. We prove that the proposed estimator is invariant to distributional shifts on the instruments and worstcase optimal whenever these shifts are sufficiently strong. These results hold even in the under-identified case where the instruments are not sufficiently rich to identify the causal function.

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.

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