In many causal inference problems the parameter of interest is the regression causal effect, defined as the conditional mean difference in the potential outcomes given covariates. In this paper we discuss how sufficient dimension reduction can be used to aid causal inference, and we propose a new estimator of the regression causal effect inspired by minimum average variance estimation. The estimator requires a weaker common support condition than propensity score-based approaches, and can be used to estimate the average causal effect, for which it is shown to be asymptotically super-efficient. Its finite-sample properties are illustrated by simulation.

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