Impact of sufficient dimension reduction in nonparametric estimation of causal effect

We consider the estimation of causal treatment effect using nonparametric regression or inverse propensity weighting together with sufficient dimension reduction for searching low-dimensional covariate subsets. A special case of this problem is the estimation of a response effect with data having ignorable missing response values. An issue that is not well addressed in the literature is whether the estimation of the low-dimensional covariate subsets by sufficient dimension reduction has an impact on the asymptotic variance of the resulting causal effect estimator. With some incorrect or inaccurate statements, many researchers believe that the estimation of the low-dimensional covariate subsets by sufficient dimension reduction does not affect the asymptotic variance. We rigorously establish a result showing that this is not true unless the low-dimensional covariate subsets include some covariates superfluous for estimation, and including such covariates loses efficiency. Our theory is supplemented by some simulation results.