Treatment Effect Estimation with Adjustment Feature Selection

In causal inference, it is common to select a subset of observed covariates, named the adjustment features, to be adjusted for estimating the treatment effect. For real-world applications, the abundant covariates are usually observed, which contain extra variables partially correlating to the treatment (treatment-only variables, e.g., instrumental variables) or the outcome (outcome-only variables, e.g., precision variables) besides the confounders (variables that affect both the treatment and outcome). In principle, unbiased treatment effect estimation is achieved once the adjustment features contain all the confounders. However, the performance of empirical estimations varies a lot with different extra variables. To solve this issue, variable separation/selection for treatment effect estimation has received growing attention when the extra variables contain instrumental variables and precision variables.

In this paper, assuming no mediator variables exist, we consider a more general setting by allowing for the existence of post-treatment and post-outcome variables rather than instrumental and precision variables in observed covariates. Our target is to separate the treatment-only variables from the adjustment features. To this end, we establish a metric named Optimal Adjustment Features (OAF), which empirically measures the asymptotic variance of the estimation. Theoretically, we show that our OAF metric is minimized if and only if adjustment features consist of the confounders and *