Variable selection for doubly robust causal inference

Confounding control is crucial and yet challenging for causal inference based on observational studies. Under the typical unconfoundness assumption, augmented inverse probability weighting (AIPW) has been popular for estimating the average causal effect (ACE) due to its double robustness in the sense it relies on either the propensity score model or the outcome mean model to be correctly specified. To ensure the key assumption holds, the effort is often made to collect a sufficiently rich set of pretreatment variables, rendering variable selection imperative. It is well known that variable selection for the propensity score targeted for accurate prediction may produce a variable ACE estimator by including the instrument variables. Thus, many recent works recommend selecting all outcome predictors for both confounding control and efficient estimation. This article shows that the AIPW estimator with variable selection targeted for efficient estimation may lose the desirable double robustness property. Instead, we propose controlling the propensity score model for any covariate that is a predictor of either the treatment or the outcome or both, which preserves the double robustness of the AIPW estimator. Using this principle, we propose a two-stage procedure with penalization for variable selection and the AIPW estimator for estimation. We show the proposed procedure benefits from the desirable double robustness property. We evaluate the finite-sample performance of the AIPW estimator with various variable selection criteria through simulation and an application.