  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.
