  In this paper, we propose a propensity score adapted variable selection
procedure to select covariates for inclusion in propensity score models, in
order to eliminate confounding bias and improve statistical efficiency in
observational studies. Our variable selection approach is specially designed
for causal inference, it only requires the propensity scores to be
$\sqrt{n}$-consistently estimated through a parametric model and need not
correct specification of potential outcome models. By using estimated
propensity scores as inverse probability treatment weights in performing an
adaptive lasso on the outcome, it successfully excludes instrumental variables,
and includes confounders and outcome predictors. We show its oracle properties
under the "linear association" conditions. We also perform some numerical
simulations to illustrate our propensity score adapted covariate selection
procedure and evaluate its performance under model misspecification. Comparison
to other covariate selection methods is made using artificial data as well,
through which we find that it is more powerful in excluding instrumental
variables and spurious covariates.
