Effects of correlated covariates on the efficiency of matching and inverse probability weighting estimators for causal inference

In observational studies the overall aim when fitting a model for the propensity score is to reduce bias for an estimator of the causal effect. For this purpose guidelines for covariate selection for propensity score models have been proposed in the causal inference literature. To make the assumption of an unconfounded treatment plausible researchers might be tempted to include many, possibly correlated, covariates in the propensity score model. In this paper we study how the efficiency of matching and inverse probability weighting estimators for average causal effects change when the covariates are correlated. We investigate the case with multivariate normal covariates and linear models for the propensity score and potential outcomes and show results under different model assumptions. We show that the correlation can both increase and decrease the large sample variances of the estimators, and that the corrrelation affects the efficiency of the estimators differently, both with regard to direction and magnitude. Moreover, the strength of the confounding towards the outcome and the treatment plays an important role.