In this paper, we propose a robust method to estimate the average treatment effects in observational studies when the number of potential confounders is possibly much greater than the sample size. We first use a class of penalized M-estimators for the propensity score and outcome models. We then calibrate the initial estimate of the propensity score by balancing a carefully selected subset of covariates that are predictive of the outcome. Finally, the estimated propensity score is used to construct the inverse probability weighting estimator. We prove that the proposed estimator, which has the sample boundedness property, is root-n consistent, asymptotically normal, and semiparametrically efficient when the propensity score model is correctly specified and the outcome model is linear in covariates. More importantly, we show that our estimator remains root-n consistent and asymptotically normal so long as either the propensity score model or the outcome model is correctly specified. We provide valid confidence intervals in both cases and further extend these results to the case where the outcome model is a generalized linear model. In simulation studies, we find that the proposed methodology often estimates the average treatment effect more accurately than the existing methods. We also present an empirical application, in which we estimate the average causal effect of college attendance on adulthood political participation. Open-source software is available for implementing the proposed methodology.
This paper concerns robust inference on average treatment effects following model selection. Under selection on observables, we construct confidence intervals using a doubly-robust estimator that are robust to model selection errors and prove their uniform validity over a large class of models that allows for multivalued treatments with heterogeneous effects and selection amongst (possibly) more covariates than observations. The semiparametric efficiency bound is attained under appropriate conditions. Precise conditions are given for any model selector to yield these results, and we specifically propose the group lasso, which is apt for treatment effects, and derive new results for high-dimensional, sparse multinomial logistic regression. Both a simulation study and revisiting the National Supported Work demonstration show our estimator performs well in finite samples.