The propensity score plays an important role in causal inference with observational data. Once the propensity score is available, one can use it to estimate a variety of causal effects in a unified setting. Despite this appeal, a main practical difficulty arises because the propensity score is usually unknown, has to be estimated, and extreme propensity score estimates  can lead to distorted inference procedures . To address these limitations , this article proposes  to estimate the propensity score by fully exploiting its covariate balancing property. We call the resulting estimator the integrated propensity score (IPS) as it is based on integrated moment conditions. In sharp contrast with other methods that balance only some specific moments of covariates, the IPS aims to balanceall functions of covariates. Further, the IPS estimator is data-driven, does not rely on tuning parameters such as bandwidths, admits an asymptotic linear representation, and is \sqrt{n . We derive the asymptotic properties of inverse probability weighted estimators for the average, distributional and quantile treatment effects based on the IPS , and illustrate their relative performance via Monte Carlo simulations and three empirical applications. An implementation of the proposed methods is provided in the new package \texttt{IPS} for \texttt{R}.
The propensity score plays an important role in causal inference with observational data. However, it is well documented that under slight model misspecifications, propensity score estimates based on maximum likelihood can lead to unreliable treatment effect estimators . To address this practical limitation , this article proposes a new framework for estimating propensity scores that mimics randomize control trials (RCT) in settings where only observational data is available. More specifically, given that in RCTs the joint distritbution of covariates are balanced between treated and not-treated groups, we propose to estimate the propensity score by maxizing the covariate distribution balance. The proposed propensity score estimators, which we call the integrated propensity score (IPS) , are data-driven, do not rely on tuning parameters such as bandwidths, admit an asymptotic linear representation, and can be used to estimate many different treatment effect measures in a unified manner . We derive the asymptotic properties of inverse probability weighted estimators for the average, distributional and quantile treatment effects based on the IPS and illustrate their relative performance via Monte Carlo simulations and three empirical applications. An implementation of the proposed methods is provided in the new package \texttt{IPS} for \texttt{R}.
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