[Paper Review] Optimal Covariate Balancing Conditions in Propensity Score Estimation

Jianqing Fan, Kosuke Imai+4 more|arXiv (Cornell University)|Jan 1, 2021

Advanced Causal Inference Techniques50 references4 citations

TL;DR

This paper proposes an optimal Covariate Balancing Propensity Score (CBPS) estimator that achieves double robustness and local semiparametric efficiency by deriving optimal balancing conditions for IPTW. It shows that the proposed method minimizes asymptotic bias and variance under model misspecification, with extensions via sieve estimation for global efficiency under weaker assumptions.

ABSTRACT

Inverse probability of treatment weighting (IPTW) is a popular method for estimating the average treatment effect (ATE). However, empirical studies show that the IPTW estimators can be sensitive to the misspecification of the propensity score model. To address this problem, researchers have proposed to estimate propensity score by directly optimizing the balance of pre-treatment covariates. While these methods appear to empirically perform well, little is known about how the choice of balancing conditions affects their theoretical properties. To fill this gap, we first characterize the asymptotic bias and efficiency of the IPTW estimator based on the Covariate Balancing Propensity Score (CBPS) methodology under local model misspecification. Based on this analysis, we show how to optimally choose the covariate balancing functions and propose an optimal CBPS-based IPTW estimator. This estimator is doubly robust; it is consistent for the ATE if either the propensity score model or the outcome model is correct. In addition, the proposed estimator is locally semiparametric efficient when both models are correctly specified. To further relax the parametric assumptions, we extend our method by using a sieve estimation approach. We show that the resulting estimator is globally efficient under a set of much weaker assumptions and has a smaller asymptotic bias than the existing estimators. Finally, we evaluate the finite sample performance of the proposed estimators via simulation and empirical studies. An open-source software package is available for implementing the proposed methods.

Motivation & Objective

• To address the sensitivity of IPTW estimators to propensity score model misspecification in average treatment effect (ATE) estimation.
• To theoretically characterize the asymptotic bias and efficiency of CBPS-based IPTW estimators under local model misspecification.
• To derive optimal balancing functions that minimize asymptotic bias and achieve local semiparametric efficiency.
• To extend the method using sieve estimation to achieve global efficiency under weaker parametric assumptions.
• To provide a theoretically grounded, open-source implementation for practical causal inference.

Proposed method

• Derives the asymptotic bias and variance of the IPTW estimator under local model misspecification to inform optimal balancing condition selection.
• Proposes an optimal CBPS-based IPTW estimator that is doubly robust—consistent if either the propensity score or outcome model is correctly specified.
• Establishes local semiparametric efficiency when both models are correctly specified, using optimal estimating equations.
• Introduces a sieve-based extension to relax parametric assumptions, achieving global efficiency under weaker regularity conditions.
• Uses estimating equations to enforce covariate balance by optimizing the choice of balancing functions in the propensity score model.
• Employs a two-step estimation procedure: first, estimate the optimal balancing functions; second, apply inverse probability weighting with the resulting propensity scores.

Experimental results

Research questions

• RQ1 How does the choice of covariate balancing functions affect the asymptotic bias and efficiency of IPTW estimators in the presence of local model misspecification?
• RQ2 What conditions on the balancing functions yield a doubly robust and locally efficient IPTW estimator?
• RQ3 Can the proposed method achieve global efficiency under weaker parametric assumptions than existing CBPS methods?
• RQ4 How does the proposed estimator compare in finite samples to existing IPTW and doubly robust estimators in terms of bias, variance, and coverage?
• RQ5 What is the theoretical justification for the optimality of the selected balancing functions in minimizing asymptotic bias?

Key findings

• The proposed optimal CBPS-based IPTW estimator is doubly robust, remaining consistent for the ATE if either the propensity score or outcome model is correctly specified.
• The estimator achieves local semiparametric efficiency when both the propensity score and outcome models are correctly specified.
• The asymptotic bias of the proposed estimator is smaller than that of existing estimators under local model misspecification.
• The sieve-based extension of the method achieves global efficiency under weaker assumptions than traditional parametric CBPS.
• Simulation studies confirm that the proposed estimator has lower bias and improved coverage compared to standard IPTW and GLM-based estimators.
• Empirical studies demonstrate the robustness and practical utility of the method in real-world causal inference settings.