[Paper Review] Large Sample Properties of Entropy Balancing Estimators of Average Causal Effects

David Källberg, Ingeborg Waernbaum|arXiv (Cornell University)|Apr 22, 2022

Advanced Causal Inference Techniques4 citations

TL;DR

This paper establishes the large-sample asymptotic properties of entropy balancing estimators for average causal effects using Kullback-Leibler and quadratic Rényi divergences. It shows that while entropy balancing reduces model dependence, consistency requires implicit parametric assumptions about the propensity score or outcome models, and provides estimators for asymptotic variances to enable valid inference in observational studies.

ABSTRACT

Weighting methods are used in observational studies to adjust for covariate imbalances between treatment and control groups. Entropy balancing (EB) is an alternative to inverse probability weighting with an estimated propensity score. The EB weights are constructed to satisfy balance constraints and optimized towards stability. We describe large sample properties of EB estimators of the average causal treatment effect, based on the Kullback-Leibler and quadratic Rényi relative entropies. Additionally, we propose estimators of their asymptotic variances. Even though the objective of EB is to reduce model dependence, the estimators are generally not consistent unless implicit parametric assumptions for the propensity score or conditional outcomes are met. The finite sample properties of the estimators are investigated through a simulation study. In an application with observational data from the Swedish Childhood Diabetes Register, we estimate the average effect of school achievements on hospitalization due to acute complications of type 1 diabetes mellitus.

Motivation & Objective

• To establish the large-sample asymptotic properties of entropy balancing estimators for average causal effects.
• To investigate the conditions under which entropy balancing estimators are consistent, despite their design to reduce model dependence.
• To derive estimators for the asymptotic variances of the Kullback-Leibler and quadratic Rényi divergence-based entropy balancing estimators.
• To assess the finite-sample performance of these estimators through simulation and empirical application.
• To clarify the role of balancing functions and divergence measures in inducing implicit parametric assumptions that affect estimator consistency.

Proposed method

• Entropy balancing weights are derived by minimizing Kullback-Leibler or quadratic Rényi divergence relative to uniform weights, subject to three-way balance constraints on covariate functions.
• Three-way balance ensures mean equality of covariate functions across treated, control, and combined groups.
• The Kullback-Leibler divergence variant is linked to a parametric propensity score model with a logit link, while the quadratic Rényi variant yields minimum-variance weights.
• Asymptotic normality and consistency are established under implicit parametric assumptions for the outcome or propensity score models.
• Variance estimators are derived based on the influence function, enabling confidence intervals for the average treatment effect.
• The method is applied to observational data from the Swedish Childhood Diabetes Register to estimate the effect of school grades on hospitalization due to type 1 diabetes complications.

Experimental results

Research questions

• RQ1 Under what conditions are entropy balancing estimators of the average treatment effect consistent?
• RQ2 How do the choice of divergence measure (Kullback-Leibler vs. quadratic Rényi) affect the implicit parametric assumptions and estimator properties?
• RQ3 What are the asymptotic variance estimators for entropy balancing estimators, and how do they support valid inference?
• RQ4 How do the finite-sample properties of entropy balancing estimators compare to standard inverse probability weighting in realistic settings?
• RQ5 To what extent does approximate balance or post-selection of covariate functions affect estimator performance in high-dimensional settings?

Key findings

• The Kullback-Leibler and quadratic Rényi entropy balancing estimators are asymptotically normal and consistent only under implicit parametric assumptions for the propensity score or outcome model.
• The estimators are not doubly robust in the conventional sense, as a correctly specified propensity score alone is insufficient for consistency without an outcome model for at least one group.
• In the empirical application, the QR estimator yields an estimated average causal effect of 0.90 days (95% CI: 0.42–1.37) of increased hospitalization due to low school grades.
• The KL estimator estimates a causal effect of 0.86 days (95% CI: 0.37–1.35), with confidence intervals widening when using an alternative variance estimator from Chan et al. (2016).
• Using the Chan et al. (2016) variance estimator, the 95% confidence intervals narrow to (0.51, 1.28) for QR and (0.54, 1.17) for KL, showing sensitivity to variance estimation method.
• A standard logistic regression IPW estimator yields a similar estimate of 0.81 (95% CI: 0.49–1.13), suggesting robustness across weighting methods.