Distributional Balancing for Causal Inference: A Unified Framework via Characteristic Function Distance
Diptanil Santra, Guanhua Chen, Chan Park
TL;DR
The paper tackles confounding in observational causal inference by introducing a unified nonparametric distributional balancing framework based on the characteristic function distance (CFD). It shows that CFD encompasses popular distances like MMD and energy distance as special cases and develops CFD-based balancing weights obtained from a constrained quadratic program, with a three-way balancing variant to align treated, control, and full samples. The authors establish $\sqrt{n}$-consistency for the CFD weighting estimator under mild regularity, argue that bootstrap can be invalid, and advocate subsampling for valid inference, including extensions to instrumental variables for estimating LATE. Through simulations and a real-data application, the method demonstrates robustness to model misspecification and competitive performance relative to state-of-the-art approaches, offering a versatile, one-shot weighting approach that does not rely on outcome models. The work contributes a principled, general framework that unifies distributional balance, enables IV extensions, and provides practical guidance for inference in causal studies with complex covariate structures.
Abstract
Weighting methods are essential tools for estimating causal effects in observational studies, with the goal of balancing pre-treatment covariates across treatment groups. Traditional approaches pursue this objective indirectly, for example, via inverse propensity score weighting or by matching a finite number of covariate moments, and therefore do not guarantee balance of the full joint covariate distributions. Recently, distributional balancing methods have emerged as robust, nonparametric alternatives that directly target alignment of entire covariate distributions, but they lack a unified framework, formal theoretical guarantees, and valid inferential procedures. We introduce a unified framework for nonparametric distributional balancing based on the characteristic function distance (CFD) and show that widely used discrepancy measures, including the maximum mean discrepancy and energy distance, arise as special cases. Our theoretical analysis establishes conditions under which the resulting CFD-based weighting estimator achieves $\sqrt{n}$-consistency. Since the standard bootstrap may fail for this estimator, we propose subsampling as a valid alternative for inference. We further extend our approach to an instrumental variable setting to address potential unmeasured confounding. Finally, we evaluate the performance of our method through simulation studies and a real-world application, where the proposed estimator performs well and exhibits results consistent with our theoretical predictions.