Recovering Counterfactual Distributions via Wasserstein GANs

TL;DR

This paper introduces a robust method for estimating counterfactual distributions using Wasserstein GANs, overcoming limitations of traditional quantile-based approaches especially under support mismatch and multimodal outcomes.

Contribution

It proposes a novel Wasserstein GAN-based estimator grounded in Optimal Transport, with formal identification and improved stability over existing methods.

Findings

WGAN-based estimator remains consistent under heavy-tailed contamination

It effectively recovers complex bimodal distributions

Outperforms traditional methods in stability and accuracy

Abstract

Standard Distributional Synthetic Controls (DSC) estimate counterfactual distributions by minimizing the Euclidean $L_2$ distance between quantile functions. We demonstrate that this geometric reliance renders estimators fragile: they lack informative gradients under support mismatch and produce structural artifacts when outcomes are multimodal. This paper proposes a robust estimator grounded in Optimal Transport (OT). We construct the synthetic control by minimizing the Wasserstein-1 distance between probability measures, implemented via a Wasserstein Generative Adversarial Network (WGAN). We establish the formal point identification of synthetic weights under an affine independence condition on the donor pool. Monte Carlo simulations confirm that while standard estimators exhibit catastrophic variance explosions under heavy-tailed contamination and support mismatch, our WGAN-based approach remains consistent and stable. Furthermore, we show that our measure-based method correctly recovers complex bimodal mixtures where traditional quantile averaging fails structurally.