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Recovering Counterfactual Distributions via Wasserstein GANs

Xinran Liu

TL;DR

The paper develops a Wasserstein GAN framework for Distributional Synthetic Controls, replacing $L_2$-based quantile matching with a transport-based objective $W_1$ to recover counterfactual outcome distributions. It establishes identification under a latent-factor/transport structure with a scaled-isometry condition, and shows consistency and asymptotic normality of the estimator under a sieve-type regime. The empirical results demonstrate superior robustness to heavy-tailed contamination, support mismatch, and multimodal counterfactuals, including a bimodal recovery example and a multivariate extension, with a Kansas structural-break stress test illustrating practical applicability. The approach provides a principled, geometry-aware method for distributional causal inference that extends naturally to high-dimensional outcomes and circumvents the gradient/overlap issues that plague quantile-based methods.

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.

Recovering Counterfactual Distributions via Wasserstein GANs

TL;DR

The paper develops a Wasserstein GAN framework for Distributional Synthetic Controls, replacing -based quantile matching with a transport-based objective to recover counterfactual outcome distributions. It establishes identification under a latent-factor/transport structure with a scaled-isometry condition, and shows consistency and asymptotic normality of the estimator under a sieve-type regime. The empirical results demonstrate superior robustness to heavy-tailed contamination, support mismatch, and multimodal counterfactuals, including a bimodal recovery example and a multivariate extension, with a Kansas structural-break stress test illustrating practical applicability. The approach provides a principled, geometry-aware method for distributional causal inference that extends naturally to high-dimensional outcomes and circumvents the gradient/overlap issues that plague quantile-based methods.

Abstract

Standard Distributional Synthetic Controls (DSC) estimate counterfactual distributions by minimizing the Euclidean 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.
Paper Structure (42 sections, 5 theorems, 61 equations, 10 figures, 4 tables, 2 algorithms)

This paper contains 42 sections, 5 theorems, 61 equations, 10 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Deep Generative Models for Causal Inference
  4. The Econometric Framework
  5. Setup and Notation
  6. Estimation and Aggregation
  7. Structural Identification
  8. Theorem 1 (Counterfactual Recovery).
  9. Proof.
  10. Geometric Robustness and Theoretical Comparison
  11. Gradient Existence under Support Mismatch.
  12. Proof.
  13. Estimation and Computation
  14. The Empirical Minimax Formulation
  15. Network Architecture and Gradient Penalty
  16. ...and 27 more sections

Key Result

Proposition 1

Let $P_{1}$ be the treated distribution and $P_{\lambda} = \sum_{j=2}^{J+1} \lambda_j P_{j}$ be the synthetic mixture. Assume the supports are disjoint, i.e., $\text{supp}(P_{1}) \cap \text{supp}(P_{\lambda}) = \emptyset$, and that the distributions are absolutely continuous with densities $p_1$ and

Figures (10)

  • Figure 1: Visual Comparison of Clean and Contaminated Target Distributions. The panels depict the density of the treated outcome in period $t=2$ ($N=300$) under varying degrees of heavy-tailed contamination. The solid blue line represents the clean, ground-truth distribution, while the dashed orange line shows the observed contaminated distribution. As $\epsilon$ increases from $1\%$ to $4\%$, the Wasserstein-1 distance between the clean and contaminated samples ($\widehat{\Delta W}_1$) rises monotonically. The outliers are strictly localized in the tail, leaving the central mode structurally intact.
  • Figure 2: Estimator Robustness to Heavy-Tailed Contamination. The plot displays the Average RMSE of the estimated weights ($\hat{\bm{\lambda}}$) over $N_{SIM}=100$ simulations. The Baseline $L_2$ estimator (blue) exhibits high variance and bias immediately upon contamination ($\text{RMSE} \approx 0.39$), whereas the proposed WGAN estimator (orange) maintains structural fidelity ($\text{RMSE} \approx 0.06$) across all levels.
  • Figure 3: Estimator Variance vs. Degree of Support Mismatch ($\gamma$). The plot displays the Average Variance of the estimated weights. The $L_2$-CDF estimator (blue) exhibits asymptotic instability as the supports become disjoint ($\gamma \to 0.9$). The WGAN-E estimator (orange) maintains bounded variance across the entire domain.
  • Figure 4: Mean Weight Assignment under Support Mismatch. The panels depict the average weight assigned to each donor ($J=4$). The $L_2$ solver (left) yields erratic estimates. The WGAN-E solver (right) converges to the uniform prior ($\lambda_j \approx 0.25$) as the transport cost becomes uniform across donors.
  • Figure 5: Recovery of Bimodal Probability Mass Function. The top panel displays the ground truth target distribution. The middle panel shows the synthetic distribution from the $W_2$-Quantile benchmark, which fails to capture the bimodality and produces a unimodal artifact. The bottom panel shows the WGAN-E synthesis, which successfully recovers the two distinct modes via a correct linear mixture of donors.
  • ...and 5 more figures

Theorems & Definitions (18)

  • Proposition 1: Non-vanishing Gradients
  • Theorem 1: Global Identification
  • proof
  • Theorem 2: Consistency
  • proof
  • Remark 1: Overcoming the Curse of Dimensionality
  • Theorem 3: Asymptotic Normality
  • proof
  • Remark 2: Curse of Dimensionality
  • Theorem 4: Validity of Permutation Inference
  • ...and 8 more