Distributional Discontinuity Design

1. Problem Statement

Traditional Regression Discontinuity Designs (RDD) and Regression Kink Designs (RKD) primarily focus on estimating mean treatment effects (the jump or change in slope of the conditional mean at a cutoff). While effective for average impacts, this approach often masks treatment heterogeneity. A treatment might leave the mean unchanged while drastically altering the variance, skewness, or tail behavior of the outcome distribution.

Existing methods for distributional effects, such as Quantile Treatment Effects (QTE), face practical challenges:

• They produce a curve of effects rather than a single summary measure, making communication difficult.
• They often require strong rank-invariance assumptions to interpret individual-level effects.
• They can be computationally complex to implement with uniform inference.

The authors propose a framework to estimate the entire distributional shift at a discontinuity or kink without relying on rank invariance, providing a single, interpretable metric of causal impact.

2. Methodology

Core Concept: The Wasserstein Distance

The authors define the causal estimand as the 2-Wasserstein distance ($W_2$) between the limiting conditional counterfactual outcome distributions just above and below the cutoff (or at the kink).

• Definition: For counterfactual distributions $P_1$ (treated) and $P_0$ (untreated) at the cutoff $x_0$, the estimand is:
  $$\Psi = W_2(P_1|x_0, P_0|x_0) = \left( \int_0^1 (Q_1(u) - Q_0(u))^2 du \right)^{1/2}$$
  where $Q_a(u)$ is the quantile function of the outcome under treatment $a$.
• Interpretation: This measures the minimal "transport cost" to transform the untreated distribution into the treated one. It captures shifts in location, scale, and shape simultaneously.

Key Theoretical Results

1. Relation to Mean Effects: The Wasserstein effect $\Psi$ weakly upper bounds the absolute average treatment effect (ATE), $|\tau|$.
   • $|\tau| \leq \Psi$.
   • Equality holds if and only if the treatment effect is purely additive (a constant shift across all quantiles).
   • The difference $\Psi^2 - \tau^2$ quantifies the heterogeneity of the treatment effect.
2. Orthogonal Decomposition (L-Moments): The authors derive a decomposition of $\Psi^2$ into squared differences of L-moments (linear combinations of order statistics, which are robust analogues to conventional moments).
   • $\Psi^2 = \sum_{k=1}^{\infty} (2k-1)(\lambda_k^{(1)} - \lambda_k^{(0)})^2$.
   • This allows for a "Distributional $R^2$," quantifying how much of the total distributional shift is explained by changes in location ($k=1$), scale ($k=2$), skewness ($k=3$), etc.
3. Directionality: The authors define a Wasserstein Dominance metric ($\rho \in [-1, 1]$) to determine if the treated distribution stochastically dominates the untreated one, or if effects cancel out (crossing quantiles).

Extensions to Fuzzy Designs and Kinks

• Fuzzy RDD: The framework extends to fuzzy designs (where the cutoff affects the probability of treatment but does not guarantee it). The estimand is defined for the sub-population of local compliers, identified via a local Wald ratio of the distribution functions.
• Distributional Kink Designs: The authors extend the framework to Regression Kink Designs (RKD). Instead of a jump in the distribution, they define the Wasserstein derivative ($\Psi'$) at the kink. This measures the instantaneous rate of probability mass flow through the kink.
• Identification: New identification results are derived for fuzzy kink designs, extending the work of Wang and Zhang (2025) to distributional outcomes.

Estimation and Inference

• Estimator: A plug-in estimator using local polynomial quantile regression with bias correction (following Chiang et al., 2019).
• Inference Challenges: Since $\Psi$ is a quadratic functional, the limiting distribution changes as the true effect approaches zero (non-regular estimation).
• Solutions:
  • Hypothesis Testing: Two methods are proposed: (1) An eigenvalue-based Monte Carlo test (approximating a Gaussian chaos), and (2) A conservative test using Chebyshev inequalities.
  • Confidence Intervals: Two approaches are offered: (1) Constructing intervals based on uniform confidence bands for quantile effects, and (2) A conservative interval that widens near the null to account for the non-regularity.

3. Key Contributions

1. New Causal Framework: Introduces "Distributional Discontinuity Designs," shifting the focus from mean jumps to the total Wasserstein distance between distributions.
2. Interpretability: Provides a single, scale-interpretable number ($\Psi$) that summarizes the total magnitude of distributional change, complementing traditional mean effects.
3. Decomposition Tool: The L-moment decomposition offers a novel way to diagnose why distributions differ (e.g., is the effect driven by variance or skewness?), without requiring rank invariance.
4. Fuzzy Kink Identification: Establishes new identification results for fuzzy treatment assignment in kink designs, a previously under-explored area in distributional causal inference.
5. Robust Inference: Develops valid statistical inference procedures that account for the non-standard asymptotic behavior of quadratic functionals near the null hypothesis.

4. Empirical Results

The authors re-analyze two natural experiments to demonstrate the utility of their method:

• Case 1: Incumbency Advantage (RDD)

  • Data: US House elections (Lee, 2008).
  • Finding: The mean effect (7.1% vote share increase) and the Wasserstein effect (7.5%) were very close.
  • Insight: The L-moment decomposition showed that ~88% of the distributional distance was explained by the mean (L-location), confirming that the incumbency effect was largely a uniform shift with little heterogeneity.

• Case 2: Intergovernmental Grants (RKD)

  • Data: Swedish municipal grants and public employment (Lundqvist et al., 2014).
  • Finding: The mean effect was statistically indistinguishable from zero. The Wasserstein derivative was also not significantly different from zero.
  • Insight: However, the L-moment decomposition revealed that the small observed distributional distance was driven almost entirely by higher-order moments (scale and skewness), not the mean. This suggests that while the average effect is null, the grants may have caused significant variability or outlier effects in specific municipalities, which traditional mean-based analysis would miss.

5. Significance

This paper bridges the gap between optimal transport theory and causal inference. It provides applied researchers with a rigorous, interpretable tool to detect and quantify complex treatment effects that are invisible to mean-based analyses. By decomposing the Wasserstein distance into L-moments, it offers a "diagnostic" capability to understand the nature of heterogeneity (e.g., is a policy increasing inequality or changing the shape of the distribution?). The framework is generalizable to other quasi-experimental designs (like Difference-in-Differences) and randomized trials, offering a new standard for reporting distributional causal effects.