Regression Discontinuity Design with Distribution-Valued Outcomes

This article introduces Regression Discontinuity Design (RDD) with Distribution-Valued Outcomes (R3D), extending the standard RDD framework to settings where the outcome is a distribution rather than a scalar. Such settings arise when treatment is assigned at a higher level of aggregation than the outcome-for example, when a subsidy is allocated based on a firm-level revenue cutoff while the outcome of interest is the distribution of employee wages within the firm. Since standard RDD methods cannot accommodate such two-level randomness, I propose a novel approach based on random distributions. The target estimand is a "local average quantile treatment effect", which averages across random quantiles. To estimate this target, I introduce two related approaches: one that extends local polynomial regression to random quantiles and another based on local Fr\'echet regression, a form of functional regression. For both estimators, I establish asymptotic normality and develop uniform, debiased confidence bands together with a data-driven bandwidth selection procedure. Simulations validate these theoretical properties and show existing methods to be biased and inconsistent in this setting. I then apply the proposed methods to study the effects of gubernatorial party control on within-state income distributions in the US, using a close-election design. The results suggest a classic equality-efficiency tradeoff under Democratic governorship, driven by reductions in income at the top of the distribution.

disco: Distributional Synthetic Controls

The method of synthetic controls is widely used for evaluating causal effects of policy changes in settings with observational data. Often, researchers aim to estimate the causal impact of policy interventions on a treated unit at an aggregate level while also possessing data at a finer granularity. In this article, we introduce the new disco command, which implements the Distributional Synthetic Controls method introduced in Gunsilius (2023). This command allows researchers to construct entire synthetic distributions for the treated unit based on an optimally weighted average of the distributions of the control units. Several aggregation schemes are provided to facilitate clear reporting of the distributional effects of the treatment. The package offers both quantile-based and CDF-based approaches, comprehensive inference procedures via bootstrap and permutation methods, and visualization capabilities. We empirically illustrate the use of the package by replicating the results in Van Dijcke et al. (2024).

Free Discontinuity Design: With an Application to the Economic Effects of Internet Shutdowns

Thresholds in treatment assignments can produce discontinuities in outcomes, revealing causal insights. In many contexts, like geographic settings, these thresholds are unknown and multivariate. We propose a non-parametric method to estimate the resulting discontinuities by segmenting the regression surface into smooth and discontinuous parts. This estimator uses a convex relaxation of the Mumford-Shah functional, for which we establish identification and convergence. Using our method, we estimate that an internet shutdown in India resulted in a reduction of economic activity by over 50%, greatly surpassing previous estimates and shedding new light on the true cost of such shutdowns for digital economies globally.