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

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