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.

翻译：处理分配中的阈值可能在结果中产生间断，从而揭示因果洞见。在许多情境中（如地理环境），这些阈值是未知且多变量的。我们提出一种非参数方法，通过将回归曲面分割为平滑部分和间断部分来估计由此产生的间断。该估计量采用Mumford-Shah泛函的凸松弛，我们对此建立了识别性和收敛性。运用我们的方法，我们估算出印度一次互联网中断导致经济活动减少超过50%，大幅超越此前估计，并为这类中断对全球数字经济造成的真实成本提供了全新视角。