We introduce a multivariate local-linear estimator for multivariate regression discontinuity designs in which treatment is assigned by crossing a boundary in the space of running variables. The dominant approach uses the Euclidean distance from a boundary point as the scalar running variable; hence, multivariate designs are handled as uni-variate designs. However, the distance running variable is incompatible with the assumption for asymptotic validity. We handle multivariate designs as multivariate. In this study, we develop a novel asymptotic normality for multivariate local-polynomial estimators. Our estimator is asymptotically valid and can capture heterogeneous treatment effects over the boundary. We demonstrate the effectiveness of our estimator through numerical simulations. Our empirical illustration of a Colombian scholarship study reveals a richer heterogeneity (including its absence) of the treatment effect that is hidden in the original estimates.

翻译：我们针对多变量断点回归设计引入了一种多变量局部线性估计量，其中处理分配由跨越运行变量空间中的边界决定。主流方法将到边界点的欧几里得距离作为标量运行变量，从而将多变量设计处理为单变量设计。然而，距离运行变量与渐近有效性的假设不相容。我们以多变量方式处理多变量设计。在本研究中，我们为多变量局部多项式估计量建立了一种新颖的渐近正态性理论。我们的估计量在渐近意义上有效，并能够捕捉边界上的异质性处理效应。通过数值模拟，我们证明了估计量的有效性。我们对哥伦比亚奖学金研究的实证分析揭示了处理效应中更丰富的异质性（包括其缺失性），而这些在原估计中未被显现。