We study a multivariate regression discontinuity design in which treatment is assigned by crossing a boundary in the space of multiple running variables. We document that the existing bandwidth selector is suboptimal for a multivariate regression discontinuity design when the distance to a boundary point is used for its running variable, and introduce a multivariate local-linear estimator for multivariate regression discontinuity designs. Our estimator is asymptotically valid and can capture heterogeneous treatment effects over the boundary. We demonstrate that our estimator exhibits smaller root mean squared errors and often shorter confidence intervals in numerical simulations. We illustrate our estimator in our empirical applications of multivariate designs of a Colombian scholarship study and a U.S. House of representative voting study and demonstrate that our estimator reveals richer heterogeneous treatment effects with often shorter confidence intervals than the existing estimator.

翻译：本文研究了一种多元断点回归设计，其中处理分配通过跨越多个运行变量空间中的边界实现。我们证明，当使用到边界点的距离作为运行变量时，现有带宽选择方法在多元断点回归设计中并非最优，并为此引入了一种多元局部线性估计量。该估计量具有渐近有效性，且能捕捉边界上的异质性处理效应。数值模拟表明，我们的估计量具有更小的均方根误差，且置信区间通常更短。我们通过哥伦比亚奖学金研究的多元设计案例和美国众议院投票研究的实证应用，展示了该估计量的实际效果：与现有估计量相比，我们的方法能揭示更丰富的异质性处理效应，且置信区间通常更短。