  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.
