  Most research on regression discontinuity designs (RDDs) has focused on
univariate cases, where only those units with a "forcing" variable on one side
of a threshold value receive a treatment. Geographical regression discontinuity
designs (GeoRDDs) extend the RDD to multivariate settings with spatial forcing
variables. We propose a framework for analysing GeoRDDs, which we implement
using Gaussian process regression. This yields a Bayesian posterior
distribution of the treatment effect at every point along the border. We
address nuances of having a functional estimand defind on a border with
potentially intricate topology, particularly when defining and estimating
causal estimands of the local average treatment effect (LATE). The Bayesian
estimate of the LATE can also be used as a test statistic in a hypothesis test
with good frequentist properties, which we validate using simulations and
placebo tests. We demonstrate our methodology with a dataset of property sales
in New York City, to assess whether there is a discontinuity in housing prices
at the border between two school district. We find a statistically significant
difference in price across the border between the districts with $p$=0.002, and
estimate a 20% higher price on average for a house on the more desirable side.
