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.
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