Geographic Regression Discontinuity Design

• Geographic Regression Discontinuity Design is a method where treatment is assigned based on spatial location relative to a boundary, yielding a treatment-effect function along the border.
• It generalizes one-dimensional RD by utilizing bivariate scores and applying local-polynomial or distance-based estimators to capture discontinuities across irregular borders.
• The approach supports diverse estimation strategies while emphasizing the importance of accounting for border geometry, potential misspecification, and robustness through falsification tests.

to=arxiv_search.search 大发彩票快三json {"query":"Geographic Regression Discontinuity Design boundary discontinuity design GeoRDD arXiv", "max_results": 10} to=arxiv_search.search 皇轩json {"query":"(Butts, 2021) OR (Cattaneo et al., 8 May 2025) OR (Cattaneo et al., 12 May 2025) OR (Rischard et al., 2018) OR (Kendall et al., 2021) Geographic Regression Discontinuity Design", "max_results": 10} Geographic Regression Discontinuity Design (GeoRDD) is a regression discontinuity design in which treatment assignment is determined by location relative to a geographic boundary rather than by a scalar cutoff. In the modern boundary-discontinuity formulation, units have a bivariate score $X_i=(X_{1i},X_{2i})^\top$, the support is partitioned into control and treatment regions $A_0$ and $A_1$, the assignment boundary is $B=\mathtt{bd}(A_0)\cap \mathtt{bd}(A_1)$, and the observed outcome is $Y_i=(1-T_i)Y_i(0)+T_iY_i(1)$ with $T_i=1(X_i\in A_1)$. The central estimand is typically not a single scalar jump but a treatment-effect function along the boundary, $\tau(x)=E[Y_i(1)-Y_i(0)\mid X_i=x]$ for $x\in B$. This places GeoRDD within the broader class of multi-score or boundary discontinuity designs, while preserving the standard RD logic of local comparison at a discontinuous assignment rule (Cattaneo et al., 2021, Cattaneo et al., 8 May 2025).

1. Conceptual structure and place within the RD family

GeoRDD generalizes canonical one-dimensional RD by replacing a scalar threshold with a continuous border. In the standard sharp RD reviewed in "Regression Discontinuity Designs" (Cattaneo et al., 2021), treatment is assigned by $T_i=1(X_i\ge c)$, and the sharp estimand is

$$\tau_\mathtt{SRD}=E[Y_i(1)-Y_i(0)\mid X=c] =\lim_{x\downarrow c}E[Y_i\mid X_i=x]-\lim_{x\uparrow c}E[Y_i\mid X_i=x].$$

In GeoRDD, the cutoff is an infinite collection of points on a boundary rather than a single $A_0$0, so the natural object is a curve-valued estimand indexed by boundary location rather than a scalar effect (Cattaneo et al., 2021, Cattaneo et al., 8 May 2025).

This distinction is substantive, not merely notational. A geographic boundary can contain straight segments, curved segments, and kinks; treatment effects may vary along urban and rural stretches, along populated and unpopulated stretches, or across administratively distinct subsegments. A common misconception is that GeoRDD automatically identifies one border effect. The recent boundary-discontinuity literature instead treats the treatment effect as a function $A_0$1 over the assignment boundary and then regards scalar summaries as additional choices requiring explicit weighting or aggregation rules (Cattaneo et al., 8 May 2025, Rischard et al., 2018).

The broader RD literature also distinguishes two interpretive frameworks that transfer directly to geography. Under the continuity framework, identification comes from continuity of conditional expectations of potential outcomes at the boundary. Under the local randomization framework, one assumes a sufficiently narrow window around the boundary within which side-of-border assignment is as-if random. In GeoRDD, the first framework targets a limit effect at the boundary, while the second targets an average effect within a border neighborhood (Cattaneo et al., 2021).

2. Identification logic and boundary-specific estimands

In the boundary-discontinuity formulation, the primary continuity-based estimand is

$A_0$2

identified by

$A_0$3

where $A_0$4 for $A_0$5. The formal assumptions are the standard RD ones adapted to a bivariate score: treatment changes discontinuously across a known boundary, the conditional mean functions are smooth within regions and continuous up to the boundary from each side, the score density is regular near the boundary, and bandwidth shrinks while local sample size grows (Cattaneo et al., 8 May 2025).

A parallel scalar representation uses signed distance to a boundary point. For each $A_0$6, define

$A_0$7

and then

$A_0$8

with

$A_0$9

The boundary-discontinuity literature proves $A_1$0 for all $A_1$1, so distance-to-boundary constructions are identified for the same local effect. The crucial caveat is not identification per se but approximation error and misspecification bias under realistic boundary geometry (Cattaneo et al., 8 May 2025).

An older Bayesian nonparametric GeoRDD formulation expresses the same idea in spatial coordinates. There, units have locations $A_1$2, treatment is $A_1$3, and the pointwise border effect is

$A_1$4

That paper emphasizes that a scalar “local average treatment effect” in GeoRDD is not unique: it depends on how one aggregates $A_1$5 along the border. It therefore studies uniform weighting, density weighting, inverse-variance weighting, and projected finite-population averaging, and shows that some apparently natural border averages are sensitive to border topology (Rischard et al., 2018).

This suggests a second common misconception: signed distance or border length alone does not exhaust the geometry of the design. A boundary can be statistically and substantively heterogeneous even when the treatment rule is sharp.

3. Estimation strategies

Two estimation strategies now dominate the modern boundary-discontinuity literature. The first uses the full bivariate score directly. The second reduces the design to a univariate RD in signed distance to the boundary (Cattaneo et al., 8 May 2025, Cattaneo et al., 12 May 2025).

For the bivariate local-polynomial approach, the treatment effect estimator at a boundary point $A_1$6 is

$A_1$7

where, for $A_1$8,

$A_1$9

This is the direct two-dimensional analogue of local-polynomial RD, with separate local fits on each side of the boundary. The R package rd2d implements this approach in rd2d() and supplies pointwise and uniform inference, heteroskedasticity-robust and cluster-robust options, and bandwidth selectors via rdbw2d() (Cattaneo et al., 12 May 2025).

For the distance-based approach, one estimates

$B=\mathtt{bd}(A_0)\cap \mathtt{bd}(A_1)$0

where, for $B=\mathtt{bd}(A_0)\cap \mathtt{bd}(A_1)$1,

$B=\mathtt{bd}(A_0)\cap \mathtt{bd}(A_1)$2

This is simpler because it reuses standard univariate RD machinery, and rd2d.dist() implements it directly (Cattaneo et al., 12 May 2025).

The major methodological result is that the two estimators are not interchangeable when the assignment boundary has kinks or other irregularities. "Estimation and Inference in Boundary Discontinuity Designs" (Cattaneo et al., 8 May 2025) shows that distance-based methods can exhibit an irreducible large misspecification bias of order $B=\mathtt{bd}(A_0)\cap \mathtt{bd}(A_1)$3 near such features, regardless of polynomial order, because reducing $B=\mathtt{bd}(A_0)\cap \mathtt{bd}(A_1)$4 to a scalar distance can induce non-differentiability in the conditional expectation as a function of distance. By contrast, the bivariate estimator uses the original coordinates and does not suffer from that drawback. The practical recommendation in the software paper is therefore to prefer the location-based method whenever possible, and to use distance-based methods more cautiously, especially near corners, kinks, intersections, or jagged political borders (Cattaneo et al., 8 May 2025, Cattaneo et al., 12 May 2025).

A distinct line of work replaces local-polynomial fitting with optimization. "Optimized Regression Discontinuity Designs" (Imbens et al., 2017) proposes the finite-sample minimax linear estimator

$B=\mathtt{bd}(A_0)\cap \mathtt{bd}(A_1)$5

with weights chosen to minimize worst-case conditional mean squared error under a bounded-second-derivative class. The method explicitly extends to multiple running variables and arbitrarily shaped treatment regions $B=\mathtt{bd}(A_0)\cap \mathtt{bd}(A_1)$6, which makes it directly relevant to GeoRDD with $B=\mathtt{bd}(A_0)\cap \mathtt{bd}(A_1)$7 and irregular administrative borders. Its key assumption is multivariate smoothness,

$B=\mathtt{bd}(A_0)\cap \mathtt{bd}(A_1)$8

and its advantage is bias-aware, uniform inference conditional on the realized sample geometry (Imbens et al., 2017).

4. Temporal, Bayesian, and hierarchical extensions

A central econometric extension is the geographic difference-in-discontinuities design. In a standard cross-sectional GeoRDD with signed distance $B=\mathtt{bd}(A_0)\cap \mathtt{bd}(A_1)$9, the benchmark model is

$Y_i=(1-T_i)Y_i(0)+T_iY_i(1)$0

and identification of $Y_i=(1-T_i)Y_i(0)+T_iY_i(1)$1 requires continuity of both $Y_i=(1-T_i)Y_i(0)+T_iY_i(1)$2 and $Y_i=(1-T_i)Y_i(0)+T_iY_i(1)$3 at the boundary. "Geographic Difference-in-Discontinuities" (Butts, 2021) argues that in geographic settings this is often too strong because borders may bundle compound treatments, sorting, and border-specific confounds. With two periods, the paper instead models

$Y_i=(1-T_i)Y_i(0)+T_iY_i(1)$4

where $Y_i=(1-T_i)Y_i(0)+T_iY_i(1)$5 is a time-invariant discontinuity capturing persistent sorting, persistent effects of other policies, and other fixed border-specific differences. First differencing then removes $Y_i=(1-T_i)Y_i(0)+T_iY_i(1)$6, and under the stated continuity conditions,

$Y_i=(1-T_i)Y_i(0)+T_iY_i(1)$7

The identifying gain is therefore a shift from requiring a “clean” border in cross section to requiring contamination that is stable over time (Butts, 2021).

Bayesian nonparametric GeoRDD uses Gaussian processes instead of local polynomials. "A Bayesian Nonparametric Approach to Geographic Regression Discontinuity Designs" (Rischard et al., 2018) models treatment and control surfaces as two independent Gaussian processes,

$Y_i=(1-T_i)Y_i(0)+T_iY_i(1)$8

typically with squared exponential kernel

$Y_i=(1-T_i)Y_i(0)+T_iY_i(1)$9

and then defines the border “cliff height” as the posterior difference between the two smooth surfaces at sentinel points on the border. This yields a posterior distribution for $T_i=1(X_i\in A_1)$0 at every point along the border and makes scalar summaries linear functionals of a multivariate normal posterior (Rischard et al., 2018).

A related but broader Bayesian literature is relevant as a methodological template. "Bayesian nonparametric discontinuity design" (Hinne et al., 2019) frames discontinuity analysis as Bayesian model comparison between a continuous and a discontinuous response surface using Gaussian process regression. It explicitly states that the framework extends naturally to multiple assignment variables and that GeoRDD is the special case in which the two-dimensional assignment variable represents a spatial location. "Bayesian analysis of regression discontinuity designs with heterogeneous treatment effects" (Tao et al., 14 Apr 2025) is not a GeoRDD paper, but it develops a hierarchical Gaussian-process RD model with subgroup-specific effects $T_i=1(X_i\in A_1)$1. The paper itself treats adaptation to GeoRDD as a natural extrapolation: border segments, municipalities, counties, ecological zones, or border-by-time cells can serve as known sub-populations, allowing partial pooling of sparse local effects along a border (Hinne et al., 2019, Tao et al., 14 Apr 2025).

5. Inference, falsification, and robustness

The general RD literature recommends local-polynomial estimation, data-driven bandwidth choice, and robust bias-corrected inference rather than conventional confidence intervals when MSE-optimal bandwidths are used. Those principles transfer directly to GeoRDD. In the boundary-discontinuity package literature, pointwise inference is based on studentized local-polynomial estimators, and uniform confidence bands are formed over a discretized grid of boundary points with critical values obtained from the covariance structure across those points (Cattaneo et al., 2021, Cattaneo et al., 12 May 2025).

The rd2d framework provides both pointwise and uniform inference along the boundary. For the bivariate method, pointwise intervals are formed from $T_i=1(X_i\in A_1)$2 and its estimated variance, and robust bias correction uses a higher-order local polynomial with default $T_i=1(X_i\in A_1)$3 and $T_i=1(X_i\in A_1)$4. The package also implements integrated and pointwise bandwidth criteria, side-specific bandwidths, and standardization of coordinate scales (Cattaneo et al., 12 May 2025).

Validation and falsification remain essential because geography makes border continuity difficult to defend. The general RD review emphasizes covariate continuity, placebo outcomes, placebo cutoffs, donut-hole tests, bandwidth sensitivity, and graphical analysis. In GeoRDD, these become balance tests across the boundary, placebo borders or shifted borders, exclusion of observations closest to the border, and sensitivity to buffer width or segment choice (Cattaneo et al., 2021).

A stronger robustness contribution appears in "Robust inference for geographic regression discontinuity designs: assessing the impact of police precincts" (Kendall et al., 2021). That paper studies GeoRDD with spatial point process outcomes and defines border-window and boundary-point estimands in terms of event intensities rather than conditional means. More importantly, it develops a resampling-based testing procedure using matched “null streets” entirely within a precinct. The method is explicitly robust to violations of traditional continuity or local-randomization assumptions, but the paper is equally explicit that it provides valid hypothesis testing of no effect under weaker assumptions, not a general solution to unbiased effect estimation under assumption failure (Kendall et al., 2021).

A further complementary development is global testing. "Global Testing in Multivariate Regression Discontinuity Designs" (Samiahulin, 3 Feb 2026) asks not whether a discontinuity exists at a specific point, but whether one exists anywhere along a multivariate boundary. The null is

$T_i=1(X_i\in A_1)$5

against

$T_i=1(X_i\in A_1)$6

The proposed procedure uses multivariate machine learning to estimate the sign of local discontinuities and then aggregates through signed distance into a univariate RD statistic. The paper presents this as a complement to local multivariate estimators, motivated by the fact that pointwise methods can exhibit severe size distortions in moderate samples because observations are sparse near any particular boundary point (Samiahulin, 3 Feb 2026).

6. Applications, limitations, and current research directions

GeoRDD has been applied to school district borders, media markets, police precincts, elections, and policy eligibility frontiers. The Bayesian GeoRDD application to New York City housing markets estimates a statistically significant discontinuity at the border between school districts 19 and 27, with calibrated $T_i=1(X_i\in A_1)$7, and interprets the estimated log-price discontinuity as roughly a 20% higher price on the more desirable side (Rischard et al., 2018). The point-process GeoRDD application to New York City precinct borders, by contrast, shows that naive border comparisons can suggest massive precinct effects on arrest rates, while robust resampling-based inference implies at most a small amount of precinct-level variation and no strong citywide evidence of large systematic precinct effects (Kendall et al., 2021).

Several limitations are persistent across the literature. Borders may coincide with multiple institutions or policy changes; sorting across borders may violate continuity; spillovers and interference are rarely modeled explicitly; and treatment geography can be irregular enough that scalar distance reductions become misleading. A misconception that GeoRDD is merely one-dimensional RD with signed distance is therefore too weak for many empirical settings. The recent boundary-discontinuity literature treats border geometry as part of the statistical problem itself, not as a preprocessing nuisance (Cattaneo et al., 8 May 2025, Cattaneo et al., 12 May 2025).

Current research directions are correspondingly structured around geometry, heterogeneity, and robustness. One line develops software and theory for pointwise and uniform estimation along continuous boundaries (Cattaneo et al., 8 May 2025, Cattaneo et al., 12 May 2025). Another line develops globally valid testing procedures for moderate samples (Samiahulin, 3 Feb 2026). Another extends RD logic to panel or pre/post border settings through difference-in-discontinuities (Butts, 2021). Bayesian work contributes flexible spatial response-surface models, model comparison for continuity versus discontinuity, and hierarchical borrowing across groups or segments (Rischard et al., 2018, Hinne et al., 2019, Tao et al., 14 Apr 2025).

The resulting view of Geographic Regression Discontinuity Design is technically specific. It is a local causal design defined on a boundary, not merely near a border; its natural estimand is usually a treatment-effect function $T_i=1(X_i\in A_1)$8 rather than a single scalar; scalar summaries depend on explicit weighting choices; and empirical credibility depends not only on local smoothness but also on border geometry, sorting, co-occurring treatments, and the choice between continuity-based, local-randomization, temporal-differencing, optimization-based, or Bayesian formulations.