Geographic Regression Discontinuity Designs

Why a Geographic Boundary Can Identify an Effect

A policy may apply on one side of a geographic border and not the other. A comparison of all residents in the two jurisdictions is unlikely to be causal: the places may differ in income, services, history, or population. A geographic regression discontinuity instead focuses on units close to the border. Its central claim is that, at a given point along the border, locations just across it would otherwise have had similar outcomes, absent the policy difference.

The design is local. It does not establish that the two jurisdictions are comparable everywhere, nor does it automatically identify an average effect for either jurisdiction. It estimates a contrast at the boundary, or a weighted summary of such contrasts over the boundary locations represented in the analysis.

Definition: A geographic regression discontinuity design uses a geographic boundary that changes treatment eligibility or exposure. For unit \(i\), let \(Z_i\) indicate which side of the boundary the unit occupies, let \(D_i\) denote treatment received, and let \(Y_i\) denote the outcome. Define signed distance \(R_i\) as distance to the boundary, negative on the \(Z_i=0\) side and positive on the \(Z_i=1\) side, so the border is \(R_i=0\). Let \(S_i\) identify the unit's position along the boundary, such as its nearest border segment or projected border coordinate.

For a straight boundary, signed distance can be the perpendicular distance to the line. For a curved boundary, it is commonly the shortest signed distance to the boundary, together with the point or segment \(S_i\) where that distance is measured. The side convention is arbitrary, but it must be used consistently. Units must be assigned according to the relevant geographic rule; distance to a convenient line that does not match the actual policy boundary is not a valid running variable.

The coordinate \(R_i\) determines how close a unit is to the assignment boundary. The coordinate \(S_i\) matters because the border is not one homogeneous location: housing, terrain, roads, and local economies may vary along it. The comparison should therefore allow outcome levels and, where appropriate, treatment effects to vary by border location. A single distance-only fit can be misleading if it confounds a change across the border with a change along it.

Identification at the Border

Let \(Y_i(1)\) and \(Y_i(0)\) be unit \(i\)'s potential outcomes under treatment and no treatment. In a sharp geographic design, treatment is determined by side: \(D_i=Z_i\). The causal effect of interest at a particular boundary location \(s\) is the difference between the limiting mean outcomes on the two sides as distance approaches zero.

Identification conditions for a sharp geographic design: (1) Treatment follows the geographic assignment rule at the boundary, so \(D_i=Z_i\). (2) At each relevant boundary location \(s\), the conditional means of both potential outcomes vary continuously with signed distance through zero. (3) Units cannot precisely sort their locations or recorded coordinates to choose a side in a way that also changes potential outcomes discontinuously. (4) No other cause of the outcome has a discontinuity at the same boundary and location. There must also be observations with useful support on both sides near the boundary.

These conditions permit background characteristics and potential outcomes to vary substantially over space. They rule out a jump at the border in the untreated potential outcome or treated potential outcome, not all geographic variation. Under the conditions, the observed outcome on each side converges to the corresponding potential-outcome mean at \(R_i=0\). Therefore, the jump in observed outcomes is the local treatment effect:

\tau(s) = \lim_{r\to 0^+} E[Y_i\mid R_i=r,S_i=s] - \lim_{r\to 0^-} E[Y_i\mid R_i=r,S_i=s].

To see why, on the positive side in a sharp design, \(Y_i=Y_i(1)\), while on the negative side \(Y_i=Y_i(0)\). Continuity gives the two limits \(E[Y_i(1)\mid R_i=0,S_i=s]\) and \(E[Y_i(0)\mid R_i=0,S_i=s]\), respectively. Their difference is the conditional average treatment effect at that boundary location. This identifies an effect for locations near the border, not necessarily for residents far from it.

Estimating and Summarizing Local Contrasts

In practice, analysts select a bandwidth \(h\) and use observations with \(|R_i|\leq h\), fitting local regressions that allow separate trends in distance on each side. The basic idea is to estimate each side's outcome at \(R_i=0\), rather than compare raw averages of all units inside the bandwidth. A typical local linear specification includes side-specific intercepts and slopes in \(R_i\), and accounts for boundary position using segment indicators, local geographic controls, or suitably flexible functions of \(S_i\). The coefficient on \(Z_i\), under this parameterization, estimates the jump at the border for the specified model and weighting scheme.

If effects differ along the border, report segment-specific contrasts or define a transparent weighted average, such as an average over prespecified border segments. A pooled estimate is not automatically the simple average of the location-specific effects: its implicit weights can depend on the sampling density, bandwidth, kernel weights, and model specification. State what boundary locations the estimate represents.

Bandwidth and support diagnostics are especially important here. Map the observations and the boundary; inspect how many observations lie on each side and in each relevant segment; and check whether the fitted values at zero rely on extrapolation. Compare reasonable bandwidths and specifications. The earlier material on robust bias-corrected regression discontinuity inference is relevant to uncertainty from local polynomial estimation; geographic dependence additionally requires standard errors that account for spatially correlated outcomes. Treating nearby units as independent can substantially understate uncertainty.

Worked Example: A Sharp Effect Across Two Border Segments

Hypothetical data. A municipal program begins at a county border, and every eligible address receives the program. Analysts estimate local linear outcome limits at the border separately for two prespecified segments. The outcome is monthly clinic visits per 1,000 residents. For segment A, the estimated limit is 12.4 visits on the treated side and 10.1 on the untreated side. For segment B, the corresponding limits are 11.8 and 9.9 visits.

The estimated local jumps are:

\hat{\tau}(A)=12.4-10.1=2.3 \quad\text{visits per 1,000 residents per month},

and

\hat{\tau}(B)=11.8-9.9=1.9 \quad\text{visits per 1,000 residents per month}.

If the target summary gives the two segments equal weight, its estimate is \((2.3+1.9)/2=2.1\) visits per 1,000 residents per month. This is an equal-segment average, not necessarily the effect for all residents of either county.

The calculation is causal only if treatment actually follows the border rule, potential-outcome means are continuous through the border within each segment, no other relevant change occurs at that border, and the local fits have support on both sides. These conditions are not established by the arithmetic. In particular, if segment A contains a hospital on only one side, the segment A contrast could reflect that discontinuity rather than the program.

Fuzzy Geographic Assignment

Sometimes crossing the border changes the probability of treatment but does not determine treatment perfectly. For example, eligibility may increase program take-up, while some eligible units do not participate and some ineligible units obtain the service through another route. This is a fuzzy geographic design. Let \(\Delta_Y(s)\) be the outcome limit from the positive side minus the negative-side limit, and let \(\Delta_D(s)\) be the corresponding jump in treatment probability.

Fuzzy geographic RD result: If the border changes treatment take-up, potential outcomes and potential treatment status satisfy the required continuity conditions at the border, the border affects the outcome only through treatment, and treatment is monotone in the border assignment (no units are induced to do the opposite of the assignment's direction), then, where \(\Delta_D(s)\ne 0\), the ratio \(\Delta_Y(s)/\Delta_D(s)\) identifies a local average treatment effect for units whose treatment is changed by the border assignment. The exclusion and monotonicity conditions are substantive assumptions, not consequences of a strong first stage.

The ratio follows from the same local comparison. Under continuity, the outcome jump is the assignment-induced change in mean outcomes at the border. Under exclusion, assignment affects outcomes only by changing treatment. Under monotonicity, the treatment-probability jump is the proportion of local compliers, and the outcome jump is that proportion multiplied by their average treatment effect. Dividing the outcome jump by the treatment jump recovers that complier effect. If the first-stage jump is very small, the ratio is unstable and the design provides weak evidence about the treatment effect.

Worked Example: A Fuzzy Border and Its Wald Estimate

Hypothetical data. On the policy side of a border, the estimated treatment rate at the boundary is 0.72; on the comparison side it is 0.22. The outcome is a standardized health score, and its estimated boundary limits are 5.6 on the policy side and 4.7 on the comparison side.

The first-stage discontinuity is \(\hat{\Delta}_D=0.72-0.22=0.50\), or a 50-percentage-point increase in treatment. The outcome discontinuity is \(\hat{\Delta}_Y=5.6-4.7=0.9\) score points. The fuzzy RD Wald estimate is:

\hat{\tau}_{\mathrm{FRD}} = \frac{\hat{\Delta}_Y}{\hat{\Delta}_D} = \frac{0.9\ \text{score points}}{0.50} = 1.8\ \text{score points per treatment}.

Under the fuzzy-design assumptions, this estimates the local average effect for compliers at the boundary: units whose treatment status is changed by being on the policy side. It is not the average effect for every unit near the border. A 50-percentage-point first stage is substantial in this illustration, but its standard error and uncertainty still need to be assessed. The design also requires exclusion: the side assignment must not affect the score through other border-specific channels.

Diagnostics and Geographic Failure Modes

A credible analysis combines design knowledge with empirical checks. No single diagnostic proves continuity, but several can reveal reasons the comparison is implausible.

Map assignment and support. Confirm that the observed border matches the actual eligibility rule. Map locations, treatment, and outcomes; look for sparse areas, gaps, or one-sided observations near particular segments.
Check for sorting and measurement problems. Addresses may be recorded imprecisely, geocoding may place units on the wrong side, or people and firms may choose locations in response to the policy. These features can make observed distance or side a poor proxy for assignment. The earlier material on sorting at the threshold and heaping and measurement error near the cutoff applies in geographic form.
Examine predetermined characteristics. Estimate their boundary jumps using the same geographic structure. A discontinuity in a characteristic fixed before the policy is evidence against local comparability, though no detected jump does not prove the assumption.
Look for competing border changes. Jurisdiction borders often coincide with differences in taxes, schools, policing, service access, or data collection. A coincident change violates the interpretation of the outcome jump as the effect of the policy of interest unless it is part of the treatment being defined.
Assess heterogeneity along the border. A pooled jump can conceal segments with very different effects or even different signs. Compare segment estimates and ensure the reported summary reflects a defensible target population.
Check bandwidth and inference choices. Results that change sharply with small bandwidth changes, polynomial order, or removal of a segment may indicate inadequate support or model dependence. Account for spatial correlation and limited numbers of independent geographic areas.

Worked Example: A Covariate Discontinuity as a Warning

Hypothetical data. In a border analysis of a new health service, the outcome jump is estimated to be 0.8 score points. As a diagnostic, analysts estimate the boundary jump in a predetermined neighborhood-income measure. Its local limit is 52.0 thousand dollars on the policy side and 46.5 thousand dollars on the other side.

The covariate jump is \(52.0-46.5=5.5\) thousand dollars. Because neighborhood income is predetermined, this visible difference raises concern that the areas immediately across the border are not comparable, or that another geographic difference coincides with the border. The diagnostic does not prove that the estimated 0.8-point outcome jump is biased by exactly 5.5 units; the variables have different units, and the covariate result alone does not quantify confounding. Analysts should investigate the geography and institutional history, examine other predetermined characteristics, and assess whether adjustment changes the estimate. Covariate adjustment cannot make a fundamentally discontinuous counterfactual outcome smooth by itself.

How the Design Differs From Alternatives

A simple border comparison averages outcomes on the two sides, often over wide areas. It may be useful descriptively, but it combines the policy difference with broad differences between jurisdictions. Geographic RD narrows the comparison to locations near the border and estimates the side-specific limits, relying on continuity rather than assuming that distant areas are comparable.

Matching or regression adjustment can balance observed covariates, but by themselves they do not remove bias from unobserved geographic differences. A geographic RD can be more credible when the boundary creates a sharp assignment change and potential outcomes would otherwise vary smoothly across it. Conversely, if another determinant jumps at the same border, or there is poor local support, a more elaborate local regression does not rescue identification. An instrumental-variable interpretation in the fuzzy case additionally depends on exclusion and monotonicity.

The method's limitations follow from its local design. Estimates may not generalize far from the border; effects may vary by segment; spatial dependence complicates standard errors; and a policy border may bundle several institutional differences. The strongest analysis makes the target boundary locations explicit, explains the assignment mechanism, reports support and sensitivity diagnostics, and treats the identifying assumptions as arguments to assess rather than facts guaranteed by the estimator.

Common pitfall: “The two sides are geographically close, so they must be equivalent except for treatment.” Proximity alone does not establish continuity. A river, major road, school-district line, or jurisdictional service change can create a discontinuity in potential outcomes. The correction is to evaluate the local assignment rule, competing border changes, support, sorting, and predetermined characteristics at the relevant boundary locations.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In a sharp geographic design, what does the outcome jump at the border estimate, and what conditions give it that interpretation?
A fuzzy design has an outcome jump of 1.2 points and a treatment-probability jump of 0.40. What is the Wald estimate, and to whom does it apply under the identifying assumptions?
Why can a single pooled border estimate be difficult to interpret when treatment effects differ along the boundary?
A predetermined covariate has a clear jump at the border. Does this prove the treatment estimate is biased by a known amount? Explain.
Why might ordinary standard errors be inappropriate when many nearby geographic units are observed?

Solutions. (1) It estimates the local average treatment effect at the boundary, under treatment assignment by side, continuity of both potential-outcome means, no precise sorting that creates a discontinuity, no competing border discontinuity, and adequate support. (2) The estimate is \(1.2/0.40=3.0\) outcome points per treatment. Under the fuzzy-design assumptions, it applies to local compliers whose treatment changes with border assignment. (3) The pooled estimate's implicit weights may reflect where observations are concentrated and how the model is fitted, rather than a clear average over boundary locations. (4) No. The jump is a warning about local comparability, but it does not specify the amount of outcome bias; investigate the cause and other evidence. (5) Nearby outcomes may be spatially correlated, so treating observations as independent can understate uncertainty.