Why a Cutoff Effect Is Local
A regression-discontinuity (RD) design can identify a causal effect for units near an assignment threshold, under its design assumptions. But a policy decision may concern people far from that threshold, or a population with a different mix of characteristics. The central question is therefore not only whether the cutoff estimate is credible, but also which people it describes and what additional assumptions would extend it.
Let \(X\) be the running variable, \(c\) the cutoff, \(D\) treatment receipt, and \(Y\) the outcome. Write \(Y(1)\) and \(Y(0)\) for the outcomes a unit would have under treatment and no treatment. For a sharp RD, treatment status changes deterministically at \(c\). The causal effect identified at the threshold is the limit of the conditional average treatment effect as \(X\) approaches \(c\), provided the relevant potential-outcome means are continuous there and units cannot precisely sort around the cutoff.
The notation \(x\downarrow c\) denotes approaching from above and \(x\uparrow c\) approaching from below. The equality to the treatment effect at \(X=c\) relies on the RD assumptions, including continuity of the potential-outcome regression functions at the cutoff. It does not imply that the effect is the same at \(X=c+20\), or in a population with a different distribution of age, baseline risk, or other effect modifiers.
In a fuzzy RD, crossing the threshold changes the probability of treatment rather than determining treatment for everyone. Under the fuzzy-RD identification assumptions, including a nonzero first-stage jump, exclusion at the threshold, monotonicity, and continuity, the local Wald ratio identifies an average treatment effect for compliers at the cutoff. That is a still more specific population: units whose treatment receipt is changed by eligibility near \(c\). Neither sharp nor fuzzy RD automatically identifies an average effect for people far from the threshold.
Transporting an Effect Requires More Than a Precise Estimate
For a target population \(T\), a common decision quantity is the average treatment effect \(E[Y(1)-Y(0)\mid T]\). This differs from the RD estimand whenever the target includes people unlike those at the threshold or treatment effects vary across people. A very precise estimate of \(\tau_{\mathrm{RD}}(c)\) can still be a poor guide to the target effect.
One useful strategy is to identify measured characteristics \(Z\) that modify treatment effects. Suppose the effect conditional on \(Z=z\) is the same in the cutoff population and target population, and suppose the effect can be estimated for every \(z\) represented in the target. Then the target average can be obtained by averaging those conditional effects over the target distribution of \(Z\):
This standardization requires substantive conditions. First, conditional effect transportability must hold: after conditioning on the chosen \(Z\), the treatment effect does not differ systematically between the cutoff and target populations. Second, there must be overlap: each relevant target value of \(Z\) must be represented near the cutoff well enough to estimate its conditional effect. Third, the target distribution \(F_T\) must be measured or specified correctly. If \(Z\) does not capture important effect modifiers, or target values fall outside source support, the weighted estimate is not justified by this formula.
Even with these conditions, transportability of effects is not the same as transportability of baseline outcomes. The target may have a different untreated outcome level but the same treatment effect, or the same baseline level but a different treatment effect. The analyst must specify which causal quantity is being transported and which features are assumed to make it comparable.
Worked Example: Reweighting Conditional Effects to a Target Population
Consider a hypothetical sharp RD evaluating a support program. The outcome is a standardized test score, measured in points, and the cutoff effect is estimated separately for two age groups. Suppose the estimated conditional effects at the cutoff are 10 points for younger students and 4 points for older students. A district considering the program has a target population that is 30% younger and 70% older.
If the conditional effects apply equally to the target students in each age group, and both groups are represented near the cutoff, the target average effect is the weighted mean:
The arithmetic gives a transported estimate of 5.8 test-score points. The calculation is not itself evidence that transportability holds. It depends on the assumptions that age groups capture relevant effect differences, that the age-specific cutoff effects also apply to target students in those groups, and that there is adequate source support for both groups. If older target students differ in important ways from older students near the cutoff, the 4-point estimate may not apply to them.
The unweighted average, \((10+4)/2=7\) points, would describe a population split evenly between the two groups, not this target district. Weighting must reflect the target population rather than the source sample composition.
Effect Variation Across the Running Variable
A frequent target is not a distinct set of demographic groups but people farther from the threshold on the running variable itself. A cutoff RD estimates a limit at \(c\); it does not reveal the treatment-effect curve throughout the range of \(X\). Extrapolating farther from \(c\) therefore requires additional information or restrictions, such as a credible model for how effects change with \(X\), or separate causal evidence at other values.
For example, suppose the causal effect is modeled as linear in the running variable over a stated range: \(\tau(x)=\beta_0+\beta_1(x-c)\). The cutoff effect is \(\beta_0\), while the effect at \(x=c+20\) is \(\beta_0+20\beta_1\). This calculation is valid only if the linear effect model is appropriate across that range and its parameters are identified with data or defensible external evidence. A smooth fitted outcome curve on either side of the cutoff does not, by itself, establish that the treatment effect follows this model.
Worked Example: A Running-Variable Extrapolation
Suppose a hypothetical RD concerns a running score with cutoff \(c=50\). The estimated cutoff effect is 12 fewer hospital days, and an analysis using additional design information proposes the effect model \(\tau(x)=12-0.20(x-50)\), where effects are in hospital days. What does this model imply at score 70?
Substitute \(x=70\):
The model implies an effect of 8 fewer hospital days at score 70. This is not an estimate supplied by the cutoff jump alone: it also relies on the assumed slope of \(-0.20\) hospital days per score point and on applying that slope 20 points away. If the observed data only support a narrow neighborhood of 50, the value at 70 is an extrapolation and could be highly model-dependent. A responsible report would show how the target estimate changes under plausible alternative slopes and would avoid presenting 8 days as an assumption-free RD result.
Fuzzy RD and the Population of Compliers
External validity needs special care for fuzzy RD. The fuzzy-RD effect at the cutoff is an average effect for local compliers, not necessarily for all people near the threshold and certainly not automatically for all people in a policy target population. The complier group is defined by how treatment receipt responds to threshold eligibility. Its characteristics and treatment effects can differ from those of always-takers, never-takers, or people whose treatment status is unaffected by the threshold.
A change in the share of compliers across populations does not alone determine whether the treatment effect transports. What matters is whether the treatment effects for the relevant target units are represented by the local-complier effect, or whether effects can be estimated and appropriately averaged for target-relevant groups. A local effect for one complier population is not a general treatment effect merely because the intervention is the same.
Worked Example: Why a Fuzzy-RD Effect Does Not Automatically Generalize
In a hypothetical fuzzy RD, eligibility raises treatment receipt from 0.20 to 0.60 at the cutoff, a first-stage jump of \(0.40\). The outcome mean has a cutoff jump of 3 units. Under the fuzzy-RD assumptions, the Wald estimate is:
Thus the estimated effect is 7.5 outcome units for compliers at the cutoff. Suppose a decision-maker asks for the average effect among all eligible people far above the cutoff. The 7.5-unit estimate does not answer that question unless additional assumptions link those target people's effects to the local-complier effects. For instance, target patients may be more likely to comply because their circumstances differ, and the treatment may work differently for them. The first-stage jump and Wald arithmetic establish neither equal effects across compliance types nor equal effects across locations in \(X\).
A useful next step is to describe who the local compliers are, compare their measured characteristics with the target population, and assess whether the proposed transport assumptions are plausible. If the target includes types or covariate profiles absent among local compliers, the local estimate cannot identify their treatment effects without added assumptions or evidence.
Diagnostics, Alternatives, and Limits
External-validity assessment is strongest when it makes the source population, target population, effect modifiers, and extrapolation assumptions explicit. Several checks can reveal when transport is especially fragile:
- Compare covariate distributions. Tabulate or plot key baseline characteristics for units near the cutoff and the target population. Large differences suggest that a single local effect may not represent the target.
- Check overlap. Identify target subgroups that are rare or absent near the cutoff. Weighting cannot recover effects for unsupported groups merely by assigning them large weights.
- Assess effect heterogeneity. Estimate cutoff effects across prespecified, substantively relevant subgroups when data permit. Wide uncertainty may make target standardization unstable even if overlap is present.
- Vary extrapolation assumptions. Compare plausible effect models or slopes and report how target estimates change. Sensitivity is evidence that the target conclusion depends on modeling choices.
- Separate identification from precision. Narrow confidence intervals for the cutoff effect address sampling uncertainty in that local estimand; they do not account for uncertainty about transportability.
Alternatives may be more informative when the target is distant from the cutoff. Additional RD cutoffs can provide local effects at multiple points, if each cutoff is credible and the estimates are comparable. A randomized study or another design with causal support in the target population may directly address the target effect. A model-based extrapolation can also be useful, but its assumptions should be stated and examined rather than treated as consequences of the RD design. These approaches answer different questions: a cutoff design prioritizes local causal identification, whereas a target-population study prioritizes relevance to that population.
The main limitation is fundamental: data concentrated near one cutoff contain limited direct information about effects far away. If treatment effects vary with \(X\), with unmeasured characteristics, or with compliance type, the target effect may not be identified from the local RD alone. More elaborate adjustment does not remove this limitation unless it adds credible identifying information. Report the local estimand as local, label any transported quantity as assumption-dependent, and specify which evidence supports the extension.
A Practical Assessment
Specify the population and causal effect relevant to the decision, including its range of the running variable when applicable.
State whether it is a sharp-RD effect at the cutoff or, for fuzzy RD, a local-complier effect at the cutoff.
Examine observed effect modifiers and determine whether the target is represented near the cutoff.
Describe conditional effect transportability, overlap, or the functional form used for extrapolation, and explain why each is plausible.
Show how the target estimate changes under reasonable alternatives and distinguish this uncertainty from the standard error of the local estimate.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A sharp RD estimates an effect at \(X=c\). Does this alone identify the average treatment effect for people with \(X\) at least 15 units above \(c\)? Why or why not?
- Conditional cutoff effects are 6 units in group A and 2 units in group B. A target population is 25% group A and 75% group B. Under valid conditional effect transportability and overlap, what is the target average effect?
- In a fuzzy RD, the outcome jump is 2.4 units and the treatment-probability jump is 0.30. Compute the local-complier Wald estimate and state the population it describes.
- What does it mean if a target subgroup has no comparable observations near the cutoff?
Solutions. (1) No. The RD identifies a local effect at the cutoff under its assumptions; effects may vary with \(X\), so the farther-away target requires additional evidence or extrapolation assumptions. (2) The weighted effect is \((0.25)(6)+(0.75)(2)=1.5+1.5=3.0\) units, provided the stated transportability and overlap conditions hold. (3) The Wald estimate is \(2.4/0.30=8.0\) outcome units for local compliers at the cutoff, under the fuzzy-RD assumptions; it is not automatically the average effect for everyone in a target population. (4) There is a lack of overlap: the cutoff data do not support estimating that subgroup's effect directly, and weighting cannot create the missing causal information.