Why a Cutoff Defines a Local Causal Question
Some programs use a numerical score, age, or location to determine who is eligible for an intervention. People just above and just below the threshold may be similar in many respects, but the rule changes their eligibility. Regression discontinuity (RD) designs use this threshold-based assignment to study effects near the cutoff. The first step is to state precisely what runs across the threshold, what the rule assigns, and what causal effect is of interest.
A cutoff does not, by itself, make a comparison causal. It defines an assignment mechanism and a local target. Whether the observed data identify that target depends on additional assumptions, which must be kept distinct from the definitions of the design.
The running variable may be a test score, a date of birth, or signed distance from a boundary. Its scale and direction must be explicit: if higher values are on the eligible side, \(Z_i=1(R_i\ge c)\) is a natural convention. If lower values qualify, the inequality is reversed. State what happens at exactly \(c\), too. In the convention above, a unit with \(R_i=c\) is assigned to the eligible side.
Assignment, Receipt, and Potential Outcomes
Eligibility and treatment receipt are not synonyms. A rule may assign an offer, entitlement, or access, while a person may decline it; conversely, some ineligible people may obtain a similar service elsewhere. To describe this distinction, let \(D_i(z)\) be unit \(i\)'s treatment receipt under assignment status \(z\), for \(z=0,1\). The observed receipt is \(D_i=D_i(Z_i)\). Let \(Y_i(d)\) be the outcome unit \(i\) would have under treatment status \(d\), for \(d=0,1\). Under consistency, the observed outcome is \(Y_i=Y_i(D_i)\).
These potential outcomes describe treatment receipt, not merely eligibility. The individual treatment effect is \(Y_i(1)-Y_i(0)\), the difference between the unit's outcome under treatment and its outcome without treatment. Only one of these two outcomes is observed for a given unit. In a causal analysis, the unobserved potential outcome cannot be read directly from the data.
“Sharp” and “fuzzy” describe the link between the threshold rule and treatment receipt, not whether outcomes change abruptly, whether the cutoff is measured without error, or whether a causal interpretation is automatically justified. In a fuzzy design, it is useful to distinguish the assignment indicator \(Z_i\) from the treatment indicator \(D_i\) throughout the analysis.
The Cutoff-Specific Effect
An RD design focuses on an effect for units at, or arbitrarily close to, the cutoff rather than an average effect for the entire population. In a sharp design, the natural causal target is the average treatment effect for the population at the cutoff. Formally, when the conditional distribution at the cutoff is well-defined, write:
For a continuously measured running variable, the event \(R_i=c\) may have probability zero. The conditional expectation is then understood through the population distribution at that value, or as a limit from values close to \(c\), when that limit exists. The target is local: it does not automatically describe units far from the threshold or the effect of offering the intervention to everyone.
For a fuzzy design, crossing the cutoff changes eligibility or assignment, not treatment receipt with certainty. The causal target is consequently tied to units near the cutoff whose treatment is affected by that assignment change, rather than simply being the average treatment effect for every person at \(R_i=c\). The precise assumptions and identification formula for that fuzzy-design target are separate questions. In particular, a difference in assignment status is not itself a treatment effect.
It is also important to distinguish a causal target from an observed outcome discontinuity. The observed-data contrast at the cutoff compares outcomes on the two sides:
Here the first limit approaches \(c\) from above and the second approaches it from below. This contrast is a feature of the observed outcome distribution. It becomes informative about a causal effect only under additional design assumptions; the definition alone does not establish that those assumptions hold or that \(\Delta_Y(c)=\tau(c)\).
Worked Examples
Worked Example: A Sharp Eligibility Threshold
Hypothetical setting. A training program offers a place to applicants with an entrance score of at least 70. Let \(R_i\) be the score, \(c=70\), and \(D_i=1\) if the applicant attends the program. Suppose every eligible applicant attends and no ineligible applicant attends. The rule is \(Z_i=1(R_i\ge70)\), and, by the stated setting, \(D_i=Z_i\): this is sharp assignment.
Suppose the causal question concerns applicants at the threshold, and a hypothetical description of that marginal population gives the following potential outcomes for employment (1 for employed and 0 for not employed):
| Applicant | \(Y_i(1)\) | \(Y_i(0)\) | \(Y_i(1)-Y_i(0)\) |
|---|---|---|---|
| A | 1 | 0 | 1 |
| B | 1 | 1 | 0 |
| C | 0 | 0 | 0 |
| D | 1 | 0 | 1 |
The average treatment effect in this hypothetical cutoff population is
Thus, in this constructed example, treatment increases the employment rate by 0.500, or 50.0 percentage points, on average for the four applicants in the cutoff population. This calculation illustrates the estimand; it does not show how to recover it from ordinary observed data, because each applicant's two potential outcomes cannot both be observed.
Worked Example: Eligibility With Incomplete Treatment Receipt
Hypothetical setting. A scholarship is offered to students whose score is at least 80. Let \(R_i\) be the score, \(c=80\), and \(Z_i=1(R_i\ge80)\) indicate an offer. Some eligible students decline, while some ineligible students obtain equivalent support from another source. In a hypothetical sample of 25 students just above the cutoff, 20 receive the scholarship; in a sample of 25 just below, 6 receive it.
The treatment receipt rates in the two groups are
The difference in these sample rates is \(0.800-0.240=0.560\), or 56.0 percentage points. Eligibility is associated with a higher rate of scholarship receipt, but it does not determine receipt exactly; this is fuzzy rather than sharp assignment. The calculated difference describes treatment uptake in these hypothetical groups. It does not, by itself, calculate a causal effect of the scholarship on an outcome.
For this design, one must keep three questions separate: whether the score rule changes eligibility at 80, whether crossing 80 changes treatment receipt, and what outcome effect is attributable to treatment for the population affected by that assignment change. The example supplies information about the first two, not enough information to answer the third.
Worked Example: A Geographic Threshold and an Outcome Jump
Hypothetical setting. A health service is available to residents within 5 km of a clinic. Let \(R_i\) be signed distance from the clinic boundary, with negative values inside the service area and positive values outside; let \(c=0\). Eligibility is \(Z_i=1(R_i\le0)\). Some eligible residents do not use the clinic, and some residents outside the boundary use other providers, so eligibility is not the same as treatment receipt.
Suppose a hypothetical comparison of residents near the boundary gives a mean of 14 clinic-related visits per year on the eligible side and 10 on the ineligible side. The observed difference is
This is a descriptive difference in mean outcomes across the boundary. It is not automatically the effect of receiving clinic care: eligibility and receipt differ, and the arithmetic alone does not rule out other reasons outcomes might differ across the boundary. The example also shows why defining the direction of the running variable matters. Since smaller \(R_i\) values are eligible here, the rule uses \(R_i\le0\), not \(R_i\ge0\).
Why an Outcome Jump Is Not a Definition of the Effect
The distinction between an observed outcome contrast and a causal effect follows directly from the potential-outcomes definition. Under consistency, the observed outcome can be written as
In a sharp design, \(D_i=Z_i\). On the eligible side the observed outcome is therefore \(Y_i(1)\), while on the ineligible side it is \(Y_i(0)\). But the two sides can also contain different distributions of potential outcomes, even in the absence of treatment. More explicitly, at a value \(r\) where the relevant conditional means are defined:
This expression is a mixture of outcomes under treatment and without treatment. A change in the observed mean across a cutoff could reflect the change in treatment status, a change in which kinds of units lie on each side, or both. Establishing when the observed discontinuity identifies the causal target requires assumptions about behavior and potential outcomes near the threshold. The basic setup and the causal target should therefore be stated before any estimation or causal interpretation.
Common Mistakes
- Calling eligibility treatment. An offer or entitlement is \(Z_i\); actual receipt is \(D_i\). They coincide by definition only under sharp assignment.
- Calling any threshold comparison causal. A difference in observed outcomes is descriptive until the assumptions that justify a causal interpretation are supported.
- Describing the target as a population-wide effect. The RD target concerns units at or close to \(c\); generalizing far from the cutoff requires additional justification.
- Leaving the rule ambiguous at the boundary. State which side includes \(R_i=c\) and whether higher or lower values confer eligibility.
- Using a fuzzy design's assignment change as the treatment effect. An assignment change can alter treatment uptake without directly giving the outcome effect of treatment.
Check Your Understanding
Use the definitions and distinctions in this tutorial to answer the following questions.
- A program assigns eligibility when an age measure is at least 65, but some eligible people do not enroll. Identify the running variable, cutoff, assignment indicator, and treatment receipt indicator. Is assignment sharp or fuzzy?
- In a sharp design, what causal quantity does \(\tau(c)=E[Y_i(1)-Y_i(0)\mid R_i=c]\) describe? Why is it local?
- A geographic rule gives eligible residents access to a service, but use is voluntary. Explain why eligibility and receipt must be recorded separately.
- Two hypothetical groups near a cutoff have different average outcomes. What additional distinction must be made before interpreting that difference causally?
- Why can an observed outcome discontinuity fail to equal the treatment effect at the cutoff, even when the treatment rule changes exactly at the cutoff?