Regression Discontinuity Assignment Rules and Local Effects

Why Assignment Rules Matter

A program may be assigned using a score, age, income measure, or other observed quantity: people just above a threshold qualify, while people just below it do not. This creates a distinctive setting for causal analysis. The threshold determines who is assigned treatment, but it does not by itself tell us which causal effect an analysis can recover or whether that effect describes everyone in the population.

Regression discontinuity (RD) designs begin by specifying the assignment rule and the outcomes that would occur under each treatment state. In a sharp RD design, crossing the cutoff determines treatment exactly. The central causal target is local: it concerns units at, or in the limit near, the cutoff. That target need not equal the average treatment effect across a broader population.

Running Variable, Cutoff, and Sharp Assignment

Definition: The running variable $R_i$ is the measured variable that determines eligibility or treatment assignment for unit $i$. The cutoff $c$ is the threshold value of $R_i$ at which assignment changes. In the conventional sharp RD rule, treatment status is $$ D_i = \mathbf{1}\{R_i \geq c\}, $$ where $D_i=1$ denotes treatment and $D_i=0$ denotes no treatment. The indicator equals one when its condition is true and zero otherwise.

The inequality is part of the rule: here, a unit whose running variable equals the cutoff is assigned treatment. Some applications instead specify treatment for values strictly greater than the cutoff. The assignment convention at equality must be stated, especially when the running variable is discrete and units can have $R_i=c$.

“Sharp” means that the rule assigns treatment status without exceptions: every unit on the treated side receives treatment, and every unit on the untreated side does not. If eligibility and actual treatment receipt do not coincide perfectly, this is not a sharp assignment rule. The treatment of noncompliance belongs to a different design.

Worked Example: Applying a Threshold Rule

Suppose a hypothetical training program is assigned to applicants with an assessment score of at least 70. The running variable is the score, measured in points; the cutoff is $c=70$. Apply the rule $D_i=\mathbf{1}\{R_i\geq70\}$ to applicants scoring 68, 70, and 73.

For a score of 68, the condition $68\geq70$ is false, so $D_i=0$. For a score of 70, $70\geq70$ is true, so $D_i=1$. For a score of 73, $73\geq70$ is also true, so $D_i=1$.

Thus, under this stated convention, applicants at the cutoff are assigned treatment. The calculation describes assignment, not necessarily participation in a different setting where eligible applicants can decline or ineligible applicants can obtain the program.

Potential Outcomes and Individual Effects

To define a causal effect, distinguish an outcome under treatment from the outcome the same unit would have without treatment. Let $Y_i(1)$ be unit $i$'s potential outcome under treatment and $Y_i(0)$ its potential outcome without treatment. These are defined for every unit, although only one is observed. The observed outcome is the potential outcome corresponding to the unit's assigned treatment:

$$ Y_i = D_iY_i(1) + (1-D_i)Y_i(0). $$

This equation is the consistency relation: if $D_i=1$, then $Y_i=Y_i(1)$; if $D_i=0$, then $Y_i=Y_i(0)$. It does not assert that both potential outcomes are observed. The individual treatment effect is

\tau_i = Y_i(1)-Y_i(0).

For example, if the outcome is monthly earnings measured in dollars, $\tau_i=250$ means that this unit's earnings would be $250 higher under treatment than without treatment. In observed data for a single unit, the unassigned potential outcome is missing, so the individual effect is not directly calculated by subtracting two recorded outcomes for that unit.

Worked Example: Observed Outcomes Under a Sharp Rule

Consider a hypothetical program with cutoff $c=50$ and rule $D_i=\mathbf{1}\{R_i\geq50\}$. The outcome is a test score in points. Suppose the potential outcomes below are stipulated for illustration; in an actual study, both potential outcomes for each person would not be observed.

Unit	$R_i$ (points)	$D_i$	$Y_i(0)$ (points)	$Y_i(1)$ (points)	Observed $Y_i$ (points)
A	48	0	62	66	62
B	50	1	70	75	75
C	53	1	72	76	76

For unit A, $48<50$, so $D_A=0$, and the consistency relation gives $Y_A=0(66)+(1-0)(62)=62$ points. For unit B, $50\geq50$, so $D_B=1$, and $Y_B=1(75)+0(70)=75$ points. For unit C, $53\geq50$, so $D_C=1$, and $Y_C=76$ points.

Because both potential outcomes were supplied for this constructed example, the individual effects can also be computed: $\tau_A=66-62=4$ points, $\tau_B=75-70=5$ points, and $\tau_C=76-72=4$ points. In observed data, the corresponding untreated outcomes for B and C would be unavailable; the table includes them only to make the potential-outcome definition explicit.

The Cutoff-Specific Effect

The RD question is not usually “What is the effect for every unit?” It is “What is the treatment effect for units at the assignment threshold?” When the running variable is continuous, the probability of observing a unit at exactly $R_i=c$ may be zero. In that case, the cutoff target is naturally stated as a limit of conditional effects for units whose running-variable values approach the cutoff.

Definition: If the conditional mean treatment effect has a well-defined limit at $c$, the cutoff-specific treatment effect is $$ \tau(c)=\lim_{r\to c} E[Y_i(1)-Y_i(0)\mid R_i=r]. $$ It describes the mean effect for units at the cutoff, understood as the limit from nearby running-variable values. It is not, by definition, the average effect for all units.

This definition concerns potential outcomes. The assignment rule tells us that outcomes observed just below the cutoff come from $Y_i(0)$, while outcomes observed at or above it come from $Y_i(1)$. But comparing those observed outcomes on opposite sides does not automatically reveal the cutoff-specific causal effect: the groups may differ in their potential outcomes for reasons other than treatment. Whether the observed change identifies $\tau(c)$ requires additional conditions. The assignment rule alone is not enough.

Worked Example: A Local Effect Is Not a Population Average

Suppose, in a hypothetical setting, the conditional mean treatment effect at the cutoff $c=100$ is 5 points. Assume effects are 2 points for 80% of a broader population far from the cutoff and 9 points for the remaining 20%. Calculate the population average treatment effect and compare it with the cutoff effect.

The population average treatment effect is the mean of individual effects over the population. With the stated proportions and effects, it is

\text{ATE}=0.80(2)+0.20(9)=1.6+1.8=3.4\text{ points}.

The cutoff-specific effect is 5 points, whereas the broader population average is 3.4 points. Both quantities can be valid causal targets, but they answer different questions. The 5-point effect concerns units at the threshold; the 3.4-point effect averages across the stated population. The difference arises because treatment effects vary across units and the population gives different weight to those effects than the local cutoff target does.

Why an Observed Jump Is Not Automatically Causal

It is useful to see algebraically what the sharp rule implies and what it does not. Define $m_1(r)=E[Y_i(1)\mid R_i=r]$ and $m_0(r)=E[Y_i(0)\mid R_i=r]$, wherever these conditional means are defined. Under the sharp rule, for values just above $c$, the observed conditional mean is $m_1(r)$; just below $c$, it is $m_0(r)$. Therefore, the difference between the right and left limits of observed outcome means is

\lim_{r\downarrow c} E[Y_i\mid R_i=r] - \lim_{r\uparrow c} E[Y_i\mid R_i=r] = \lim_{r\downarrow c}m_1(r) - \lim_{r\uparrow c}m_0(r),

when the indicated limits exist. The expression uses a treated potential-outcome mean on the right and an untreated potential-outcome mean on the left. It equals the cutoff-specific effect only under additional conditions connecting the potential-outcome means across the cutoff. For instance, if both potential-outcome conditional means vary continuously at $c$, this contrast equals $m_1(c)-m_0(c)$. That condition is not implied merely by having a deterministic assignment rule.

This distinction matters because an outcome can change at the threshold even when treatment has no effect. A separate abrupt change in the untreated outcome at the same score could also contribute to the observed jump. The assignment rule defines which potential outcome is observed on each side; it does not, by itself, rule out such changes.

Worked Example: Separating a Baseline Change From a Treatment Effect

Consider a hypothetical sharp RD design with cutoff $c=10$. Suppose the conditional mean untreated outcome is 100 units just below the cutoff and 110 units just above it. Suppose the treatment effect at the cutoff is 4 units, so the treated potential-outcome mean just above the cutoff is $110+4=114$ units.

The observed mean just below the cutoff is 100 units, because those units are untreated. The observed mean just above it is 114 units, because those units are treated. The observed jump is therefore

114-100=14\text{ units}.

But the stipulated treatment effect is only 4 units. The remaining 10 units of the observed jump come from the change in the untreated potential-outcome mean, $110-100=10$. Thus $14=10+4$: the observed change combines a baseline change and a treatment effect. This constructed example shows why interpreting every observed threshold jump as the causal effect requires an additional identifying argument.

Scope, Interpretation, and Common Pitfalls

The cutoff-specific effect is most directly informative about units whose running-variable values are close to the threshold. It may be highly relevant when a policy decision concerns applicants near an eligibility boundary. It does not automatically describe people far from that boundary, other eligibility groups, or the effect of expanding the program to a different population.

To move from a local effect to a broader population effect requires further justification, such as a defensible assumption that effects are sufficiently similar across the relevant groups, or additional evidence that identifies effects away from the cutoff. Without such justification, report the RD target as local rather than presenting it as a universal program effect.

Confusing eligibility with receipt. A sharp rule means the assignment rule determines treatment exactly. If eligibility does not guarantee receipt, assignment and treatment receipt differ.
Leaving the equality case unspecified. A rule using $R_i\geq c$ assigns a unit exactly at $c$ to treatment. State the convention rather than assuming it is obvious.
Calling the cutoff effect the ATE. The cutoff target concerns units at the threshold; the ATE averages effects over a specified population.
Treating an observed jump as self-evidently causal. The assignment rule produces a switch in observed potential outcomes, but it does not alone establish that no other potential-outcome change occurs at the cutoff.

Key takeaway: A sharp RD design is defined by a deterministic threshold assignment rule. Potential outcomes define the causal effect, and the central RD target is local to the cutoff. The rule alone neither makes that local effect equal to a broader population average nor guarantees that an observed outcome jump identifies it.

Check Your Understanding

Answer each question by referring to the assignment rule and causal target, not just to the observed outcomes.

A program treats units when $R_i\geq 25$. What are the running variable and cutoff, and how is a unit with $R_i=25$ assigned?
In a sharp RD design, what does it mean for treatment assignment to be sharp? How does this differ from merely using a threshold for eligibility?
Explain why only one of $Y_i(1)$ and $Y_i(0)$ is observed for a given unit, and write the consistency relation for $Y_i$.
What population does the cutoff-specific effect describe, and why need it not equal the population average treatment effect?
Why does a deterministic assignment rule alone not prove that an observed outcome jump is the causal effect at the cutoff?