Continuity-Based Identification in Sharp Regression Discontinuity

Why a Jump Can Identify a Causal Effect

In a sharp regression discontinuity (RD) design, treatment is assigned deterministically according to whether a running variable crosses a cutoff. Units just below and just above the threshold can therefore differ in treatment while being close in the running variable. The central identification question is whether the change in their expected outcomes at the threshold reflects treatment, rather than some other feature that also changes there.

The key idea is to compare the one-sided limits of the conditional outcome mean at the cutoff. If, in the absence of treatment, expected outcomes would vary smoothly through the threshold, then a jump in the observed conditional mean can be attributed to the treatment assignment. This argument identifies an effect at the cutoff; it does not, by itself, identify an average effect for units far from the threshold.

Setup and Identification Assumptions

For unit $i$, let $X_i$ be the running variable, $c$ the cutoff, $D_i$ the treatment indicator, and $Y_i$ the observed outcome. A sharp assignment rule is

D_i=\mathbf{1}\{X_i\geq c\},

where $\mathbf{1}\{\cdot\}$ equals 1 when the condition inside braces holds and 0 otherwise. Thus units with $X_i<c$ are untreated, and units with $X_i\geq c$ are treated.

Write $Y_i(1)$ for unit $i$'s potential outcome under treatment and $Y_i(0)$ for its potential outcome without treatment. Consistency means that the observed outcome is the potential outcome corresponding to the treatment actually received:

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

As usual, this potential-outcome setup assumes a well-defined treatment and no interference between units: one unit's outcome does not depend on other units' treatment assignments. Define the potential-outcome conditional mean functions

\mu_d(x)=E[Y_i(d)\mid X_i=x], \qquad d\in\{0,1\}.

The observed conditional mean is $m(x)=E[Y_i\mid X_i=x]$. The identification argument uses the following assumptions.

Sharp assignment: treatment follows the stated cutoff rule, with no noncompliance that makes treatment differ from $D_i$.
Support on both sides: units can have running-variable values arbitrarily close to $c$ from below and from above. This makes both one-sided limits meaningful from the population distribution.
Continuity of potential-outcome means: $\mu_0(x)$ and $\mu_1(x)$ each have a continuous extension to $c$. In particular, their limits from either side equal their values at $c$.

Continuity here is a restriction on conditional means, not a claim that every unit's potential outcome is a continuous function of $X_i$. It is also not a claim that $\mu_0(c)=\mu_1(c)$; treatment may create a jump between the two potential-outcome means.

Definition: The sharp RD estimand at cutoff $c$ is the right-hand limit of the observed conditional outcome mean minus its left-hand limit: $$ \tau_{\mathrm{RD}}(c) = \lim_{x\downarrow c}m(x)-\lim_{x\uparrow c}m(x). $$ Here $x\downarrow c$ denotes approaching $c$ from above, and $x\uparrow c$ denotes approaching from below. Under the assumptions stated above, this jump identifies the average treatment effect conditional on the running variable being at the cutoff, $\mu_1(c)-\mu_0(c)$.

Why the Limits Identify an Effect at the Cutoff

The argument follows from what is observed on each side of the assignment threshold. Below $c$, all units are untreated, so their observed conditional mean is the untreated potential-outcome mean. At and above $c$, all units are treated, so their observed conditional mean is the treated potential-outcome mean. Taking limits brings those two means to the same running-variable value, $c$.

Express the observed mean below the cutoff.
When $x<c$, sharp assignment gives $D_i=0$. Consistency therefore implies $m(x)=\mu_0(x)$.

Express the observed mean above the cutoff.
When $x\geq c$, sharp assignment gives $D_i=1$. Consistency therefore implies $m(x)=\mu_1(x)$ on that side.

Take the one-sided limits.
Continuity of the potential-outcome conditional means gives a left limit of $\mu_0(c)$ and a right limit of $\mu_1(c)$.

In equations, for $x<c$, $m(x)=\mu_0(x)$, and for $x>c$, $m(x)=\mu_1(x)$. The value assigned to a unit exactly at $c$ does not change either one-sided limit. Consequently,

\begin{aligned} \lim_{x\downarrow c}m(x)-\lim_{x\uparrow c}m(x) &=\lim_{x\downarrow c}\mu_1(x)-\lim_{x\uparrow c}\mu_0(x)\\ &=\mu_1(c)-\mu_0(c)\\ &=E[Y_i(1)-Y_i(0)\mid X_i=c]. \end{aligned}

The second equality uses continuity of the two conditional mean functions at $c$; the final equality follows from linearity of conditional expectation. This is the identification result. It says that the jump identifies the conditional average treatment effect at the cutoff. Treatment effects may vary across units: the result does not require $Y_i(1)-Y_i(0)$ to be constant.

The conditional expectation at an exact value of a continuous running variable should be understood through the continuous version of the conditional mean, or equivalently through its limit at $c$. The support assumption matters because, without observations arbitrarily close on both sides, these limits cannot be learned from the population distribution near the cutoff.

Worked Examples

Worked Example: A Constant Jump in a Score Outcome

Suppose a hypothetical program is assigned to applicants with a score of at least 70. The outcome is a later assessment score, measured in points. Let $X_i$ be the application score, $c=70$, and suppose the population conditional means of the potential outcomes are

\mu_0(x)=50+2(x-70), \qquad \mu_1(x)=62+2(x-70).

These functions are continuous at 70, and assume the running variable has support arbitrarily close to 70 on both sides. The assignment rule is sharp, and no other policy or measurement change occurs at the threshold. Thus the identification assumptions apply.

Just below the cutoff, the observed conditional mean is $\mu_0(x)$, so its left-hand limit is

\lim_{x\uparrow 70}m(x)=\mu_0(70)=50+2(70-70)=50\text{ points}.

Just above the cutoff, the observed conditional mean is $\mu_1(x)$, so its right-hand limit is

\lim_{x\downarrow 70}m(x)=\mu_1(70)=62+2(70-70)=62\text{ points}.

The jump is $62-50=12$ points. Under the stated assumptions, the identified average treatment effect for applicants at the cutoff is 12 assessment-score points. This conclusion concerns the cutoff, not applicants with much lower or higher scores.

Worked Example: Different Slopes on the Two Sides

Consider a hypothetical grant assigned to firms with an eligibility index of at least 40. Let the outcome be annual investment measured in thousands of dollars, and suppose

\mu_0(x)=80+1.5(x-40)-0.04(x-40)^2, $$ $$ \mu_1(x)=87+0.5(x-40)-0.02(x-40)^2.

Assume both potential-outcome mean functions are continuous at 40, firms are observed arbitrarily close to 40 on each side, and treatment follows the sharp rule with no other change at the threshold. The slopes and curvature differ, but continuity does not require the two functions to have matching slopes or to be linear.

At the cutoff, $x-40=0$. The left-hand limit of the observed conditional mean is therefore

\lim_{x\uparrow 40}m(x)=\mu_0(40) =80+1.5(0)-0.04(0)^2 =80\text{ thousand dollars}.

The right-hand limit is

\lim_{x\downarrow 40}m(x)=\mu_1(40) =87+0.5(0)-0.02(0)^2 =87\text{ thousand dollars}.

Hence $\tau_{\mathrm{RD}}(40)=87-80=7$ thousand dollars, or $7,000 per firm at the cutoff. The different slopes do not enter the limit difference because the terms involving $x-40$ vanish as $x$ approaches 40. Away from the cutoff, the treatment effect implied by these functions can differ from $7,000.

Worked Example: A Concurrent Threshold Change Breaks the Interpretation

Suppose a hypothetical training program is assigned at $c=0$, and the outcome is monthly earnings in hundreds of dollars. Without either the program or a separate policy change, assume the untreated potential-outcome mean would be

\mu_0(x)=200+3x.

Suppose the program adds 5 hundreds of dollars to the conditional mean at every value of $x$. But a separate reporting policy also adds 8 hundreds of dollars to recorded earnings for units with $x\geq 0$, whether or not they receive the program. For the units near the cutoff, the observed conditional mean is then $200+3x$ below zero and $200+3x+5+8$ at or above zero.

The left-hand limit is

\lim_{x\uparrow 0}m(x)=200+3(0)=200.

The right-hand limit and the observed jump are

\lim_{x\downarrow 0}m(x)=200+3(0)+5+8=213, \qquad 213-200=13\text{ hundreds of dollars}.

The treatment effect is only $5$ hundreds of dollars; the remaining $8$ hundreds come from the separate threshold change. The jump is 13 hundreds of dollars, or $1,300, so it does not identify the program effect. In this example the potential-outcome mean under no treatment is not continuous through the cutoff once the reporting policy is included. The continuity assumption fails, and the observed jump combines two changes.

What the Assumptions Do—and Do Not—Say

Continuity is the substantive condition that separates a treatment effect from other threshold-related changes. A smooth change in the distribution of observed characteristics with $X_i$ is compatible with RD identification; the key concern is a discontinuity in the potential-outcome conditional means that would occur independently of treatment. If another policy, measurement rule, or eligibility consequence changes at the same cutoff, the observed jump may combine its effect with the treatment effect.

The assumptions are not established merely by calculating a jump. They describe why that jump has a causal interpretation. In particular, an observed difference in conditional means is a statistical feature of the outcome distribution. Calling it a treatment effect requires the sharp assignment rule, support on both sides, and continuity of both potential-outcome conditional means.

Nor does identification mean that the effect is representative of every treated unit. The estimand is local to the threshold: it is the conditional average effect at $X_i=c$, interpreted as a limit. Extrapolating it to the full treated population requires additional assumptions that are not part of continuity-based sharp RD identification.

Common Mistake

Wrong reasoning: “The outcome mean is 12 points higher just above the cutoff than just below it, so treatment raises the outcome by 12 points for everyone who receives treatment.”

Correction: Under the identification assumptions, the limit difference identifies the conditional average treatment effect at the cutoff—not the effect for every treated unit. The causal conclusion also depends on continuity: if another factor causes an outcome-mean jump at the same threshold, the observed jump need not equal the treatment effect. A precise conclusion names the cutoff, the outcome units, and the assumptions supporting the interpretation.

Key takeaway: In a sharp RD design, the observed conditional-mean jump identifies $\mu_1(c)-\mu_0(c)$ when treatment changes deterministically at the cutoff, the running variable has support on both sides, and both potential-outcome conditional means are continuous there.

Check Your Understanding

Answer each question using the definitions and identification argument above.

In a sharp RD design, which side of the cutoff supplies the left-hand limit of the observed conditional outcome mean, and which side supplies the right-hand limit?
Why does continuity of $\mu_0(x)$ and $\mu_1(x)$ matter for interpreting the limit difference causally?
Does the identification result require every unit to have the same treatment effect? Explain briefly.
A policy's observed conditional mean has a left-hand limit of 40 units and a right-hand limit of 46 units. What is the identified effect at the cutoff if all identification assumptions hold, and what does it not establish?
Suppose an unrelated outcome-reporting rule also changes at the cutoff. Which part of the identification argument is threatened, and why?