Sharp RD Identification by Continuity

Why a Jump at the Cutoff Can Be Causal

In a regression discontinuity design, treatment changes deterministically when a running variable crosses a known cutoff. Outcomes may vary with the running variable even without treatment. The causal question is therefore not whether outcomes change as the running variable changes, but whether the outcome regression has a jump exactly where treatment assignment changes.

The identification argument compares the outcome regression just above the cutoff with the regression just below it. If the mean potential outcomes would each have changed smoothly through the cutoff in the absence of a treatment switch, then their background variation has no jump to explain the observed discontinuity. The remaining jump identifies the treatment effect at the cutoff.

Definition: Let $R$ be a running variable, $c$ a cutoff, and $D$ the treatment indicator. In a sharp regression discontinuity design, treatment assignment follows the deterministic rule $D=\mathbf{1}\{R\geq c\}$: units at or above the cutoff are treated, and units below it are untreated. Let $Y(1)$ and $Y(0)$ denote the potential outcomes under treatment and no treatment, respectively. The observed outcome is $Y=DY(1)+(1-D)Y(0)$.

Write the conditional potential-outcome means as $m_d(r)=E[Y(d)\mid R=r]$, for $d=0,1$. The sharp RD estimand is the difference in these means at the cutoff, $\tau(c)=m_1(c)-m_0(c)$, when those conditional means are defined there. If $R$ is continuously distributed, the probability of observing $R=c$ may be zero. In that case, the relevant quantities are defined by their one-sided limits at the cutoff, rather than by an average over observations exactly at $c$.

The Identification Result

The key conditions are that assignment is sharp as specified and that each conditional potential-outcome mean is continuous at $c$. Continuity here means that the mean outcome under each treatment state approaches its value at the cutoff from either side. It does not require outcomes to be constant in $R$, nor does it require the treatment effect to be constant away from the cutoff.

Theorem: Suppose $D=\mathbf{1}\{R\geq c\}$, the observed outcome satisfies consistency, $Y=DY(1)+(1-D)Y(0)$, and the conditional means $m_1(r)=E[Y(1)\mid R=r]$ and $m_0(r)=E[Y(0)\mid R=r]$ are continuous at $c$. Then the jump in the observed conditional outcome mean identifies the treatment effect at the cutoff: $$ \lim_{r\to c^+} E[Y\mid R=r]-\lim_{r\to c^-} E[Y\mid R=r] = m_1(c)-m_0(c) = \tau(c). $$

The equality follows by considering the two sides separately. For $r>c$, sharp assignment gives $D=1$, so the observed outcome is $Y(1)$. For $r<c$, it gives $D=0$, so the observed outcome is $Y(0)$. Thus the right-side outcome regression is $m_1(r)$, and the left-side regression is $m_0(r)$. Taking limits and applying continuity yields

\begin{aligned} \lim_{r\to c^+}E[Y\mid R=r] &=\lim_{r\to c^+}E[Y(1)\mid R=r] =m_1(c),\\ \lim_{r\to c^-}E[Y\mid R=r] &=\lim_{r\to c^-}E[Y(0)\mid R=r] =m_0(c). \end{aligned}

Subtracting the second limit from the first gives $m_1(c)-m_0(c)$, which is the conditional average treatment effect at the cutoff. This argument identifies the effect at $c$, not the effect for every unit and not, without additional assumptions, an average effect for the full population. It relies on the fact that just below the cutoff reveals the untreated outcome regression that continues to the cutoff, while just above reveals the treated outcome regression that continues to the cutoff.

Worked Interpretations

Worked Example: A Test-Score Eligibility Threshold

Suppose a hypothetical program is assigned to every student whose qualifying score $R$ is at least $c=70$. The outcome $Y$ is a later test score, measured in points. Assume assignment is followed exactly, outcomes are recorded consistently, and the conditional mean of each potential test score is continuous at 70. Suppose the hypothetical one-sided outcome-regression limits are 76 points from above and 70 points from below.

Because assignment is sharp, students just above 70 reveal the treated potential-outcome mean at the cutoff, and students just below 70 reveal the untreated mean there. The identified effect is

\widehat{\text{limit difference}}=76-70=6\text{ points}.

The causal interpretation is that the average effect of program assignment for units at the cutoff is 6 test-score points, under the stated continuity and assignment conditions. The result is local: it does not establish a six-point effect for students with scores far from 70.

Worked Example: A Cutoff and Hospitalization Risk

Consider a hypothetical policy that provides a preventive service to everyone with a risk score $R\geq c=50$. Let $Y$ indicate hospitalization during a specified follow-up period, so conditional means are probabilities. Suppose the limit of the observed hospitalization probability from just above 50 is 0.09, and the limit from just below 50 is 0.12. Assume treatment is assigned exactly by the threshold, the outcome is consistently defined, and each potential-outcome hospitalization probability is continuous at 50.

The right limit identifies the treated potential-outcome probability at the cutoff, while the left limit identifies the untreated probability. Subtracting gives

$$ 0.09-0.12=-0.03. $$

Thus the treatment effect at the cutoff is a 0.03 lower probability of hospitalization, or a decrease of 3 percentage points. This is a difference in probabilities, not a 3 percent relative reduction. For comparison, the relative change from 0.12 to 0.09 is $(0.09-0.12)/0.12=-0.25$, or a 25% reduction relative to the untreated probability. The RD estimand itself is the 3-percentage-point difference.

Worked Example: Smooth Background Trends on Both Sides

Suppose a hypothetical training program is assigned when $R\geq c=10$, and the outcome is monthly earnings in thousands of dollars. Assume sharp assignment and continuity of both potential-outcome means at 10. Suppose, near the cutoff, the conditional means follow

m_0(r)=40+2(r-10),\qquad m_1(r)=45+2(r-10).

Both potential outcomes have a smooth upward trend of 2 thousand dollars per unit of $R$. Evaluate their limits at the cutoff:

m_0(10)=40+2(10-10)=40,\qquad m_1(10)=45+2(10-10)=45.

Accordingly, the left-side observed mean approaches 40 thousand dollars and the right-side observed mean approaches 45 thousand dollars. Their difference is $45-40=5$ thousand dollars per month. The smooth trend does not bias this limit contrast: it is present in both potential-outcome regressions and does not create a jump at 10. The identified cutoff effect is therefore 5 thousand dollars per month.

What the Assumptions Rule Out

Continuity is a substantive identifying assumption, not a consequence of observing a cutoff. It rules out a discontinuous change at $c$ in the mean potential outcome under either treatment state. In particular, if untreated outcomes would jump at the cutoff even without treatment, part of the observed jump could reflect that change rather than treatment. The same concern applies to a discontinuity in the treated potential-outcome mean.

For example, suppose the untreated mean just below the cutoff is 40, but the untreated mean just above it would be 43 even if no one received treatment. If the treated mean just above is 48, the observed jump is $48-40=8$, while the treatment effect at the cutoff is $48-43=5$. The extra 3 units come from the discontinuity in the untreated potential outcome. The observed jump no longer equals the causal effect.

Key takeaway: In a sharp RD design, deterministic assignment makes the right-side outcome limit a treated mean and the left-side limit an untreated mean. Continuity of both conditional potential-outcome means lets those limits represent the two potential outcomes at the cutoff. Their difference identifies the treatment effect there.

Common Mistakes

Calling the limit difference a population-wide effect. The identified contrast is local to the cutoff. Effects elsewhere can differ.
Assuming that any outcome jump is causal. The causal interpretation requires continuity of both potential-outcome means at the cutoff, as well as sharp assignment and consistent outcomes.
Subtracting in the wrong order. The RD contrast is the right-side limit minus the left-side limit. With treatment assigned above the cutoff, this is treated minus untreated.
Confusing continuity with a flat outcome regression. The means may slope with $R$; continuity requires no jump at the cutoff, not a zero slope.
Interpreting a difference in probabilities as a relative percent change. A contrast of $-0.03$ is a decrease of 3 percentage points; a relative percentage requires dividing by a specified baseline probability.

Identify the assignment rule.
Confirm that treatment is assigned exactly according to $D=\mathbf{1}\{R\geq c\}$.

Name the two observed limits.
The limit from above is the treated outcome mean; the limit from below is the untreated outcome mean.

State the continuity requirement.
Require each conditional potential-outcome mean to approach its value at the cutoff from both sides.

Subtract and interpret locally.
Take the right limit minus the left limit, attach the outcome's units, and describe the resulting effect as applying at the cutoff.

Check Your Understanding

Use the assumptions and limit argument above to answer the following questions.

In a sharp RD design with treatment assigned for $R\geq c$, which potential-outcome mean is revealed by the observed regression just below $c$, and why?
Suppose the right-side limit is 12.4 units and the left-side limit is 10.1 units. What is the estimated cutoff contrast, including its sign and units if the outcome is measured in units?
Why can a smoothly increasing outcome mean on both sides of the cutoff be compatible with identification by the limit difference?
Suppose the untreated potential-outcome mean itself has a jump at the cutoff. Explain why the observed outcome jump need not equal the treatment effect there.
Does identification of $\tau(c)$ imply that the treatment effect is the same for units far from $c$? Explain.