Why Continuity Identifies The Cutoff Effect

Why the Limits Matter

In a sharp regression discontinuity design, treatment assignment changes deterministically at a threshold of a running variable. Outcomes just below and just above that threshold can differ for two reasons: treatment may change the outcome, and the running variable may be related to the outcome even without treatment. The central identification question is whether the second source of difference disappears as we approach the threshold from either side.

The answer is supplied by continuity. If the average potential outcome under each treatment state changes smoothly through the cutoff, then its value just below the cutoff approaches the same limit as its value just above. On the left, observed outcomes reveal the untreated mean; on the right, they reveal the treated mean. The difference between those limits is therefore the treatment effect at the cutoff. This is an identification result: it says what population quantity the limiting contrast represents, not how to estimate it from a finite sample.

The earlier material on Sharp Regression Discontinuity Identification and The Local Estimand In Regression Discontinuity motivates the design and its local target. Here the key step is to derive that target directly from the observed conditional means.

Setup and Assumptions

Definition: Let \(R\) be a running variable with cutoff \(c\), and let treatment be assigned by \(D=1(R\geq c)\). For each unit, let \(Y(1)\) and \(Y(0)\) be the potential outcomes under treatment and no treatment. The observed outcome is \(Y=D Y(1)+(1-D)Y(0)\). Define \(m_d(r)=E[Y(d)\mid R=r]\) for \(d\in\{0,1\}\). The sharp regression discontinuity effect at the cutoff is \(\tau(c)=m_1(c)-m_0(c)\), using continuous versions of these conditional means at \(c\).

If \(R\) has no point mass at \(c\), then \(P(R=c)\) may be zero, so a conditional expectation at exactly \(R=c\) is not directly determined by observations at that single value. In this setting, \(m_d(c)\) means the value at \(c\) of a version of the conditional mean that is continuous there. Equivalently, the target is defined through the corresponding one-sided limits. If there is a point mass at the cutoff, the target and assignment at the cutoff require additional care; the derivation below is framed in terms of limits from either side.

The identification argument uses the following conditions.

Sharp assignment and consistency: units with \(R<c\) receive \(D=0\), and units with \(R\geq c\) receive \(D=1\); the observed outcome equals the potential outcome for the treatment actually received.
Two-sided support: values of \(R\) occur arbitrarily close to \(c\) on both sides. Otherwise, one or both limits cannot be learned from nearby observations.
Continuity of potential-outcome means: \(m_0(r)\) and \(m_1(r)\) are finite and continuous at \(c\). This is a condition on conditional means, not a claim that every individual's potential outcomes are smooth functions of \(R\).
No other discontinuous change that affects the outcome mean: the interpretation of the jump as the treatment effect requires that no separate policy, eligibility rule, or measurement change also shifts outcomes at \(c\). Such a change would generally violate continuity of the relevant potential-outcome means.

These conditions do not require treatment to be as-if randomly assigned throughout the full range of \(R\). They require the two potential-outcome means to meet continuously at the threshold, so the cutoff comparison is local.

Deriving the Identification Equation

Let \(m(r)=E[Y\mid R=r]\) be the observed conditional outcome mean. Consistency and sharp assignment imply that below the cutoff the observed outcome is \(Y(0)\), while above the cutoff it is \(Y(1)\). Thus, on the two sides,

m(r)= \begin{cases} m_0(r), & r<c,\\ m_1(r), & r\geq c. \end{cases}

Take the limit from below. Because the observed mean equals \(m_0(r)\) on that side, and \(m_0\) is continuous at \(c\), the left-hand limit is \(m_0(c)\). Take the limit from above: the observed mean equals \(m_1(r)\), and continuity makes its right-hand limit \(m_1(c)\). Subtracting gives the identification equation:

\lim_{r\downarrow c} E[Y\mid R=r] - \lim_{r\uparrow c} E[Y\mid R=r] = m_1(c)-m_0(c) = \tau(c).

Here \(r\downarrow c\) denotes approaching \(c\) from values above it, and \(r\uparrow c\) denotes approaching from below. The order of subtraction is right limit minus left limit, matching treated minus untreated. The treatment assignment convention at the single point \(R=c\) does not determine either one-sided limit.

The continuity condition does the essential work. Without it, the left limit of the observed mean need not equal \(m_0(c)\), or the right limit need not equal \(m_1(c)\). The observed jump could then combine a treatment effect with a discontinuity in the potential-outcome mean. The algebra of the assignment rule would remain true, but the causal interpretation would not follow.

Worked Example: A Smooth Common Trend

Suppose a hypothetical sharp program uses cutoff \(c=50\). Conditional mean potential outcomes, in outcome units, are \(m_0(r)=64+2(r-50)\) and \(m_1(r)=72+2(r-50)\) near the cutoff. Assume treatment assignment is exactly \(D=1(R\geq50)\), both means are continuous at 50, and there is no other policy change at that value.

For \(r<50\), observed outcomes follow the untreated mean. The left limit is \(\lim_{r\uparrow50}m(r)=64+2(50-50)=64\) outcome units. For \(r\geq50\), outcomes follow the treated mean. The right limit is \(\lim_{r\downarrow50}m(r)=72+2(50-50)=72\) outcome units.

Therefore the cutoff effect is \(72-64=8\) outcome units. The common slope does not create a jump: its contribution tends to zero at \(r=50\). The calculation identifies the treatment effect at the cutoff, not necessarily the average effect for units far from 50.

The Role of Local Trends

The two observed means need not be equal a short distance from the cutoff when treatment has no effect. The running variable may be associated with outcomes on both sides. Continuity says that the no-treatment mean, for example, does not jump as \(R\) passes through \(c\); it does not say that the mean is flat. The limiting operation removes a smooth trend because the distance to the cutoff shrinks to zero.

Worked Example: Different Slopes on the Two Sides

In a second hypothetical design, let \(c=10\), with \(m_0(r)=50+3(r-10)\) and \(m_1(r)=56-2(r-10)\) near the threshold. Suppose assignment is sharp, both potential-outcome means are continuous at 10, and no other outcome-affecting rule changes there.

At \(r=9.9\), the observed untreated mean is \(50+3(9.9-10)=49.7\) outcome units. At \(r=10.1\), the observed treated mean is \(56-2(10.1-10)=55.8\) outcome units. The difference between these two nearby values is \(55.8-49.7=6.1\) outcome units. It is not the limiting cutoff effect: the observations are each 0.1 units away from the threshold, and their smooth trends have not vanished.

Taking the limits instead gives \(\lim_{r\uparrow10}m(r)=50+3(10-10)=50\) and \(\lim_{r\downarrow10}m(r)=56-2(10-10)=56\). Hence \(\tau(10)=56-50=6\) outcome units. The different slopes are compatible with the result because each side's mean is still continuous at the cutoff.

This example also clarifies why the jump is not a comparison of two arbitrary groups selected from a broad interval. It is the difference between the two sides' limiting means at the same running-variable value. In finite data, those limits must be estimated; the equation itself does not prescribe an estimation procedure or quantify sampling uncertainty.

When the Interpretation Fails

A discontinuity in an untreated potential-outcome mean can masquerade as a treatment effect. Suppose another program begins at the same cutoff, or the outcome is recorded differently on opposite sides. Then even with sharp assignment to the treatment of interest, the observed jump may include that second change. Continuity cannot be taken for granted merely because the assignment rule is deterministic.

Worked Example: A Competing Policy at the Threshold

Consider a hypothetical cutoff \(c=0\). Treatment of interest is assigned sharply at zero. Just below zero, the untreated outcome mean approaches 40 units. Just above zero, a separate policy also begins, raising the untreated potential-outcome mean to 45 units; under the treated state and the second policy, the mean is 52 units. Suppose the treatment effect at the cutoff under the post-threshold policy is therefore \(52-45=7\) units.

The observed left limit is 40 units because units just below zero receive neither change. The observed right limit is 52 units because units just above zero receive both changes. The observed jump is \(52-40=12\) units. The jump exceeds the treatment effect of 7 units by the separate 5-unit policy shift.

The failure is explicit: the untreated mean does not have a common limit at the cutoff; it changes from 40 to 45 units. The required continuity condition is false, so the 12-unit observed jump does not identify the 7-unit treatment effect. The same logic applies to any coincident discontinuity that alters outcomes, not only a second public policy.

A credible application therefore needs a substantive argument for why potential-outcome means would evolve smoothly absent the treatment change. Relevant warning signs include a second rule triggered at the same threshold, changes in measurement or sample composition, and a lack of observations close to the cutoff on one side. Evidence that units can precisely sort around the threshold can also undermine the plausibility of continuity, because the units immediately above and below may no longer be comparable in their potential outcomes. Such checks can reveal concerns, but no single diagnostic proves the continuity assumption.

What the Result Does—and Does Not—Say

The main alternatives clarify the scope of this identification result. A broad before-and-after comparison generally mixes the treatment change with any smooth relationship between \(R\) and the outcome. A global regression that imposes one simple trend may likewise impose more structure than is needed for the limiting argument. A randomized experiment can identify an effect through random assignment over its study population, whereas sharp regression discontinuity relies on the local continuity and support conditions at the threshold. In a fuzzy design, treatment receipt does not switch perfectly at the cutoff, so the simple sharp equation above does not apply as stated.

The result is local: it identifies \(m_1(c)-m_0(c)\), the difference in mean potential outcomes at the cutoff. It does not by itself identify an average treatment effect for the full population, an effect at distant values of \(R\), or an individual treatment effect. Nor does identification guarantee precision. If observations near the cutoff are sparse or outcomes are highly variable, a finite-sample estimate of the limits may be uncertain even when the assumptions are credible.

Keep the distinction between identification and estimation clear. Continuity and the assignment rule establish which causal quantity the population jump represents. A statistical method must still estimate the one-sided limits from observed data, and its uncertainty and sensitivity to modeling choices must be assessed separately.

Key takeaway: Under sharp assignment, two-sided support, and continuity of both potential-outcome means at the cutoff, the right-hand limit of the observed outcome mean equals the treated mean at the cutoff, the left-hand limit equals the untreated mean, and their difference identifies the cutoff treatment effect.

Check Your Understanding

Answer each question using the limit argument, then compare with the solutions below.

In a sharp design, the left and right limits of the observed mean are 18 and 23 points. Under the required assumptions, what is the cutoff effect, and in which direction is it expressed?
Why does continuity of \(m_0(r)\) matter for interpreting the left-hand limit?
Suppose \(m_0(r)=30+4(r-c)\) and \(m_1(r)=35-(r-c)\) near \(c\). What is the cutoff effect?
If a second policy creates a 3-unit upward jump in the untreated potential-outcome mean at the cutoff, can the observed jump alone identify the treatment effect? Explain.

Solutions. (1) The effect is \(23-18=5\) points, expressed as the treated mean minus the untreated mean at the cutoff. (2) Below the cutoff, the observed mean equals \(m_0(r)\); continuity ensures its left-hand limit is \(m_0(c)\), rather than a different limiting value. (3) At \(r=c\), the two means are 30 and 35, so the effect is \(35-30=5\) outcome units. (4) No. The untreated mean itself is discontinuous, violating the continuity condition; the observed jump can combine the treatment effect with the 3-unit policy shift.