Continuity Assumptions and Local Identification

Why Continuity Identifies an Effect at the Cutoff

In a regression discontinuity (RD) design, treatment assignment changes at a known cutoff of a running variable. Outcomes can differ on the two sides for two reasons: the treatment assignment changes, and the running variable itself may be related to outcomes. The central identifying idea is that, if the relevant potential-outcome functions do not jump at the cutoff, their values just below and just above it would otherwise be comparable. A discontinuity in observed outcomes can then be attributed to the treatment change, under the design’s assumptions.

This reasoning is deliberately local. The limits immediately to either side of the cutoff identify an effect at the cutoff; continuity there does not say that effects are constant elsewhere or that the cutoff effect applies to people far from the threshold.

Definition: Let $R$ be the running variable, $c$ the cutoff, and $Z=1\{R\ge c\}$ the threshold-based assignment indicator. Write $Y(1)$ and $Y(0)$ for the potential outcomes under treatment and no treatment, and $D$ for treatment received. A function $g(r)$ is continuous at $c$ when its left and right limits exist and agree: $$ \lim_{r\uparrow c}g(r)=\lim_{r\downarrow c}g(r)=g(c). $$ In RD identification, the key continuity conditions concern potential-outcome or treatment-related functions, not the observed outcome function, which may have a jump because assignment changes at $c$.

Sharp RD: Continuity of Potential Outcomes

In a sharp RD design, treatment receipt follows the threshold exactly: $D=Z$. Define the conditional potential-outcome means $m_d(r)=E[Y(d)\mid R=r]$, for $d\in\{0,1\}$. The cutoff treatment effect is

\tau(c)=E[Y(1)-Y(0)\mid R=c]=m_1(c)-m_0(c).

The identifying condition is that both $m_1(r)$ and $m_0(r)$ are continuous at $c$. In particular, the untreated potential-outcome mean must not itself have a discontinuity at the cutoff, and neither may the treated potential-outcome mean. With treatment assigned at the threshold, the observed conditional mean just below the cutoff follows $m_0(r)$, while just above it follows $m_1(r)$. Their difference in limits therefore isolates the treatment effect at $c$.

Theorem: In a sharp RD design with $D=1\{R\ge c\}$, if the one-sided limits of $E[Y(0)\mid R=r]$ and $E[Y(1)\mid R=r]$ exist and each is continuous at $c$, then $$ \lim_{r\downarrow c}E[Y\mid R=r]-\lim_{r\uparrow c}E[Y\mid R=r] =E[Y(1)-Y(0)\mid R=c]. $$ Here $r\downarrow c$ denotes approaching from above and $r\uparrow c$ denotes approaching from below.

Worked Example: A Sharp RD Cutoff Effect

Suppose a hypothetical program is assigned to everyone with a score $R\ge 70$, and the outcome is a test score measured in points. Assume that the conditional potential-outcome means near 70 are

E[Y(0)\mid R=r]=12+0.04(r-70),\qquad E[Y(1)\mid R=r]=18+0.02(r-70).

These functions are continuous at 70. Under sharp assignment, the observed mean just below 70 approaches the untreated mean, and the observed mean just above 70 approaches the treated mean. Substitution gives

\tau(70)=\lim_{r\downarrow70}E[Y\mid R=r] -\lim_{r\uparrow70}E[Y\mid R=r] =18-12=6\text{ points}.

Under the stated design and continuity assumptions, the estimated causal target is an average increase of 6 test-score points for individuals at the score cutoff. The calculation does not establish that the program increases scores by 6 points at every score value.

Fuzzy RD: Continuity and a Local Wald Ratio

In a fuzzy RD design, crossing the threshold changes the probability of treatment but does not determine treatment receipt for every unit. Let $D(1)$ and $D(0)$ be the treatment statuses a unit would have under assignment above and below the cutoff, respectively. The first stage is the discontinuity in the conditional treatment probability:

\Delta_D(c)=\lim_{r\downarrow c}E[D\mid R=r] -\lim_{r\uparrow c}E[D\mid R=r].

The reduced-form discontinuity, $\Delta_Y(c)$, is defined in the same way using $Y$ in place of $D$. If the first stage is nonzero, the fuzzy RD Wald ratio is $\Delta_Y(c)/\Delta_D(c)$. Its causal interpretation requires more than continuity of the overall mean outcome: treatment assignment must affect the outcome only through treatment received (the exclusion restriction); assignment must not make anyone less likely to receive treatment (monotonicity); and the relevant potential-outcome and treatment-related functions must be continuous at the cutoff. The ratio then identifies a local average treatment effect for compliers at the cutoff, not the effect for everyone.

For clarity, define principal strata by the pair $(D(0),D(1))$. Never-takers have $(0,0)$, always-takers have $(1,1)$, and compliers have $(0,1)$. Monotonicity rules out defiers, the $(1,0)$ type. One sufficient continuity condition for the fuzzy RD argument is continuity at $c$ of the probabilities of these strata and of their conditional potential-outcome moments $E[Y(d)1\{G=g\}\mid R=r]$, for each treatment level $d$ and stratum $g$. These conditions prevent the composition and relevant potential outcomes of the strata from jumping at the cutoff.

Theorem: Under a nonzero first stage, exclusion, monotonicity, and continuity of the relevant stratum probabilities and potential-outcome moments at $c$, the fuzzy RD Wald ratio identifies $$ \frac{\Delta_Y(c)}{\Delta_D(c)} =E[Y(1)-Y(0)\mid G=\text{complier},R=c]. $$ This is a local complier average treatment effect: it applies to compliers at the cutoff.

Why the Fuzzy RD Ratio Works

The justification is a cancellation argument. Just below the cutoff, $Z=0$; just above it, $Z=1$. Continuity allows the one-sided limits of the relevant potential-outcome and stratum functions to be compared at the same running-variable value, $R=c$. Under exclusion, the outcome under assignment $z$ is $Y(D(z))$. Thus the limiting outcome jump is the average of $Y(D(1))-Y(D(0))$ at the cutoff.

For never-takers and always-takers, $D(1)=D(0)$, so their contribution to that difference is zero. For compliers, $D(1)=1$ and $D(0)=0$, so their contribution is $Y(1)-Y(0)$. Therefore,

\Delta_Y(c)=P(G=\text{complier}\mid R=c) E[Y(1)-Y(0)\mid G=\text{complier},R=c].

The treatment-probability jump is the corresponding change in treatment status. Never-takers and always-takers again contribute zero, while each complier contributes one:

\Delta_D(c)=P(G=\text{complier}\mid R=c).

Dividing the first expression by the second cancels the proportion of compliers and leaves their mean treatment effect at the cutoff. A positive first stage is required so the denominator is not zero. This argument also shows why monotonicity matters: if defiers were present, they would contribute in the opposite direction, and the simple ratio would not generally be the complier effect stated above.

Worked Example: Interpreting a Fuzzy RD Wald Ratio

In a hypothetical eligibility program, crossing a score cutoff raises the treatment probability from 0.20 to 0.50. The outcome is a health score, and the conditional outcome mean rises from 8.40 to 8.52 points at the cutoff. Assume exclusion, monotonicity, continuity of the relevant potential-outcome and stratum functions, and a nonzero first stage.

First calculate the treatment-probability jump:

\Delta_D(c)=0.50-0.20=0.30.

Then calculate the outcome jump and divide:

\Delta_Y(c)=8.52-8.40=0.12\text{ points},\qquad \frac{\Delta_Y(c)}{\Delta_D(c)} =\frac{0.12}{0.30}=0.40\text{ points}.

Under the assumptions, the ratio identifies an average treatment effect of 0.40 health-score points for compliers at the cutoff. It is not the raw outcome jump, which is 0.12 points, and it is not automatically the effect for all eligible people or for people far from the threshold.

Local Identification Is Not Extrapolation

A continuity assumption is a statement about behavior as $R$ approaches $c$. It does not require the potential-outcome functions to be linear, constant, or even similar far from the cutoff. Consequently, an RD effect is tied to the cutoff population and the threshold contrast that identifies it. Applying that effect to a broad range of scores requires additional assumptions or evidence not supplied by continuity at $c$.

Worked Example: A Cutoff Effect That Changes With the Running Variable

Suppose a hypothetical sharp RD has cutoff $c=70$ and potential-outcome means near and away from the cutoff described by

E[Y(0)\mid R=r]=100+0.3(r-70),\qquad E[Y(1)\mid R=r]=E[Y(0)\mid R=r]+2+0.1(r-70).

The treatment effect at score 70 is

\tau(70)=2+0.1(70-70)=2.

At score 90, the same functions imply

\tau(90)=2+0.1(90-70)=2+2=4.

Both potential-outcome means are continuous at 70, so continuity supports identification of the 2-unit effect at the cutoff. It does not make the effect 2 units at score 90; the hypothetical effect there is 4 units. The RD cutoff contrast alone cannot justify that extrapolation.

What Continuity Does—and Does Not—Assume

Continuity is not the claim that outcomes are identical on both sides of the cutoff. In a sharp RD, the observed outcome mean is expected to jump when treatment changes, even when each potential-outcome mean is continuous. In a fuzzy RD, the observed mean may jump because treatment probability changes. The identifying restriction is that the relevant potential-outcome and treatment-related functions do not have their own unexplained discontinuity at the threshold.

Continuity is also about limits, not necessarily about differentiability. A function can be continuous at the cutoff without having a well-defined slope there. The key comparison is between values arbitrarily close to $c$ on the two sides. If a potential-outcome mean or, in the fuzzy case, a relevant stratum function itself jumps at $c$, that jump can be mistaken for a treatment effect, and the stated identification argument no longer applies.

The cutoff need not contain an observed unit for the limiting argument to be meaningful. The design requires observations with running-variable values approaching the cutoff from both sides so that the relevant limits can be defined and learned from data. This is distinct from assuming that all units near the cutoff are literally randomized; the continuity argument concerns the limiting potential-outcome functions.

Common Mistake: Treating a Local Effect as a Universal Effect

Wrong reasoning: “The RD identifies a treatment effect at the cutoff, so that is the treatment effect for everyone with a score above the cutoff.” This conclusion does not follow. Continuity identifies the limiting contrast at $c$; it imposes no constancy of treatment effects as $R$ moves away from $c$.

Correction: Report the estimand at the cutoff and describe its local population. For sharp RD, that is the mean treatment effect at $R=c$; for fuzzy RD, it is the mean treatment effect for compliers at $R=c$. Generalizing to other scores or populations requires additional substantive assumptions and supporting evidence.

Key takeaway: RD identifies a causal effect because, under continuity, the potential-outcome functions on either side share the same limiting values at the cutoff. In a sharp design the outcome jump is the cutoff treatment effect; in a fuzzy design the outcome jump divided by the treatment-probability jump is a local complier effect. Neither result by itself licenses extrapolation away from the cutoff.

Check Your Understanding

Use the definitions and assumptions in this tutorial to answer the following questions.

In a sharp RD, which potential-outcome functions must be continuous at the cutoff for the outcome jump to identify the cutoff effect?
Why can the observed outcome mean jump even when both potential-outcome means are continuous?
A fuzzy RD has an outcome jump of 0.18 points and a treatment-probability jump of 0.45. Calculate the Wald ratio and state its causal interpretation under the required assumptions.
What role does monotonicity play in the fuzzy RD cancellation argument?
Why does continuity at a cutoff of 50 not establish that the treatment effect at a running-variable value of 80 equals the cutoff effect?