Sharp Regression Discontinuity Identification

Why a Threshold Can Identify an Effect

When treatment is assigned by a score threshold, people just below and just above the threshold can be similar in many respects while receiving different treatments. The key identification question is whether their expected outcomes, absent the treatment change, would have varied smoothly through the threshold. If so, a sudden change in observed outcomes at the cutoff can be attributed to the treatment assignment rather than to a sudden change in the untreated outcome pattern.

This logic is specific to a sharp regression discontinuity design: treatment is determined exactly by whether the running variable crosses the cutoff. The result is not that every treated person can be compared causally with every untreated person. Instead, the design identifies a contrast at the threshold, under assumptions about potential-outcome means there.

Setup and Identification Assumptions

Let $X$ be the running variable, let $c$ be its cutoff, let $D$ indicate treatment, and let $Y$ be the observed outcome. Write $Y(1)$ and $Y(0)$ for the potential outcomes under treatment and no treatment, respectively. In a sharp design, the assignment rule is $D=1$ when $X\ge c$ and $D=0$ when $X<c$. Consistency gives $Y=DY(1)+(1-D)Y(0)$: the observed outcome equals the potential outcome corresponding to the treatment actually received.

Definition: A sharp regression discontinuity design identifies the treatment effect at cutoff $c$ from the discontinuity in the conditional mean of the observed outcome, provided that the assignment rule is sharp, outcomes are consistently defined, and the conditional means of both potential outcomes are continuous at $c$, with running-variable support on both sides of the cutoff.

More precisely, define the conditional potential-outcome means $m_1(x)=E[Y(1)\mid X=x]$ and $m_0(x)=E[Y(0)\mid X=x]$. The central smoothness assumption is that each mean has a well-defined limit at $c$ from both sides, and that its left- and right-hand limits agree. The assumption concerns means; individual potential outcomes need not be smooth functions of $X$, and treatment effects need not be constant.

There must also be observations with running-variable values arbitrarily close to $c$ from below and above. Formally, every neighborhood of $c$ must contain values with positive probability on both sides. Without this support, one or both limiting means cannot be learned from the data near the cutoff. The treatment rule must be sharp, and the outcome and treatment must be consistently defined. These are identification assumptions; they do not by themselves prescribe an estimation method or guarantee precise estimates in a finite sample.

Deriving the Outcome-Jump Formula

Let $g(x)=E[Y\mid X=x]$ denote the conditional mean of the observed outcome. Below the cutoff, the assignment rule gives $D=0$, so consistency implies $g(x)=m_0(x)$ for $x<c$. Above it, $D=1$, so $g(x)=m_1(x)$ for $x\ge c$. Taking limits toward $c$, and using continuity of the potential-outcome means, gives:

\begin{aligned} \lim_{x\to c^-}g(x)&=m_0(c),\\ \lim_{x\to c^+}g(x)&=m_1(c). \end{aligned}

Subtract the left-hand limit from the right-hand limit. The result is the difference between the two potential-outcome means at the cutoff. By linearity of conditional expectation, that difference is also the mean of the individual treatment effect conditional on $X=c$.

\begin{aligned} \tau_{\mathrm{RD}}(c) &=\lim_{x\to c^+}E[Y\mid X=x] -\lim_{x\to c^-}E[Y\mid X=x]\\ &=E[Y(1)\mid X=c]-E[Y(0)\mid X=c]\\ &=E[Y(1)-Y(0)\mid X=c]. \end{aligned}

The equality identifies a causal contrast at the cutoff; it does not assert that the observed difference between two particular people is a treatment effect. Nor does it require the untreated outcome mean to be flat. It may rise or fall with $X$, provided its conditional mean does not itself jump at $c$.

Worked Example: A Smooth Trend and an Eight-Point Jump

Suppose a hypothetical program is assigned to everyone with an eligibility score $X\ge70$. The outcome is a test score, measured in points. Near the cutoff, suppose the conditional mean functions are $m_0(x)=40+0.5(x-70)$ without the program and $m_1(x)=48+0.5(x-70)$ with it. Assume the sharp rule, consistency, support on both sides, and continuity of these potential-outcome means at 70.

At the cutoff, the two potential-outcome means are $m_0(70)=40+0.5(0)=40$ points and $m_1(70)=48+0.5(0)=48$ points. Therefore the identified effect is $48-40=8$ points.

The observed conditional mean below the cutoff follows $m_0$, and above it follows $m_1$. For illustration, at a score of 68 the untreated mean is $40+0.5(-2)=39$ points; at 72 the treated mean is $48+0.5(2)=49$ points. Their difference, $49-39=10$ points, is not the cutoff effect: it compares different score values. The limiting means at the same cutoff are 40 and 48, giving the eight-point jump.

The formulas make the continuity assumption explicit: both potential-outcome means have a smooth trend through 70. The example’s linear equations are a convenient way to show the calculation, not a requirement that real outcome functions be linear.

What Is Assumed, and What Is Estimated

The identification argument is about the limits of conditional means. In an actual finite sample, those limits are not directly observed: one typically has few, if any, observations with $X$ exactly equal to $c$. An analysis must estimate the two sides’ near-cutoff behavior and extrapolate to the threshold. That estimation step can require choices about functional form and which observations to use. Such choices are not the same as the continuity assumption and cannot make a discontinuity in an untreated potential-outcome mean disappear.

In particular, a straight-line fit on each side may be useful for estimating the limits, but the causal identification result does not require linearity. Conversely, fitting a straight line does not establish continuity. A poorly chosen functional form can estimate the jump inaccurately even when the identifying assumptions hold.

Worked Example: Unequal Slopes Do Not Change the Cutoff Contrast

In a hypothetical setting, let $c=50$. Near the threshold, suppose $m_0(x)=100+4(x-50)$ and $m_1(x)=106-2(x-50)$, with a sharp assignment rule and continuous potential-outcome means. The outcome is measured in units of a clinical score.

At the cutoff, $m_0(50)=100$ and $m_1(50)=106$, so the jump is $106-100=6$ score units. The two sides have different slopes: the untreated mean rises by 4 units per running-variable unit, while the treated mean falls by 2 units per running-variable unit. Continuity of each function still holds at 50.

For a check, at $X=49.5$, the untreated mean is $100+4(-0.5)=98$. At $X=50.5$, the treated mean is $106-2(0.5)=105$. The cross-score difference is $105-98=7$, not 6, because it includes both the treatment jump and the change in score position. As the two score values approach 50, their trend contributions vanish and the difference approaches 6.

The identification conditions are satisfied in this example because the means are defined on both sides, have no jumps of their own at 50, and treatment changes exactly at 50. Neither equal slopes nor a constant treatment effect away from the cutoff is required.

When the Jump Is Not a Treatment Effect

The crucial threat is a discontinuity at $c$ in a potential-outcome mean that would have occurred even without the treatment change. For example, if a second policy also begins at the same score and directly changes the outcome, the observed jump combines that policy’s effect with the treatment effect of interest. Similarly, a sudden change in outcome measurement at the threshold can create an observed jump without the intended causal interpretation. Continuity is a substantive claim about the setting, not a consequence of drawing a threshold on a graph.

The sharp-assignment condition matters too. If some people below the cutoff receive treatment or some above it do not, then the observed outcome means on either side are not simply $m_0(x)$ and $m_1(x)$. The sharp-design formula no longer follows. Other designs and estimands are needed when treatment assignment is imperfect.

Diagnostics can inform, but cannot prove, the assumptions. Researchers can examine whether running-variable values appear unusually concentrated or sparse near the threshold, whether the outcome pattern looks compatible with a jump rather than an abrupt change in trend, and whether variables determined before treatment show unexpected jumps. These checks may reveal concerns about sorting, co-occurring changes, or measurement. Passing them does not establish that the unobserved potential-outcome means are continuous.

Worked Example: A Coincident Policy Breaks the Interpretation

Suppose a hypothetical training program is assigned at $X\ge100$, and the outcome is monthly earnings in dollars. At the same threshold, however, a separate payment of \$200 per month begins for everyone above the cutoff. Assume the training itself raises earnings by \$50 at the cutoff, and the other components of earnings would be continuous absent both changes.

The observed right-minus-left outcome jump is $50+200=\$250$ per month. The assignment is sharp, and running-variable values exist on both sides, but the observed jump is not the training effect alone: the separate payment also changes at the threshold. The required continuity condition for the potential outcome under the relevant treatment contrast is violated by this coincident intervention.

Thus, the formula would attribute the whole \$250 jump to the treatment only if all other causes of the outcome had continuous conditional means at the cutoff. Here that interpretation is contradicted by the known policy change. The training effect cannot be isolated from this jump without additional design information or assumptions.

How This Design Differs From Alternatives

A randomized experiment identifies effects by randomized treatment allocation across its study population, subject to its own design and implementation assumptions. Sharp regression discontinuity instead uses deterministic assignment around a threshold; its identifying comparison is local to that cutoff and depends on continuity of potential-outcome means there. It does not require the entire population to be exchangeable between treated and untreated groups.

Ordinary regression adjustment or propensity score matching can compare treated and untreated units across a broader range of covariates, but those approaches rely on their own conditions for controlling confounding. A sharp threshold design uses the assignment rule and the limiting outcome means at the cutoff. The resulting causal interpretation is not interchangeable with a population-wide adjusted comparison.

The main limitation is correspondingly important: the identified effect pertains to units at the cutoff, not automatically to people with much lower or higher running-variable values. The result also depends on a credible threshold rule, adequate support near the cutoff, and the absence of other abrupt changes in potential-outcome means. More observations far from the threshold do not, by themselves, resolve a failure of continuity at it.

Common Mistakes

Comparing outcomes at different running-variable values. The causal contrast is the difference between the two limiting means at $c$, not a raw difference between a treated unit far above the cutoff and an untreated unit far below it.
Assuming the outcome trend must be flat. The potential-outcome means may slope with $X$; the requirement is that they do not jump at the cutoff.
Treating a fitted model as the identification assumption. A linear fit is an estimation choice. It does not prove continuity or make the design sharp.
Attributing every observed jump to treatment. A concurrent policy, measurement change, or other cause of an outcome discontinuity can invalidate that attribution.
Generalizing the cutoff effect to everyone. The identified contrast is at the threshold and need not equal effects elsewhere.

Key takeaway: With sharp assignment, consistent outcomes, support on both sides, and continuous conditional means of both potential outcomes at the cutoff, the right-hand limit of the observed outcome mean minus its left-hand limit equals the mean treatment effect at the cutoff.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A treatment is assigned when $X\ge20$. The observed outcome mean has left and right limits of 12 and 17 units. Under the sharp-design continuity assumptions, what is the identified effect and its direction?
Why does continuity of the untreated potential-outcome mean not require that it be constant near the cutoff?
At a cutoff, the untreated mean is 30 and the treated mean is 34. What is the identified effect? What additional conclusion about effects far from the cutoff is justified?
A second intervention changes at the same cutoff as treatment. Which part of the identification argument is threatened, and why?
Why is the difference between an observation just above the cutoff and one far below it not generally the identified cutoff effect?

Solutions. (1) The right-minus-left jump is $17-12=5$ outcome units, so the identified effect is a five-unit increase at the cutoff. (2) Continuity allows a changing mean, such as a rising or falling trend, as long as its limit from each side is the same at the cutoff. (3) The effect at the cutoff is $34-30=4$ units. No effect away from the cutoff follows from this calculation; effects may vary with $X$. (4) The potential-outcome mean may have a discontinuity due to the second intervention, so the observed jump cannot be attributed solely to the treatment of interest. (5) The two observations have different $X$ values, so their outcome difference can include both a treatment jump and changes in outcome trends across those values. The cutoff effect instead compares the two limiting means at the same threshold.