Two Effects at a Threshold
In a sharp regression-discontinuity design, crossing the cutoff determines treatment. With partial compliance, eligibility changes the chance of receiving treatment, but some eligible units do not take it up and some ineligible units receive it anyway. The outcome may still jump at the threshold, but that jump is not automatically the effect of treatment receipt. It is first the effect of crossing the eligibility threshold.
The distinction matters whenever the policy changes access, an offer, or eligibility rather than assigning treatment perfectly. For example, a scholarship cutoff might increase enrollment in a program without every eligible student enrolling. The local effect of eligibility includes the consequences of that offer for everyone near the cutoff. The local effect of enrollment is a different quantity, applying to people whose enrollment changes because eligibility changes.
Let \(R\) be the running variable and \(c\) the cutoff. Define eligibility as \(Z=1\{R\ge c\}\), where \(Z=1\) indicates eligibility. Let \(D\) indicate actual treatment receipt, with \(D=1\) for receipt and \(D=0\) otherwise, and let \(Y\) be the outcome. Write \(Y(d)\) for the outcome that would be observed under treatment status \(d\), and \(D(z)\) for the treatment status a unit would have under eligibility status \(z\).
Assumptions for the Local Complier Interpretation
The eligibility and treatment jumps are local comparisons between units just above and just below the threshold. Their causal interpretation requires more than observing two discontinuities. The conditions below specify why units on either side are comparable, what eligibility changes, and whose treatment effect the ratio represents.
- Continuity at the cutoff. In the absence of the eligibility change, the relevant conditional expectations of potential outcomes and potential treatment statuses would vary continuously through \(c\). In particular, there must not be another abrupt change at the same threshold that independently shifts the outcome. The running-variable density and support must also permit comparison arbitrarily close to the cutoff; sorting and support diagnostics help assess this design requirement.
- Relevance. Eligibility changes treatment receipt at the cutoff: \(\tau_D\ne 0\). The size and precision of this jump matter for estimation, not just its sign.
- Exclusion restriction. At the cutoff, eligibility affects the outcome only through actual treatment receipt. Formally, the outcome under eligibility \(z\) and receipt \(d\) is \(Y(z,d)=Y(d)\). If eligibility also provides a separate service or changes behavior directly, the ratio need not identify an effect of receipt.
- Monotonicity. Eligibility does not cause anyone to move away from treatment: \(D(1)\ge D(0)\) for units at the cutoff. This rules out defiers, for whom eligibility would reduce treatment receipt.
- Consistency and a well-defined treatment. Observed treatment and outcome correspond to the potential treatment and outcome under the eligibility status actually received. The treatment must be sufficiently well defined for a contrast between \(Y(1)\) and \(Y(0)\) to be meaningful.
The continuity condition is what lets the limits on either side stand in for the potential-outcome comparison at the cutoff. The other conditions explain how to interpret that comparison. If continuity or the no-other-discontinuity condition is implausible, the observed jumps may reflect more than eligibility. If exclusion fails, the outcome jump can still describe the local effect of eligibility, but dividing it by the treatment jump does not isolate the effect of receipt.
Why the Wald Ratio Is a Complier Effect
At the cutoff, compliance type is defined by the pair \((D(0),D(1))\). Under monotonicity, there are three types: compliers with \((0,1)\), who receive treatment only when eligible; always-takers with \((1,1)\); and never-takers with \((0,0)\). The excluded type \((1,0)\) consists of defiers.
Let \(\pi_C\) be the proportion of compliers among units at the cutoff. Under continuity, the first-stage discontinuity equals the local change in treatment status caused by eligibility. Always-takers and never-takers do not change status, while each complier does. Therefore, \(\tau_D=\pi_C\). Under the exclusion restriction, only compliers change their treatment-induced outcome when eligibility changes. Their average outcome change is the average treatment effect for compliers, denoted \(\tau_C=E[Y(1)-Y(0)\mid D(1)>D(0),R=c]\).
This is the local Wald-ratio justification. It is not a claim that the outcome jump alone is the treatment effect: under monotonicity and exclusion, the outcome jump combines the size of the complier group with their average treatment effect. Dividing by the first-stage jump removes the complier share. With continuous running-variable values, these expressions describe the limiting comparison at \(c\); an empirical analysis estimates the two local discontinuities and forms their ratio.
Worked Example: Separating Eligibility From Receipt
Suppose a hypothetical program uses a score cutoff. For students just below and above the cutoff, estimated limits in annual earnings are $30,000 and $31,200, respectively. The corresponding limits in program participation are 0.40 and 0.65. Assume the design and causal assumptions above are credible.
Eligibility effect. The outcome jump is $$ \tau_Y=\$31{,}200-\$30{,}000=\$1{,}200\text{ per year}. $$ Crossing the threshold increases annual earnings by an estimated $1,200 for units at the cutoff. This is the local effect of eligibility, not yet the effect of participating.
First stage. The treatment-receipt jump is $$ \tau_D=0.65-0.40=0.25. $$ Eligibility increases participation by 0.25, or 25 percentage points, at the cutoff.
Complier effect. The Wald ratio is $$ \frac{\tau_Y}{\tau_D}=\frac{\$1{,}200}{0.25}=\$4{,}800\text{ per year}. $$ Under the stated assumptions, this estimates the average annual earnings effect of participation for compliers at the cutoff: students whose participation is changed by eligibility. It is not the average effect for all students, nor does it say that each complier gains exactly $4,800.
Reading Compliance Patterns Correctly
The ratio’s interpretation depends on the pattern of take-up. Consider a hypothetical population at the cutoff with 30% compliers, 20% always-takers, and 50% never-takers. There are no defiers. If treatment increases a binary outcome by an average of 0.20 among compliers, then the local treatment effect is a 20-percentage-point increase in the outcome probability for compliers. Always-takers and never-takers do not change treatment status when eligibility changes.
Worked Example: From Compliance Shares to the Two Jumps
Use the compliance shares above, and suppose the hypothetical average treatment effect among compliers is 0.20 on a binary outcome. Calculate the first-stage and outcome discontinuities.
First stage. Only compliers change receipt, from \(D(0)=0\) to \(D(1)=1\). Thus $$ \tau_D=0.30. $$ Eligibility increases treatment receipt by 30 percentage points.
Outcome jump. By the exclusion restriction, only the compliers’ outcomes change in response to eligibility. Their share is 0.30, and their average treatment effect is 0.20, so $$ \tau_Y=(0.30)(0.20)=0.06. $$ The eligibility effect is a 6-percentage-point increase in the binary outcome probability.
Ratio. The Wald ratio is $$ \frac{0.06}{0.30}=0.20. $$ It recovers a 20-percentage-point treatment effect for compliers. The 6-point eligibility effect and the 20-point complier effect are both valid local quantities, but they answer different questions.
Estimation, Diagnostics, and Failure Modes
In practice, \(\tau_Y\) and \(\tau_D\) are estimated as separate local discontinuities at the same cutoff, using an appropriate bandwidth and local specification. Their ratio is then reported as the treatment-receipt effect estimate. The earlier material on first-stage diagnostics addresses whether the treatment jump is sufficiently clear; a near-zero or imprecisely estimated first stage makes the ratio unstable. Uncertainty in both estimated jumps must be reflected in inference for their ratio. A narrow confidence interval for the outcome jump alone does not establish a precise treatment-receipt effect.
Diagnostics support the credibility of a design; they do not prove the assumptions. Assess whether units could sort around the cutoff, whether the running variable has adequate support on both sides, and whether predetermined covariates show suspicious jumps. Consider whether another policy changes at the same score and whether the treatment definition captures the pathway from eligibility to outcome. Bandwidth and specification sensitivity also matter: an apparent jump that changes substantially under reasonable local specifications calls for caution.
Worked Example: When the Ratio Does Not Isolate Receipt
Suppose a hypothetical eligibility threshold raises participation by 0.20 and raises an outcome by 0.08 on a probability scale. However, eligibility also gives students access to a separate advising service, which directly raises the outcome probability by 0.03, apart from program participation. Calculate the ratio and assess its interpretation.
The observed Wald ratio is $$ \frac{0.08}{0.20}=0.40. $$ Numerically, eligibility raises the outcome by 8 percentage points and treatment receipt by 20 percentage points. But the exclusion restriction fails: the advising service is a direct eligibility-to-outcome pathway not captured by \(D\).
For illustration, if that direct pathway contributes 0.03 to the outcome jump and the remaining 0.05 comes through treatment, dividing the treatment-related component by the first stage gives $$ \frac{0.08-0.03}{0.20}=0.25. $$ This decomposition is only illustrative; the data do not generally identify the direct component without further information. The computed ratio of 0.40 is not justified as the effect of treatment receipt. The 0.08 outcome discontinuity remains interpretable as the total local eligibility effect, provided the continuity conditions hold.
A second failure occurs if monotonicity is false. With defiers, eligibility can increase receipt for some units and decrease it for others; the first stage is then a net change, and the simple complier-share argument no longer applies. A nonzero ratio may still be computable, but the standard local average treatment effect interpretation does not follow. Likewise, weak relevance can produce highly uncertain ratio estimates even when the point estimate looks large.
Which Effect Should Be Reported?
| Quantity | What it answers | Interpretation |
|---|---|---|
| Outcome discontinuity, \(\tau_Y\) | What is the local effect of crossing the eligibility threshold? | Effect of eligibility or offer at the cutoff, including all consequences of eligibility. |
| Treatment discontinuity, \(\tau_D\) | How much does eligibility change treatment receipt? | Local increase or decrease in treatment probability at the cutoff. |
| Wald ratio, \(\tau_Y/\tau_D\) | What is the effect of receipt for people whose receipt changes with eligibility? | Local complier average treatment effect, if relevance, exclusion, monotonicity, and continuity hold. |
A sharp design is the special case in which eligibility perfectly determines receipt at the cutoff: the first-stage jump is one, so the outcome jump and treatment effect coincide. An eligibility-only analysis reports the reduced-form outcome jump and avoids claiming that the ratio is an effect of receipt; it may be the more defensible target when exclusion is doubtful or the policy question concerns eligibility itself. The fuzzy ratio answers a narrower question and requires stronger assumptions.
Neither local quantity automatically generalizes to people far from the cutoff. The complier effect pertains to units near \(c\) whose treatment receipt responds to eligibility, and may differ from effects for always-takers, never-takers, or the broader population. Report the outcome jump and first-stage jump alongside the ratio so readers can see both the policy effect of eligibility and the strength of the treatment change underlying the receipt-effect estimate.
Check Your Understanding
Answer each question, then check the solutions below.
- At a cutoff, the outcome jumps by 900 dollars per year and treatment receipt jumps by 0.30. What are the eligibility effect and the Wald-ratio estimate, and who does the latter describe?
- What does monotonicity rule out, and why is it needed for the usual complier interpretation?
- If the treatment jump is zero, can the Wald ratio identify a complier treatment effect? Explain.
- A program has a 5-percentage-point outcome jump and a 25-percentage-point receipt jump. Under the required assumptions, calculate and interpret the ratio on the probability scale.
- Eligibility also changes access to a separate service that affects the outcome directly. Which assumption fails, and what remains interpretable?
Solutions. (1) The eligibility effect is \(900\) dollars per year. The ratio is \(900/0.30=3{,}000\) dollars per year, an estimated effect of receipt for compliers at the cutoff, not for all units. (2) Monotonicity rules out defiers, for whom eligibility reduces receipt. It ensures that the first-stage jump represents the share of compliers rather than a net of people moving in opposite directions, supporting the standard ratio derivation. (3) No. Relevance requires a nonzero treatment jump; with a zero denominator, the ratio is undefined, and eligibility has not shifted receipt at the cutoff. (4) The ratio is \(0.05/0.25=0.20\), a 20-percentage-point local treatment effect for compliers, conditional on the assumptions. (5) The exclusion restriction fails. The outcome jump can still be interpreted as the total local eligibility effect if continuity holds, but the ratio is not thereby identified as an effect of receipt.