Worked Sharp RD Analysis of Test-Score Eligibility

Why a Sharp RD Analysis Starts With the Assignment Rule

A test-score threshold can create a useful comparison when it deterministically assigns a treatment: students just above the threshold receive the program, while students just below it do not. The central question is whether the outcome has a discontinuity at the threshold. Because students near either side may differ in score, the analysis allows the outcome to vary with score and estimates the jump at the cutoff rather than comparing all eligible students with all ineligible students.

This tutorial develops a hypothetical example in which students scoring at least 70 on an entrance assessment are automatically offered a tutoring program. The outcome is a later mathematics examination score, measured in points. The design is sharp only if the assignment rule is followed exactly: every student at or above 70 is treated, and no student below 70 is treated.

Definition: Let $X$ be the running variable centered at the cutoff, with $X=0$ at the threshold, and let $D$ indicate treatment receipt. A sharp regression discontinuity design has the deterministic assignment rule $D=1[X\geq 0]$. Its cutoff effect is the difference between the right-hand and left-hand limits of the conditional mean outcome: $$ \tau_{\mathrm{SRD}}=\lim_{x\to 0^+}E[Y\mid X=x]-\lim_{x\to 0^-}E[Y\mid X=x]. $$

The RD estimand is local: it concerns students at the score threshold, not the average effect for every student in the program. A causal interpretation requires that, absent the treatment, the mean potential outcome would vary continuously through the cutoff. In practical terms, there must not be a separate change at 70 that would itself cause the outcome to jump. The rule must also be correctly recorded and enforced. Diagnostics can reveal problems with these assumptions, but cannot prove them.

Worked Example: Translating Eligibility Into an Estimand

Suppose the hypothetical tutoring rule is $D=1[S\geq 70]$, where $S$ is the entrance score. Define the centered running variable as $X=S-70$. A student with score 69 has $X=-1$ and is ineligible; a student with score 70 has $X=0$ and is eligible; a student with score 74 has $X=4$ and is eligible.

The assignment rule is therefore $D=1[X\geq 0]$. Because treatment is assigned exactly by this rule, there is no distinction between eligibility and treatment receipt in this example. The target is the change in the expected later mathematics score at $X=0$, in examination-score points: $$ \tau_{\mathrm{SRD}}=\lim_{x\to 0^+}E[Y\mid X=x]-\lim_{x\to 0^-}E[Y\mid X=x]. $$

This is not the difference between the average outcome of every student scoring at least 70 and that of every student scoring below 70. Those groups may differ substantially in their scores and other characteristics. The estimand compares the limiting outcome levels for students immediately above and below the threshold.

Local Linear Estimation at the Cutoff

A local linear RD fit estimates a separate straight line on each side of the cutoff using observations close to zero. With bandwidth $h$, a triangular kernel gives greater weight to observations nearer the cutoff and weight zero at the edge of the window. The left and right fits are evaluated at $X=0$; their intercept difference estimates the cutoff jump. Separate fits allow both the level and the score-outcome slope to differ across sides.

Estimation procedure: For each side of the cutoff, choose an intercept $a$ and slope $b$ to minimize the kernel-weighted squared residuals among observations within bandwidth $h$. With the triangular kernel $K(u)=\max(1-|u|,0)$, the side-specific criterion is $$ \sum_{i:\,X_i\text{ on the side},\,|X_i|\leq h}K(X_i/h)(Y_i-a-bX_i)^2. $$ The fitted intercept is the estimated conditional mean at the cutoff for that side. The local linear RD estimate is the right intercept minus the left intercept.

The bandwidth determines which observations can contribute to the local fit, while the kernel determines how they are weighted within that window. A narrow bandwidth emphasizes observations very close to the threshold but uses fewer data; a wider bandwidth uses more data but may rely more heavily on the assumed local linear form farther from the cutoff. A data-driven selector can provide a principled starting value, but the choice still needs sensitivity assessment.

Worked Example: Fitting the Local Lines

Consider a hypothetical dataset of 1,240 students from 80 schools. A data-driven bandwidth selector returns $h=8$ score points. The main local fit uses scores from 62 up to but not including 70 on the ineligible side, and scores from 70 through 78 on the eligible side. Thus the exact cutoff belongs to the eligible side under the rule. With triangular weights, observations at scores 62 and 78 have weight zero, while observations closer to 70 receive greater weight.

Suppose the two weighted local linear fits, with $X=S-70$, are $$ \widehat{E}[Y\mid X]=68.4+0.82X\quad(X<0), \qquad \widehat{E}[Y\mid X]=74.1+0.55X\quad(X\geq 0). $$

At the cutoff, $X=0$, so the left fitted value is $68.4+0.82(0)=68.4$ points and the right fitted value is $74.1+0.55(0)=74.1$ points. The estimated discontinuity is $$ \widehat{\tau}_{\mathrm{SRD}}=74.1-68.4=5.7\text{ points}. $$

The local linear estimate is therefore a 5.7-point higher fitted outcome immediately above the threshold than immediately below it. This numerical difference is an estimated causal effect at the cutoff only if the RD identification assumptions are credible; the regression calculation alone does not establish those assumptions.

Bandwidth Choice and Sensitivity

The bandwidth is consequential because the fitted jump can change when observations farther from the cutoff are included. A narrow window can have substantial sampling variability; a wide window can introduce approximation bias if the conditional mean curves bend in ways that separate local lines do not capture. Data-driven bandwidth selection balances these concerns under a specified criterion. It does not select a universally correct window or eliminate the need to examine sensitivity.

For the hypothetical analysis, suppose the selector chooses $h=8$. The analyst also reports estimates using nearby bandwidths of 6 and 10 points. If those estimates are respectively 5.4 and 5.9 points, the estimates are in the same general range, although the differences should be reported rather than hidden. This comparison does not replace the primary selection rule; it shows how dependent the substantive conclusion is on reasonable nearby choices.

Specification sensitivity can also include checking whether conclusions depend on a particular polynomial order or on a small set of observations. In RD work, high-order global polynomials can behave erratically near boundaries, so the local linear specification is a more defensible starting point. Here the key comparison remains the estimated limits at the cutoff, not a global curve across the entire score range.

Worked Example: Interpreting the Bandwidth and Kernel Weights

With $h=8$, a student whose centered score is $X=-4$ is four points below the threshold. The triangular weight is $$ K(-4/8)=1-|{-4/8}|=1-0.5=0.5. $$ A student at $X=-2$ has weight $K(-2/8)=1-0.25=0.75$. An observation at $X=-8$ has weight $K(-8/8)=0$, as does an observation at $X=8$. The fit can use the window endpoints as part of its stated range, but observations exactly at those endpoints contribute zero weight under this kernel.

If the bandwidth is reduced to 6, observations with $|X|>6$ are no longer in the window, and the remaining observations are relatively more concentrated near the cutoff. If it is increased to 10, more distant observations can contribute. Thus bandwidth sensitivity asks whether the estimated cutoff jump is materially altered as this local comparison is changed; it is not a test that determines whether one bandwidth is true.

Diagnostics and Robust Inference

A credible sharp RD analysis examines whether the score distribution suggests sorting at the threshold, whether predetermined covariates appear continuous there, and whether placebo cutoffs or outcomes show unexplained jumps. These checks are supporting evidence, not proof of continuity. A test with a large p-value does not establish that there is no sorting or covariate discontinuity; it means that the diagnostic did not detect one at its chosen precision.

Inference should account for the fact that local polynomial estimates can have smoothing bias. Robust bias-corrected inference estimates and adjusts for leading bias, then uses a standard error designed for that correction. If students are grouped in schools and outcomes may be correlated within school, uncertainty should also account for clustering at the school level. The standard error and confidence interval should match the dependence structure and the bias-correction procedure used.

Worked Example: Diagnostics and Robust Bias-Corrected Inference

Suppose the hypothetical analysis reports a score-density test p-value of 0.48, no detected jumps in predetermined covariates (for example, the reported age difference at the cutoff is 0.02 years with p-value 0.74), and a placebo-cutoff estimate of 0.4 examination points at score 66 with p-value 0.81. These results do not show evidence of the particular sorting, covariate, or placebo discontinuities examined. They do not prove that the continuity assumption holds, and the researcher should also consider institutional knowledge about score manipulation and other changes at the eligibility threshold.

Suppose robust bias-corrected, school-clustered inference reports an adjusted estimate of 5.5 points with $SE=1.9$ points. The approximate test statistic for a zero cutoff effect is $$ z=\frac{5.5}{1.9}=2.895. $$

A two-sided standard normal p-value for $z=2.895$ is approximately 0.0038. Under the null hypothesis of no cutoff effect and the inference assumptions, a test statistic at least this extreme in absolute value would be unusual. The p-value is not the probability that the null hypothesis is true.

The corresponding approximate 95% confidence interval is $$ 5.5\pm 1.96(1.9)=5.5\pm 3.724=[1.776,\ 9.224]\text{ points}. $$

Rounded to a sensible precision, the interval is about 1.8 to 9.2 examination points. In repeated applications of this interval procedure under its assumptions, 95% of intervals would cover the cutoff effect. It is not a statement that this particular interval has a 95% probability of containing the fixed effect. The result supports a positive local effect in this hypothetical analysis, conditional on the design assumptions and the validity of the inference procedure.

Why the Continuity Argument Identifies a Causal Effect

The core identification argument is short but important. Let $Y(1)$ and $Y(0)$ denote the potential outcomes under tutoring and no tutoring. Under sharp assignment, students just above the cutoff have observed outcome $Y(1)$, while students just below have observed outcome $Y(0)$. If the conditional means of both potential outcomes are continuous at the cutoff, taking the limits from either side recovers the two potential-outcome means at zero.

Justification: For $x\geq 0$, $Y=Y(1)$, and for $x<0$, $Y=Y(0)$. Therefore, $$ \lim_{x\to 0^+}E[Y\mid X=x]=\lim_{x\to 0^+}E[Y(1)\mid X=x]=E[Y(1)\mid X=0], $$ and $$ \lim_{x\to 0^-}E[Y\mid X=x]=\lim_{x\to 0^-}E[Y(0)\mid X=x]=E[Y(0)\mid X=0], $$ where the equalities to the cutoff means follow from continuity. Subtracting gives $$ \tau_{\mathrm{SRD}}=E[Y(1)-Y(0)\mid X=0]. $$ Thus the discontinuity identifies the average treatment effect at the cutoff, not an effect for the full score distribution.

Common Mistakes

Including the cutoff on both sides. With $D=1[X\geq 0]$, $X=0$ is assigned to the treated side. The ineligible side is $X<0$, not $X\leq 0$.
Calling the full-group difference the RD effect. Comparing all eligible with all ineligible students mixes the cutoff contrast with differences across the score distribution. The RD estimate compares fitted limits at the threshold.
Treating diagnostics as proof. A nonsignificant sorting or covariate test does not establish continuity. Interpret it as a lack of detected evidence, alongside design knowledge and other checks.
Overstating the result. A credible sharp RD estimate supports a local causal interpretation at the cutoff under its assumptions. It does not automatically establish an effect for students far from the threshold.
Reporting an ordinary standard error without regard to design. Bias correction and clustering may matter. The reported interval should be based on an inferential method aligned with the estimator and data dependence.

Key takeaway: In a sharp RD, the deterministic assignment rule creates a treatment jump at the cutoff. Separate local fits estimate the outcome limits on either side; continuity of untreated and treated potential-outcome means gives the jump a local causal interpretation. Bandwidth choices, diagnostics, and robust inference are essential parts of the analysis, not substitutes for the identifying assumptions.

Check Your Understanding

Use the test-score tutoring example to answer the following questions.

If eligibility is $D=1[S\geq 70]$, which side of the cutoff contains a student scoring exactly 70?
Why is the intercept difference from separate local linear fits the estimated cutoff effect?
With triangular weights and $h=8$, what weight does an observation at $X=2$ receive?
What does a nonsignificant density test establish, and what does it not establish?
Why does the RD estimate concern students at the cutoff rather than all students eligible for tutoring?
In the worked inference example, how should the 95% confidence interval be interpreted in context?