From an Eligibility Rule to a Cutoff Effect
A sharp regression discontinuity (RD) design uses a rule that assigns treatment deterministically at a threshold of a running variable. For example, a program might be offered to applicants whose score meets or exceeds a fixed eligibility score. A complete analysis asks more than whether outcomes differ on the two sides: it must assess whether the rule and the data support a causal comparison at the cutoff, estimate the discontinuity, and describe its uncertainty and limitations.
Let \(R_i\) be unit \(i\)'s running variable, \(c\) the cutoff, \(D_i\) its treatment status, and \(Y_i\) its observed outcome. In a sharp design, treatment follows the rule exactly: \(D_i=1\{R_i\ge c\}\). Write \(Y_i(1)\) and \(Y_i(0)\) for the unit's potential outcomes under treatment and no treatment, respectively. The target is the average treatment effect at the cutoff, not an average effect for all units.
The identification argument requires the treatment rule to switch from untreated to treated at \(c\), and the potential-outcome conditional means \(E[Y_i(0)\mid R_i=r]\) and \(E[Y_i(1)\mid R_i=r]\) to be continuous at \(c\). There must be observations with running-variable values on both sides close enough to the cutoff to learn the two limits. The interpretation also relies on units not being able to sort precisely around the cutoff in a way that disrupts comparability, and on treatment and outcome definitions being consistent. These conditions make the outcome trend that would have prevailed without the treatment smooth through the threshold; the treatment assignment is the source of the jump.
Continuity is an assumption about potential outcomes, so it cannot be established directly from observed data. Plots, density checks, predetermined covariates, and placebo analyses can reveal reasons for concern, but passing them does not prove the assumption. The earlier material on continuity identifies the cutoff effect and on sorting and covariate checks provides the relevant diagnostic ideas; here they are combined into an analysis workflow.
Start With the Rule, the Plot, and the Checks
Before estimating, verify the assignment rule from the program's actual operation: which side is eligible, whether eligibility translates into treatment for every unit, and how equality at the cutoff is handled. Plot the outcome against \(R_i-c\), marking zero as the threshold. The plot should show the data as well as any binned means and fitted curves. A visible jump can motivate estimation, but bin widths and plotting scales can make a jump look larger or smaller; the plot is a diagnostic, not the estimator.
Next, examine whether the design is credible close to the threshold. Check whether the running-variable density shows an unusual discontinuity consistent with precise sorting, whether predetermined covariates appear continuous, and whether outcomes or covariates show jumps at placebo cutoffs where no treatment rule changes. Also inspect support: there should be enough observations close to zero on both sides, not merely a large overall sample. These checks have limited power, so an apparently smooth graph or non-significant test is not proof of continuity.
Worked Example: Reviewing a Hypothetical Eligibility Threshold
Suppose a hypothetical training program is automatically assigned to applicants with an assessment score of at least 70. The running variable is \(R_i=\text{score}_i\), the cutoff is \(c=70\), and the outcome is days employed during the following year. The administrator confirms that applicants scoring 70 or higher all receive the program and those below 70 do not. Thus the rule is sharp, provided records accurately reflect treatment receipt.
An initial plot shows a gradual increase in employment on each side and a possible upward jump at 70. There are observations within a few score points on both sides. This supports proceeding to estimation, but it does not establish that the potential-outcome means would be continuous at 70.
A hypothetical predetermined covariate, prior-year employment days, has estimated local means of 180 days just below 70 and 182 days just above. The difference is \(182-180=2\) days. That small difference is compatible with continuity, but its uncertainty and the pattern across nearby bandwidths matter; it is not proof. A density plot also shows no conspicuous pile-up above 70, and placebo cutoffs at 65 and 75 show no comparable outcome jumps. Together these diagnostics do not reveal an obvious failure. The causal interpretation still depends on the continuity and no-precise-sorting assumptions.
Local Linear Estimation at the Cutoff
A common estimator fits separate weighted straight lines on the two sides of the cutoff, using observations within a chosen bandwidth \(h\). Define \(X_i=R_i-c\). For a kernel function \(K\), estimate the left and right lines by weighted least squares, with weights \(K(X_i/h)\), fitting only observations for which \(-h\le X_i<0\) on the left and \(0\le X_i\le h\) on the right. A triangular kernel, \(K(u)=1-|u|\) for \(|u|\le1\), gives more weight to observations nearer the cutoff. The estimated cutoff effect is the right-side intercept minus the left-side intercept: $$ \hat{\tau}(c)=\hat{\beta}_{+,0}-\hat{\beta}_{-,0}. $$
The intercepts are predictions at \(X_i=0\); they are not generally observed outcome means. Local linear fits allow the outcome trend to differ on either side, while focusing the estimate on the threshold. The intercept difference estimates the discontinuity if the fitted lines approximate the conditional mean functions well near zero. The derivation is direct: each side's local line estimates its conditional mean limit at \(X_i=0\); subtracting those two predictions estimates the difference between the right and left limits.
Worked Example: Calculating a Local Linear Cutoff Estimate
For a transparent arithmetic illustration, suppose a hypothetical data set has the following outcome values at distances \(X_i\) from the cutoff. The outcome is days employed. Use bandwidth \(h=4\) and triangular weights, so the weights at distances 1, 2, and 3 are \(0.75\), \(0.50\), and \(0.25\), respectively. These few values illustrate the calculation; they are not enough to support reliable inference by themselves.
| Side | \(X_i\) | Days employed \(Y_i\) | Weight |
|---|---|---|---|
| Left | -1 | 10 | 0.75 |
| Left | -2 | 8 | 0.50 |
| Left | -3 | 7 | 0.25 |
| Right | 1 | 16 | 0.75 |
| Right | 2 | 17 | 0.50 |
| Right | 3 | 19 | 0.25 |
For weighted regression of \(Y_i\) on an intercept and \(X_i\), the intercept is \((S_{yy}S_{xx}-S_xS_{xy})/(S_wS_{xx}-S_x^2)\), where \(S_w=\sum w_i\), \(S_x=\sum w_iX_i\), \(S_{yy}=\sum w_iY_i\), \(S_{xx}=\sum w_iX_i^2\), and \(S_{xy}=\sum w_iX_iY_i\). For the left side, these sums are \(S_w=1.50\), \(S_x=-2.50\), \(S_{yy}=13.25\), \(S_{xx}=5.00\), and \(S_{xy}=-20.75\). Therefore, $$ \hat{\beta}_{-,0} =\frac{(13.25)(5.00)-(-2.50)(-20.75)} {(1.50)(5.00)-(-2.50)^2} =\frac{14.375}{1.25} =11.50\text{ days}. $$
On the right, the sums are \(S_w=1.50\), \(S_x=2.50\), \(S_{yy}=25.25\), \(S_{xx}=5.00\), and \(S_{xy}=43.25\). Thus, $$ \hat{\beta}_{+,0} =\frac{(25.25)(5.00)-(2.50)(43.25)} {(1.50)(5.00)-(2.50)^2} =\frac{18.125}{1.25} =14.50\text{ days}. $$
The estimated discontinuity is \(14.50-11.50=3.00\) days. In this hypothetical calculation, the fitted right-side outcome at the cutoff is 3.00 days higher than the fitted left-side outcome. Under the design assumptions, this estimates the local causal effect of program assignment at the threshold. The calculation alone does not establish those assumptions or quantify uncertainty.
Bandwidth, Inference, and Sensitivity
Bandwidth \(h\) determines how far from the threshold observations contribute. A smaller bandwidth generally reduces approximation bias from fitting a line to a curved outcome function, but it uses fewer observations and can increase sampling variability. A larger bandwidth improves precision by including more observations, but the local linear approximation may become less credible farther from the cutoff. Bandwidth selection is therefore a bias-variance tradeoff, not a search for the one value that produces a preferred result.
Use a principled data-based bandwidth choice as a main specification, then assess sensitivity to reasonable narrower and wider choices. Report the corresponding sample sizes, estimates, and uncertainty intervals. A stable estimate across choices is reassuring, but it does not verify continuity; a large change can indicate curvature, sparse support, influential observations, or sensitivity to modeling choices. The earlier material on bandwidth choice and support diagnostics treats these issues in detail. Inference should account for the local estimation and possible smoothing bias; when appropriate, use robust bias-corrected RD intervals rather than treating an uncorrected conventional interval as definitive.
Worked Example: Interpreting Bandwidth Sensitivity
Suppose a hypothetical analysis of the employment outcome chooses \(h=4\) as its main bandwidth and obtains a cutoff estimate of 3.0 days. As a sensitivity check, it reports estimates of 2.7 days at \(h=3\) and 2.4 days at \(h=5\). The estimates differ by \(3.0-2.4=0.6\) days across the narrow and wide specifications. That is modest relative to the estimates, so the point-estimate pattern is reasonably stable over these choices.
This comparison does not say that the effect is statistically distinguishable from zero: no uncertainty estimates have been supplied. Nor does stability prove the assumptions. A useful report would also show observations on each side at each bandwidth and the intervals from an appropriate inference procedure. If the wider bandwidth had instead produced a much larger estimate, the analyst should investigate curvature, influential distant observations, and support rather than simply selecting the bandwidth that gives the desired result.
Worked Example: Interpreting an Uncertainty Interval
Suppose a separate hypothetical analysis, using an adequate sample and a suitable robust bias-corrected procedure, reports a cutoff estimate of 3.0 days and a standard error of 1.2 days. For a simple normal-reference illustration, a 95% interval is $$ 3.0\pm1.96(1.2)=3.0\pm2.352=[0.648,\;5.352]\text{ days}. $$
Rounded to two decimal places, the interval is 0.65 to 5.35 additional employment days at the cutoff. Under the procedure's repeated-sampling assumptions, intervals formed this way would cover the target effect in about 95% of repeated samples. The interval excludes zero, but its width shows substantial uncertainty about the magnitude. It does not mean there is a 95% probability that this fixed effect lies in this particular interval, and it does not address a violation of the RD identification assumptions.
Common Pitfalls and Limits of the Conclusion
- Calling a visual jump causal by itself. A plot can display a discontinuity, but identification requires the assignment rule and continuity assumptions, not just a visible difference.
- Treating diagnostic non-significance as proof. A non-significant sorting, covariate, or placebo test may reflect limited power. Evaluate plots, estimates, uncertainty, and institutional knowledge together.
- Choosing bandwidth after seeing which result is significant. This makes the reported result depend on outcome-driven specification search. Use a justified primary choice and transparent sensitivity analysis.
- Confusing a sharp rule with an imperfect threshold. The sharp design requires deterministic treatment assignment at the cutoff. If some eligible units do not receive treatment or some ineligible units do, this setup does not apply as stated.
Even a credible sharp RD estimate is local: it describes the treatment effect at, or for units close to, the cutoff under the stated assumptions. It need not generalize to applicants with much higher or lower scores, or to a different program or threshold. A careful conclusion states the outcome, units, cutoff population, estimate, uncertainty, and key qualification. For example: “Among applicants near a score of 70, program assignment is estimated to increase days employed during the following year by about 3 days, subject to the continuity and no-precise-sorting assumptions.”
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- A policy assigns treatment to everyone with \(R_i\ge c\). What makes this a sharp rather than an imperfect threshold design?
- Why does a local linear estimate subtract two intercepts instead of comparing the raw average outcomes on either side?
- A covariate continuity test is non-significant. What can and cannot be concluded?
- A 95% interval for a cutoff effect in minutes is 1.2 to 4.8. Give a correct interpretation and state one conclusion it does not justify.
- What is the main tradeoff when the analysis bandwidth is widened?
Solutions: (1) Treatment status must follow the threshold rule deterministically: no units below receive treatment and all units at or above receive it. (2) The intercepts estimate the two conditional mean limits at the cutoff, adjusting for local trends; raw means can differ because observations lie at different distances from the threshold. (3) The test did not detect a discontinuity, but that is not proof of continuity; power and other diagnostics matter. (4) Under the interval procedure's assumptions, the data are compatible with cutoff effects from 1.2 to 4.8 additional minutes, and repeated intervals formed this way would cover the target about 95% of the time. It does not establish that the effect applies far from the cutoff. (5) A wider bandwidth adds observations and can improve precision, but it can increase bias if the outcome trend is no longer well approximated locally.