Regression Discontinuity Analysis With Real Data

From Assignment Rule to a Defensible Analysis

A regression-discontinuity analysis is not just a comparison of observations above and below a threshold. Its credibility depends on a clearly specified assignment rule, an outcome that changes smoothly in the absence of treatment, and careful attention to the observations near the cutoff. This tutorial works through those decisions for sharp and fuzzy designs using hypothetical, data-like examples.

Let $X_i$ be unit $i$'s running variable, $c$ the cutoff, $R_i=X_i-c$ the centered running variable, $D_i$ treatment receipt, and $Y_i$ the outcome. The indicator $Z_i=1\{X_i\geq c\}$ records which side of the cutoff a unit falls on. A regression-discontinuity estimate concerns a local change at $R_i=0$, not generally an average effect for all units in the sample.

Definition: In a sharp regression-discontinuity design, treatment is assigned deterministically by the cutoff: $D_i=Z_i$. The cutoff effect is the difference between the right-hand and left-hand limits of the conditional mean outcome at $c$. In a fuzzy regression-discontinuity design, treatment receipt is not determined perfectly by $Z_i$, but its probability changes at the cutoff. The ratio of the outcome discontinuity to the treatment-probability discontinuity is a local Wald estimand.

The causal interpretation requires more than observing a jump. In a sharp design, the conditional means of the potential outcomes without and with treatment must be continuous at the cutoff. In a fuzzy design, the corresponding potential outcomes and potential treatment probabilities must be continuous there; the cutoff must affect the outcome only through treatment receipt (exclusion); and assignment at the cutoff must not make anyone less likely to receive treatment (monotonicity). Both designs also require a credible assignment process around the threshold: units must not precisely sort to one side in a way that makes their potential outcomes discontinuous at the cutoff.

Sharp Design: Estimate the Two Limits

Within a chosen bandwidth $h$, a common estimator fits a separate local linear regression on each side of the cutoff. The right- and left-side intercepts estimate the respective conditional mean limits at $R_i=0$; their difference estimates the discontinuity. The analysis is local: observations farther than $h$ from the threshold do not enter these fits.

Y_i=\alpha_-+\beta_-R_i+\varepsilon_i \quad ( -h\leq R_i<0), \qquad Y_i=\alpha_++\beta_+R_i+\varepsilon_i \quad (0\leq R_i\leq h), \qquad \hat{\tau}_{\mathrm{sharp}}=\hat{\alpha}_+-\hat{\alpha}_- .

The slopes allow the outcome to trend with the running variable on each side rather than treating every observation in the bandwidth as interchangeable. In applications, local polynomial fits commonly weight observations by distance from the cutoff. The chosen polynomial order, kernel, bandwidth, and inference method should be reported. Robust bias-corrected inference is designed to account for leading smoothing bias; a conventional standard error alone does not correct that bias.

Worked Example: A Sharp Eligibility Cutoff

Hypothetical data. A program offers a service to applicants scoring at least 70 on an assessment. The running variable is assessment score, $c=70$, and participation follows the rule exactly. The outcome is an annual wellbeing score, measured in points. For a primary bandwidth of 8 points, separate unweighted local linear fits use 80 observations on each side. Their centered-regressor summaries are:

Below the cutoff: $n_-=80$, $\sum R_i=-160$, $\sum R_i^2=640$, $\sum Y_i=3920$, and $\sum R_iY_i=-7680$. Above the cutoff: $n_+=80$, $\sum R_i=160$, $\sum R_i^2=640$, $\sum Y_i=4320$, and $\sum R_iY_i=8720$.

For a regression with an intercept, the fitted intercept can be calculated from the sufficient sums as $ \hat{\alpha}=[(\sum Y_i)(\sum R_i^2)-(\sum R_i)(\sum R_iY_i)]/[n\sum R_i^2-(\sum R_i)^2] $. Below the cutoff, substitution gives $ \hat{\alpha}_-=[3920(640)-(-160)(-7680)]/[80(640)-(-160)^2]=1{,}280{,}000/25{,}600=50.00 $. Above the cutoff it gives $ \hat{\alpha}_+=[4320(640)-(160)(8720)]/[80(640)-(160)^2]=1{,}369{,}600/25{,}600=53.50 $. Thus the estimated discontinuity is $53.50-50.00=3.50$ wellbeing points.

Suppose the bias-corrected analysis of the individual observations reports an estimate of 3.42 points, a robust standard error of 1.08 points, and a 95% confidence interval of 1.30 to 5.54 points. The summary calculations above reproduce the uncorrected local-linear intercept difference; the bias-corrected result comes from the additional bias adjustment. The interval is compatible with a positive local effect under the design assumptions. It does not show that every eligible applicant benefits by 3.42 points, or that the effect applies to applicants far from the cutoff.

The treatment rule is sharp by construction in this example, but that alone does not establish the continuity assumptions. Before making a causal claim, the analyst must examine whether the score distribution or predetermined characteristics show suspicious changes at 70, and whether the result is sensitive to reasonable analysis choices.

Fuzzy Design: Scale the Outcome Jump by the First Stage

In a fuzzy design, some units below the cutoff receive treatment and some above it do not. The discontinuity in $D_i$ is the first stage: the change in treatment probability at the threshold. The outcome discontinuity is the reduced-form effect of crossing the threshold. Their ratio expresses the outcome jump per unit of cutoff-induced change in treatment.

\hat{\tau}_{\mathrm{fuzzy}} = \frac{\widehat{\lim}_{r\downarrow 0}E[Y_i\mid R_i=r] -\widehat{\lim}_{r\uparrow 0}E[Y_i\mid R_i=r]} {\widehat{\lim}_{r\downarrow 0}E[D_i\mid R_i=r] -\widehat{\lim}_{r\uparrow 0}E[D_i\mid R_i=r]} = \frac{\widehat{\Delta}_Y}{\widehat{\Delta}_D}.

Here $r\downarrow 0$ denotes approaching zero from the right and $r\uparrow 0$ approaching from the left. Under continuity, exclusion, and monotonicity, with a nonzero first stage, the population ratio identifies the average treatment effect for units near the cutoff whose treatment status is changed by crossing it—the local average treatment effect for compliers. It is not automatically the average treatment effect for all units.

A short justification follows from the assignment contrast. Under exclusion, crossing the cutoff affects $Y_i$ through its effect on $D_i$, so the outcome discontinuity is the treatment effect among cutoff-induced compliers multiplied by the change in the share treated. The treatment-probability discontinuity is that change in the share treated. Dividing cancels the share:

\Delta_Y=\Delta_D\,\tau_{\mathrm{complier}} \quad\Longrightarrow\quad \tau_{\mathrm{complier}}=\frac{\Delta_Y}{\Delta_D}, \qquad \Delta_D\ne 0.

This ratio also makes first-stage strength important. When the treatment-probability jump is small, dividing by it can amplify sampling uncertainty. Inference should account for uncertainty in both discontinuities; a standard error that treats the estimated first stage as fixed can be misleading. Weak-first-stage-robust procedures, such as inversion of an Anderson–Rubin test, can yield wide or unbounded confidence sets when the data do not identify the ratio precisely.

Worked Example: A Fuzzy Training Threshold

Hypothetical data. A training program uses a score threshold of 60 to encourage enrollment, but enrollment is voluntary. The outcome is annual earnings in thousands of dollars. Local linear fits within the selected bandwidth estimate earnings limits of 31.2 below and 32.8 above the cutoff, so the reduced-form discontinuity is $32.8-31.2=1.60$ thousand dollars per year. The fitted enrollment probabilities are 0.32 below and 0.72 above, so the first stage is $0.72-0.32=0.40$.

The local Wald estimate is $ \hat{\tau}_{\mathrm{fuzzy}}=1.60/0.40=4.00 $. Its units are thousands of dollars per year per treatment, so the point estimate is an increase of $4,000 in annual earnings for cutoff-induced compliers. It is not a $4,000 effect for every person near the threshold.

Suppose a weak-first-stage-robust analysis reports a 95% confidence set from $0.7 thousand to $10.8 thousand per year. The interval is wide: it communicates that the local effect is imprecisely estimated, even though the point estimate is $4,000. The causal interpretation still depends on the fuzzy-design assumptions. In particular, if crossing the score threshold also changes access to another service that affects earnings, exclusion would fail and the Wald ratio would not isolate the effect of training enrollment.

Bandwidth, Diagnostics, and Sensitivity

The bandwidth controls a central trade-off. A narrow window compares units closer to the cutoff, where local approximations may be more credible, but uses fewer observations and can increase uncertainty. A wider window improves precision by adding observations but can make polynomial approximation bias more consequential. Bandwidth selection should use a principled method appropriate to the estimand and inference goal, rather than selecting whichever window produces the most favorable result. Report the primary bandwidth and show how conclusions change across reasonable alternatives.

Diagnostics address threats to the identifying assumptions, but none proves them. A density or sorting test can detect evidence that the running variable's distribution changes at the cutoff; failure to reject does not prove that precise manipulation is absent. Predetermined covariates should not display unexplained discontinuities. Placebo cutoffs and outcomes that treatment should not affect can expose suspicious patterns. These checks inform credibility rather than replace the design argument.

Worked Example: Reading Bandwidth and Manipulation Checks Together

Hypothetical follow-up analysis. For the sharp eligibility example, an analyst reports the following bias-corrected estimates and 95% confidence intervals, all in wellbeing-score points:

Bandwidth in score points	Estimate	95% confidence interval
5	3.10	0.40 to 5.80
8 (primary)	3.42	1.30 to 5.54
12	2.70	0.20 to 5.20

All three estimates are positive, but they vary from 2.70 to 3.42 points and the intervals differ in width. This pattern supports describing the estimated local discontinuity as positive across these specifications, while acknowledging that its magnitude is sensitive to bandwidth. It is not valid to choose the 8-point bandwidth because its interval happens to exclude zero; the primary bandwidth should be selected by the analysis plan or an appropriate selection procedure.

Suppose a density test at the cutoff gives $p=0.41$, and a joint test of predetermined covariates gives $p=0.28$. These results do not reject the respective no-discontinuity null hypotheses at a 5% significance level. They are reassuring in a limited sense, but neither p-value is the probability that manipulation or imbalance is absent. The analyst should also inspect the distributions and covariate estimates, consider the institutional process that generates the score, and report any plausible opportunity for sorting.

Reporting the Analysis and Its Limits

A transparent report makes the chain of reasoning auditable. State the running variable, the cutoff, the treatment assignment rule, the outcome and its units, and whether the design is sharp or fuzzy. Describe the sample restriction, bandwidth, local polynomial specification, weighting, and inference procedure. For a fuzzy design, report both the outcome discontinuity and first stage alongside their ratio. A ratio without its components hides whether a large estimate reflects a large outcome jump, a weak denominator, or both.

Present diagnostic evidence and sensitivity results with the same care as the main estimate. Explain what a density test can and cannot establish, note covariate and placebo patterns, and show the effect of reasonable bandwidth or specification changes. If results change materially, that is substantive uncertainty, not a reason to suppress alternative estimates. Finally, phrase the conclusion as local and conditional: the estimate concerns units near the cutoff and has a causal interpretation only to the extent that the design assumptions are credible.

Common Mistakes

Calling the fuzzy ratio an average effect for everyone. Under the required assumptions, it is a local effect for cutoff-induced compliers, not a population-wide average.
Equating a nonsignificant manipulation test with proof of no sorting. A non-rejection is not proof; consider the institutional setting and other diagnostics as well.
Reporting only the Wald ratio. Always show the outcome jump and treatment-probability jump; a weak first stage can make the ratio unstable.
Choosing a bandwidth because it gives statistical significance. Use a defensible selection approach and report sensitivity to reasonable alternatives.
Interpreting a local estimate as a universal treatment effect. The design identifies an effect at the cutoff under its assumptions, not necessarily far from it.

Key takeaway: A credible regression-discontinuity analysis joins a clear assignment rule to local estimation, defensible inference, and diagnostic evidence. The numerical discontinuity is only the beginning; its causal interpretation depends on the design assumptions and on reporting how the result responds to reasonable analysis choices.

Check Your Understanding

Answer each question using the definitions and reasoning in this tutorial.

In a sharp design, what quantity does the difference between the two fitted local-linear intercepts estimate?
A fuzzy design has an outcome jump of 1.2 points and a treatment-probability jump of 0.3. What is the Wald estimate, and what additional assumptions are needed for a local causal interpretation?
Why can a narrow bandwidth increase uncertainty even as it focuses attention closer to the cutoff?
What does a density-test p-value of 0.41 establish, and what does it not establish?
Why should a fuzzy-design report include the first-stage estimate as well as the Wald ratio?