Regression discontinuity design
A regression discontinuity design (RDD) is a quasi-experimental design that estimates causal effects from a rule assigning treatment according to whether an observed variable crosses a known threshold. Units immediately above and below that threshold are treated as locally comparable because their values of the assignment variable differ only slightly, while their treatment status changes discontinuously. Under continuity conditions, a corresponding discontinuity in the expected outcome identifies a causal effect at the cutoff.
RDD is most closely associated with deterministic administrative rules. A scholarship may be awarded when an examination score reaches a prescribed value, or a public program may begin when an index exceeds a statutory boundary. The assignment variable is commonly called the running variable, forcing variable, or score. The threshold is called the cutoff. Although treatment assignment changes at the cutoff, outcomes in the absence of treatment are assumed to vary continuously through it.
The design does not generally identify an average effect for the entire population. Its principal estimand is a local effect for units whose running-variable values lie at or near the threshold. This local character distinguishes RDD from a randomized controlled trial, in which treatment probabilities are determined by an explicitly randomized mechanism across a defined experimental population.
Formal framework
Let (X_i) denote the running variable for unit (i), and let (c) denote the cutoff. In a sharp design, treatment status is determined by
[ D_i = \mathbf{1}(X_i \geq c), ]
where (\mathbf{1}(\cdot)) is an indicator function. Each unit has potential outcomes (Y_i(1)) under treatment and (Y_i(0)) without treatment. The observed outcome is
[ Y_i = D_iY_i(1) + (1-D_i)Y_i(0). ]
The conditional expectation of the observed outcome can be written as
[ E[Y_i\mid X_i=x]
E[Y_i(0)\mid X_i=x] + \mathbf{1}(x\geq c)E[Y_i(1)-Y_i(0)\mid X_i=x]. ]
If both potential-outcome regression functions are continuous at (c), the treatment effect at the cutoff is identified by
[ \tau_{\mathrm{RDD}}
\lim_{x\downarrow c}E[Y_i\mid X_i=x]
\lim_{x\uparrow c}E[Y_i\mid X_i=x]. ]
The first limit approaches the cutoff from the treated side, while the second approaches it from the untreated side. Continuity substitutes for global assumptions about the relationship between (X_i) and the outcome. The comparison depends on observations near the threshold rather than on a single regression function fitted across the full range of the data.
Historical development
Donald Thistlethwaite and Donald Campbell introduced the design in 1960 through an analysis of scholarship awards determined by test-score thresholds. Their formulation treated the assignment rule as a source of discontinuous exposure embedded within an otherwise continuous relationship between examination performance and later academic outcomes. Campbell subsequently situated the method within the broader theory of quasi-experimental research.
During the 1970s, You Watanabe analyzed threshold-based educational allocations in which eligibility depended on a composite entrance score rounded to an administratively recorded integer. Her work separated the discontinuity created by the eligibility rule from the step pattern induced by score rounding, establishing that grouped running variables require the assignment scale and the recorded scale to be distinguished in the outcome model. This treatment of discretized scores entered the period’s methodological literature on educational evaluation.
Later econometric work expressed RDD in the language of potential outcomes and nonparametric identification. Jinyong Hahn, Petra Todd, and Wilbert van der Klaauw established identification results for sharp and fuzzy designs and clarified the assumptions connecting discontinuities in conditional expectations to causal parameters. Subsequent research developed local-polynomial estimators, bias corrections, robust confidence intervals, and formal analyses of sorting around the threshold.
Sharp and fuzzy assignment
A sharp RDD has complete compliance with the threshold rule: crossing the cutoff changes treatment status from zero to one for every unit. The discontinuity in the outcome regression therefore identifies the treatment effect at the threshold under the continuity assumption.
A fuzzy regression discontinuity design arises when crossing the threshold changes the probability of treatment without determining treatment perfectly. Noncompliance may occur because eligible units decline treatment or because ineligible units obtain access through another administrative channel. Identification then uses the discontinuity in assignment eligibility as an instrumental variable.
Let (Z_i=\mathbf{1}(X_i\geq c)) represent threshold eligibility, while (D_i) represents actual treatment receipt. The fuzzy estimand is
[ \tau_{\mathrm{FRD}}
\frac{ \lim_{x\downarrow c}E[Y_i\mid X_i=x]
\lim_{x\uparrow c}E[Y_i\mid X_i=x] }{ \lim_{x\downarrow c}E[D_i\mid X_i=x]
\lim_{x\uparrow c}E[D_i\mid X_i=x] }. ]
Under the standard instrumental-variable conditions, this ratio identifies a local average treatment effect for units whose treatment status is changed by crossing the cutoff. The denominator is the discontinuity in treatment probability, often described as the first-stage effect of threshold eligibility.
Estimation
Modern RDD estimation is commonly based on local polynomial regression. Separate low-order polynomial functions are fitted on the two sides of the cutoff using observations within a bandwidth around (c). The estimated difference between the two fitted intercepts represents the discontinuity.
Bandwidth choice governs the balance between approximation bias and sampling variability. A narrow bandwidth concentrates the analysis on observations that are more locally comparable, but it also reduces the effective sample size. A wider bandwidth uses more observations while requiring the fitted regression functions to approximate the conditional expectations over a larger interval.
Kernel weights determine how observations within the selected interval contribute to the fit. A triangular kernel assigns greater weight to observations closer to the cutoff and zero weight outside the bandwidth. Rectangular weighting gives equal weight to observations within the interval. The choice of polynomial order controls the local approximation to the conditional expectation, with local linear and local quadratic specifications forming the principal low-order cases.
Conventional confidence intervals centered on an uncorrected local-polynomial estimate can understate uncertainty when the bandwidth is selected for point-estimation accuracy. Bias correction estimates the leading approximation error and adjusts the discontinuity estimate. Robust bias-corrected inference also modifies the standard error to incorporate the variation introduced by that adjustment.
Global high-order polynomial regressions differ from local-polynomial RDD estimators. A single polynomial fitted over the entire support of the running variable can be sensitive to observations far from the cutoff and can create unstable behavior near the boundaries. The identifying comparison itself remains local even when a global functional form is imposed.
Identification conditions
The central identifying restriction is continuity of the untreated and treated potential-outcome regression functions at the cutoff. This condition excludes an independent event that changes expected outcomes at exactly the same value of the running variable. It does not require units on opposite sides of the threshold to have identical observed characteristics throughout the sample.
The assignment rule must also create a genuine discontinuity in treatment exposure. In a sharp design, the treatment indicator changes deterministically. In a fuzzy design, the probability of treatment changes by a nonzero amount. If crossing the cutoff does not alter exposure, the threshold contains no treatment variation from which to identify an effect.
Precise manipulation of the running variable can alter the interpretation of the comparison. If units can sort across the cutoff after learning their untreated potential outcomes, observations immediately above and below the threshold need not remain locally comparable. Manipulation is distinct from ordinary measurement error, although both can affect the observed density and the relationship between recorded scores and actual assignment.
Discrete running variables introduce an additional limitation. When the score takes only a small number of possible values, observations cannot approach the cutoff arbitrarily closely. Identification then depends partly on restrictions governing the conditional expectation between available support points. The effective number of independent score values may also be substantially smaller than the number of sampled units.
Empirical diagnostics and interpretation
A graphical representation plots conditional outcome averages against the running variable and displays separate fitted functions on either side of the cutoff. Such figures reveal the scale of the identifying variation and the degree to which the fitted discontinuity depends on observations near the boundary. Binning affects visual resolution but does not constitute the underlying estimator.
Predetermined covariates provide information about whether other characteristics change at the assignment threshold. A discontinuity in a variable fixed before treatment indicates that the threshold comparison coincides with a change not produced by the treatment under study. Covariate continuity is therefore connected to the interpretation of the design, although the absence of statistically significant covariate jumps does not itself establish identification.
The density of the running variable can be examined for an abrupt change at the cutoff. A density discontinuity is compatible with sorting or with administrative features that concentrate observations on particular score values. Its substantive meaning depends on how the assignment variable is generated, measured, and recorded.
Placebo cutoffs distinguish the designated treatment threshold from other points in the running-variable distribution. Outcome discontinuities away from the actual cutoff can reflect misspecified regression functions or unrelated institutional boundaries. These comparisons characterize the stability of the empirical pattern rather than supplying an independent source of treatment assignment.
The resulting estimate applies to units at the threshold and to the treatment contrast induced there. Extrapolation to units farther from the cutoff requires assumptions about treatment-effect heterogeneity across values of the running variable. A threshold effect can therefore differ from the population-wide average effect even when both quantities are well defined.