9 Regression Discontinuity Design
A regression discontinuity (RD) design is used when treatment assignment depends on whether an observed pre-treatment variable crosses a known threshold. This variable is usually called the running variable, forcing variable, or score.
The key idea is intuitive. Units just below and just above the threshold are likely to be very similar, except that units on one side receive the treatment while units on the other side do not. If this is true, a discontinuous jump in the post-treatment outcome at the threshold can be interpreted as the causal effect of the treatment.
For example, a public subsidy may be assigned only to firms whose eligibility score exceeds a given cutoff. Firms just above and just below the cutoff may be very similar in terms of size, productivity, sector, investment plans, and managerial capabilities. However, firms just above the cutoff receive the subsidy, while firms just below do not. The RD design compares these two groups locally around the cutoff.
RD design estimates causal effects by comparing the post-treatment outcomes of units that are very close to a treatment threshold but fall on opposite sides of it.
9.1 Historical Intuition
The RD design was introduced by Thistlethwaite and Campbell in their study of merit awards and subsequent academic outcomes (Thistlethwaite and Campbell 1960). The key institutional feature was that students received an award if their test score crossed a known cutoff. Students just above the cutoff received the award, while students just below the cutoff did not.
The intuition is that students very close to the cutoff are likely to be similar in both observed and unobserved characteristics. A student barely above the threshold and a student barely below it may differ only because one happened to fall on the treated side of the rule. Therefore, any discontinuous jump in later outcomes at the cutoff can be interpreted as evidence of the causal effect of receiving the award.
The Thistlethwaite and Campbell example captures the core logic of RD well. A simple comparison between all award recipients and all non-recipients would be misleading, because students with higher test scores are likely to differ systematically from students with lower scores. RD avoids this broad comparison and focuses only on units near the threshold, where treatment assignment changes sharply but underlying characteristics are expected to evolve smoothly.
9.2 The Basic Setup
Let \(S_i\) denote the running variable for unit \(i\), and let \(c\) denote the cutoff. Treatment status is denoted by \(D_i\). In a sharp RD design, treatment assignment is completely determined by the running variable:
\[ D_i = \begin{cases} 1 & \text{if } S_i \ge c,\\ 0 & \text{if } S_i < c. \end{cases} \]
In this case, crossing the threshold changes treatment status from 0 to 1 with probability one.
The parameter of interest is the treatment effect at the cutoff:
\[ \tau_{sharpRD} = \lim_{s \downarrow c} E[Y_i \mid S_i=s] - \lim_{s \uparrow c} E[Y_i \mid S_i=s]. \]
\(\tau_{sharpRD}\) identifies the average causal effect for units at, or very close to, the cutoff.
RD estimates are often highly credible near the cutoff. The price of this credibility is that the estimated effect may not generalize to units far from the threshold.
9.3 Identification
The central identifying assumption is that potential outcomes evolve smoothly through the cutoff. In other words, absent treatment, there should be no sudden change in the expected outcome exactly at the threshold.
Formally, the functions
\[ E[Y_i(0)\mid S_i=s] \quad \text{and} \quad E[Y_i(1)\mid S_i=s] \]
should be continuous at \(s=c\).
If this condition holds, any discontinuity in the observed post-treatment outcome at the cutoff can be attributed to the discontinuous change in treatment status.
A second key assumption is that units cannot precisely manipulate the running variable around the cutoff. If units can sort just above or below the threshold, then units on the two sides may no longer be comparable.
The concern in RD is not that units respond to incentives in general. The problem arises when units or decision-makers can precisely manipulate the running variable in order to place units on the preferred side of the cutoff.
For example, in the Italian university system, students usually pass an exam with a grade of at least 18 out of 30. Suppose that passing the exam is the cutoff that determines access to a subsequent opportunity. Manipulation would be a concern if professors systematically adjusted borderline grades, giving many students an 18 and almost no students a 17. In this case, students just below and just above the cutoff may no longer be comparable. The cutoff would not separate students who are almost identical except for treatment status; it would separate students whose position around the threshold may have been affected by precise manipulation of the running variable.
RD design is therefore credible only if units cannot precisely control, or have their position precisely controlled, around the cutoff.
9.4 Why RD Is Local
RD does not compare all treated and untreated units. It compares units near the cutoff.
This is important because units far from the cutoff may be very different. A student who obtains a very high exam grade may not be comparable to a student who clearly fails the exam. For example, a student receiving 28 out of 30 is likely to differ substantially from a student receiving 10 out of 30 in terms of preparation, ability, or effort.
Thus, the RD effect is usually interpreted as a local average treatment effect (LATE) at the cutoff:
\[ \tau_{sharpRD}(c). \]
The term local is essential. The estimate tells us the effect for marginal units whose treatment status changes because they are just above rather than just below the threshold.1
1 The direction can be reversed: in some RD designs, units below the cutoff are treated and those above it serve as controls. What matters is that treatment status changes discontinuously at the threshold.
9.5 Estimation
RD can be estimated using parametric or non-parametric methods.
A simple parametric specification is
\[ Y_i = \alpha + \tau_{sharpRD} D_i + f(S_i-c) + D_i g(S_i-c) + u_i, \]
where \(f(\cdot)\) and \(g(\cdot)\) are parametric functions of the centered running variable, specified without intercept terms. The function \(f(\cdot)\) describes how the outcome varies with the running variable below the cutoff, while \(g(\cdot)\) allows this relationship to differ above the cutoff. Because the running variable is centered at \(c\), \(\tau_{\text{sharp RD}}\) captures the discontinuity in the conditional mean of the outcome at the threshold.
In modern applications, however, non-parametric local polynomial methods are usually preferred. These methods focus on observations close to the cutoff and give greater weight to those nearest to it, so that identification comes from local comparisons around the threshold.
A common local linear specification estimates separate lines on the two sides of the cutoff within a selected bandwidth:
\[ |S_i-c| \le h. \]
The bandwidth \(h\) determines how close units must be to the cutoff to be included in the estimation; units outside this bandwidth are excluded from the estimation sample.
The choice of bandwidth is therefore central. Nowadays, RD practice often relies on robust bias-corrected methods and data-driven bandwidth selection, such as the approach proposed by Calonico, Cattaneo, and Titiunik (2014).
Local RD estimators also require a kernel, which determines how observations are weighted within the bandwidth.
Observations closer to the cutoff typically receive more weight, while observations farther away receive less weight. Common choices include triangular and Epanechnikov kernels.
The triangular kernel is often used in applied work because it gives the largest weight to observations closest to the cutoff.
An RD estimator must balance two objectives: approximating the outcome relationship accurately near the cutoff and using enough observations to obtain a precise estimate.
Bias is a systematic tendency to overestimate or underestimate the true effect. Variance describes how much the estimate would change across repeated samples. Ideally, we would like an estimator with both zero bias and zero variance, but in practice this is unattainable. Instead, estimation often involves a bias–variance trade-off: reducing bias may come at the cost of greater variance, while reducing variance may require accepting some bias.
The bandwidth determines the estimation window around the cutoff: - A smaller bandwidth uses observations closer to the threshold. Over this narrower range, a local linear approximation is generally more credible, reducing approximation bias. However, fewer observations are available, so estimates tend to be less precise. - A larger bandwidth includes more observations and generally reduces variance. However, if the outcome relationship is curved, fitting a straight line over a wider range may poorly approximate its value at the cutoff, increasing bias.
For example, with a test-score cutoff of 70, using scores between 69 and 71 provides a very local comparison but may leave few students. Using scores between 60 and 80 provides more data, but requires the fitted lines to approximate the outcome relationship over a much wider range.
The kernel determines how observations within the bandwidth are weighted. A uniform kernel assigns equal weight to all included observations, whereas a triangular kernel gives more weight to observations closer to the cutoff and progressively less weight to those farther away. Different kernels therefore handle the bias–variance trade-off differently, although the choice of bandwidth is typically more consequential.
Modern RD methods combine data-driven bandwidth selection with robust bias-corrected confidence intervals, which account for the uncertainty introduced by estimating and correcting approximation bias (Calonico et al. 2014). These tools improve estimation and inference, but cannot repair a design whose identifying assumptions fail.
9.6 Validity Checks
RD is visually intuitive, but it requires careful diagnostic checks.
A credible RD design should report the main estimate, graphical evidence, bandwidth sensitivity, density checks, and covariate balance around the cutoff.
9.6.1 No Manipulation of the Running Variable
Researchers should check whether units appear to manipulate the running variable around the cutoff. If many units bunch just above or just below the threshold, this may indicate sorting.
A standard diagnostic is a density test around the cutoff. If the density of the running variable jumps at the cutoff, the design may be invalid.
9.6.2 Covariate Balance
Pre-treatment covariates should be smooth at the cutoff. If predetermined characteristics jump at the threshold, treated and untreated units near the cutoff may not be comparable.
The same RD logic can therefore be applied to covariates:
\[ X_i \quad \text{should not jump at } c. \]
9.6.3 Placebo Cutoffs
Researchers can also test for discontinuities at placebo thresholds, where no treatment change occurs. If outcome jumps appear at arbitrary cutoffs, the estimated discontinuity at the true cutoff may not be causal.
9.6.4 Sensitivity to Bandwidth
Results should be checked using alternative bandwidths. If the estimate changes dramatically when the bandwidth changes, the conclusions may be fragile.
RD is often presented using a figure that plots the outcome against the running variable. The graph usually shows:
- binned averages of the outcome, used to improve visual clarity. Binning means grouping observations into small intervals of the running variable and plotting the average outcome within each interval. For example, in a close-election RD design, the running variable may be the margin of victory in a mayoral election. The graph may show the average outcome for municipalities where the winning margin was between -1 and -0.5 percentage points, -0.5 and 0, 0 and 0.5, 0.5 and 1, and so on;
- the cutoff that determines treatment assignment;
- fitted curves or lines estimated separately on each side of the cutoff;
- the estimated jump in the outcome at the threshold.
The visual evidence is not a substitute for formal estimation, but it is informative. A clear discontinuity in the outcome at the cutoff, combined with the absence of similar discontinuities in predetermined covariates, provides convincing supporting evidence for a causal interpretation of the RD estimate.
9.7 Empirical Example: Incumbency Advantage
A classic application is Lee’s study of incumbency advantage in U.S. House elections (Lee 2008). The running variable is the Democratic vote margin in an election. The cutoff is zero: candidates who barely win become incumbents, while candidates who barely lose do not.
Close elections provide a compelling RD setting. A party that barely wins and a party that barely loses are likely to be similar, but only the winner receives incumbency status.
Lee estimates the effect of party incumbency on electoral performance in the next election. The results reveal a substantial incumbency advantage: barely winning the current election increases the probability that the same party wins the district in the next election by about 40 percentage points and raises its subsequent vote share by roughly 8 percentage points.
9.8 Fuzzy RD design
The discussion so far has focused on sharp RD, the simplest version of the design. In a sharp RD, crossing the cutoff determines treatment status perfectly: units on one side of the threshold are treated, while those on the other side are untreated. In many real-world applications, however, crossing the cutoff only changes the probability of receiving treatment rather than treatment status itself. This setting is known as fuzzy RD.
Formally, in a fuzzy RD,
\[ \lim_{s \downarrow c} P(D_i=1 \mid S_i=s) - \lim_{s \uparrow c} P(D_i=1 \mid S_i=s) \neq 0, \]
but this discontinuity is smaller than one.
The fuzzy RD estimand is a ratio of two discontinuities:
\[ \tau_{fuzzyRD} = \frac{ \lim_{s \downarrow c} E[Y_i \mid S_i=s] - \lim_{s \uparrow c} E[Y_i \mid S_i=s] }{ \lim_{s \downarrow c} E[D_i \mid S_i=s] - \lim_{s \uparrow c} E[D_i \mid S_i=s] }. \]
The numerator measures the jump in the outcome at the cutoff, while the denominator measures the jump in the probability of treatment. The fuzzy RD estimand is therefore a ratio: it scales the discontinuity in the outcome by the discontinuity in treatment receipt. For example, suppose crossing the cutoff increases the average outcome by 4 units and raises the probability of receiving treatment from 30% to 70%. The treatment probability therefore increases by 0.40, so the fuzzy RD estimate is \[\frac{4}{0.40}=10.\] This means that treatment increases the outcome by 10 units, on average, for the units near the cutoff whose treatment status is changed by crossing the threshold.
For this reason, fuzzy RD is closely related to instrumental variables. In a fuzzy RD design, the indicator for being above the cutoff can be used as an instrument for actual treatment receipt. The logic is the same as in an IV design. The cutoff generates a source of exogenous variation in treatment: units just below and just above the threshold are assumed to be comparable, but those just above the cutoff are more likely to receive the treatment. Under the usual RD continuity assumptions, and under the IV assumptions of relevance, exclusion, and monotonicity, the fuzzy RD estimand identifies a LATE for units close to the cutoff whose treatment status is changed by crossing the threshold.
In other words, fuzzy RD estimates the treatment effect for compliers near the cutoff: units who receive the treatment because they are on the eligible side of the threshold, and who would not have received it otherwise.
9.9 Spatial RD design
When treatment assignment is based on a geographic discontinuity, the same logic gives rise to a spatial RD design. In this case, treatment status changes discontinuously at a border, and units located close to that border are compared: treated units on one side and untreated units on the other.
For example, a place-based policy may apply only to municipalities on one side of an administrative boundary. If locations near the boundary are similar in geography, climate, transport access, and local economic conditions, then differences in outcomes across the boundary can be attributed to the policy.
In spatial RD, the running variable may be:
- distance to the boundary;
- latitude and longitude;
- distance to the nearest border point;
- travel distance to the boundary.
Spatial RD is attractive because nearby locations often share many observed and unobserved characteristics. However, it has additional challenges.
First, several policies may change at the same boundary. If multiple institutional rules differ across the border, the observed discontinuity may capture the combined effect of these differences rather than isolate the policy of interest. This is sometimes called a compound treatment problem. To interpret the discontinuity as the effect of a specific policy, researchers must therefore impose an additional assumption, often referred to as compound treatment irrelevance: the other policies or institutional differences that also change at the boundary must not themselves generate a discontinuity in the outcome. An alternative, when the policy of interest is introduced at a known point in time along an already existing boundary, is to combine spatial RD with DiD. By comparing the discontinuity before and after the policy change, this approach can difference out pre-existing border differences, provided that these differences would have remained stable over time in the absence of the policy.
Second, the no-interference assumption may be more fragile in spatial RD designs than in traditional RD designs. Units located on opposite sides of a geographic boundary are often close to one another and may interact through local labor markets, housing markets, commuting flows, or social networks. As a result, the treatment assigned on one side of the border may affect outcomes on the other side as well.
9.10 Empirical Example: The West-German Zonenrandgebiet
A useful example of a spatial RD design is the study by von Ehrlich and Seidel (2018) on the long-run effects of place-based policy in the West-German Zonenrandgebiet.
The policy targeted municipalities located within a certain distance of the inner-German border. The policy was introduced to compensate municipalities close to the inner-German border for the economic disadvantages created by the division of Germany. The new border abruptly turned these areas into peripheral locations, cutting them off from former markets and economic connections and weakening their development prospects. The geographic eligibility rule, which defined the Zonenrandgebiet as an approximately 40-km-wide band along the inner-German border, generated a discontinuity in treatment assignment at the boundary of the assisted area: municipalities just inside the eligible border zone received support, while nearby municipalities just outside did not. The key idea of the design is therefore simple. Municipalities located very close to the same boundary are likely to be similar in many respects, except for their eligibility status. This makes it possible to compare units on opposite sides of the border and interpret discontinuous differences in outcomes as evidence of a treatment effect.
The figure below illustrates the geographic discontinuity exploited in the study. The authors find that the subsidies produced persistent increases in local economic activity, even after the program ended, although part of these gains appears to reflect the relocation of economic activity from nearby untreated areas.
9.11 Summary
An RD design exploits a discontinuous treatment rule. If units cannot precisely manipulate the running variable and potential outcomes vary smoothly at the cutoff, units just below and just above the threshold are expected to be similar in all respects except treatment status.
In a sharp RD, treatment changes deterministically at the cutoff. In a fuzzy RD, the cutoff changes the probability of treatment, and the design is interpreted similarly to an instrumental variables strategy.
RD is best understood as a design that estimates the causal effect for units close to the threshold.
Several software packages can be used to implement regression discontinuity designs, including estimation, bandwidth selection, RD plots, manipulation tests, and power analysis.
- R
- Python
- Stata