When to Use RDD

Use RDD when the assignment rule is threshold-based and units near the cutoff are plausibly similar except for treatment status. That logic works for close elections, exam cutoffs, age thresholds, and policy eligibility rules.

Do not use RDD if the threshold is arbitrary but unrelated to assignment, if there is strong evidence of sorting around the cutoff, or if the estimand of interest is inherently global rather than local.

Notation

Let:

• \(Y_i\): the observed outcome of interest, such as city GDP growth.
• \(D_i\): the treatment indicator, equal to 1 if the high-education candidate wins and 0 otherwise.
• \(X_i\): the running variable or forcing variable, such as vote margin.
• \(c\): the cutoff, such as 0 in a close election design.

1. Data Simulation

First, let’s create a dataset that mimics the election setup. - margin_victory: The vote margin for the high-education candidate (running variable). - high_educ_win: Indicator if margin > 0 (treatment). - gdp_growth: The outcome variable, constructed with a jump at the cutoff.

Before running any models, it is crucial to visualize the data. We want to check for two things: 1. Is there a clear jump at the cutoff (\(c=0\))? 2. Does the relationship look linear or non-linear away from the cutoff?

library(rdrobust)
library(ggplot2)
library(dplyr)
library(magrittr)

df <- readr::read_csv(
    paste0(
        here::here("research-guides/rdd/"),
        "assets/data/rdd_data.csv"
    ),
    show_col_types = FALSE
)

Code

ggplot(df, aes(x = margin_victory, y = gdp_growth, color = factor(high_educ_win))) +
    geom_point(alpha = 0.3) +
    geom_vline(xintercept = 0, linetype = "dashed") +
    theme_minimal() +
    labs(
        title = "Raw Data: Election Margin vs GDP Growth",
        x = "Margin of Victory (%)",
        y = "GDP Growth (%)",
        color = "Treatment"
    ) +
    theme(legend.position = "none")

The raw scatter plot should show clouds of points. Even with noise, if the effect is strong, you might see a vertical shift at \(X=0\).

2. RDD Estimation and Plotting

We use the rdrobust package because it is the modern default in the RD literature. Compared with older hand-tuned polynomial regressions, it gives a local linear estimate, chooses a data-driven bandwidth, and reports robust bias-corrected inference.

The default triangular kernel is standard because it gives the highest weight to observations closest to the cutoff and downweights observations farther away smoothly rather than dropping them abruptly.² That is a good compromise between bias and variance in a local design.

The bias correction matters because local polynomial estimators can be biased in finite samples. rdrobust corrects for that bias and uses the corrected standard errors and confidence intervals, which is why it is preferred over simpler plug-in alternatives.

# Run the RD estimation
# c=0 specifies the cutoff
fit <- rdrobust(y = df$gdp_growth, x = df$margin_victory, c = 0)
summary(fit)

Call: rdrobust

Sharp RD estimates using local polynomial regression.

Number of Obs.                  500
BW type                       mserd
Kernel                   Triangular
VCE method                       NN

                               Left        Right
Number of Obs.                  265          235
Eff. Number of Obs.              86           64
Order est. (p)                    1            1
Order bias (q)                    2            2
BW est. (h)                  30.286       30.286
BW bias (b)                  52.669       52.669
rho (h/b)                     0.575        0.575
Unique Obs.                     265          235

=====================================================================
                   Point    Robust Inference
                Estimate         z     P>|z|      [ 95% C.I. ]       
---------------------------------------------------------------------
     RD Effect     2.069     2.246     0.025     [0.236 , 3.466]     
=====================================================================

Interpretation of Output: Look for the Coeff under Robust. This is the estimated treatment effect at the cutoff. The robust confidence interval and p-value are based on bias-corrected inference, so they are the numbers to report.

Visualization with rdplot: rdplot creates a binned scatterplot with separate local fits on each side of the cutoff. It is cleaner than a raw scatter plot and is useful for checking whether the discontinuity is visible in the data before relying on the formal estimate.

# Binned scatterplot with fitted regression lines
rdplot(
    y = df$gdp_growth, x = df$margin_victory, c = 0,
    title = "RDD Plot: Effect of High-Education Mayor",
    y.label = "GDP Growth",
    p = 1,
    x.label = "Margin of Victory"
)

The vertical distance between where the left line ends (at 0) and the right line starts (at 0) is the estimated treatment effect.

1. McCrary Density Test (Manipulation Check)

This test asks whether units sort around the cutoff. In a close-election design, for example, a discontinuity in the density of the vote margin would suggest that the running variable itself is being manipulated rather than assigned quasi-randomly.

We use rddensity, the modern implementation of the McCrary-style density test. It improves finite-sample performance relative to older approaches and is now the preferred default in applied work.

# Testing for manipulation in the density of the running variable

library(rddensity)

# Run the density test
rdd_test <- rddensity(df$margin_victory, c = 0)

# Create an artificial heaping example based on the same simulated data

set.seed(123)

df_heap <- df %>%
    mutate(
        margin_victory_heaped =
            round(margin_victory / 5) * 5,
        # Treatment based on the heaped/recorded running variable
        high_educ_win_heaped = ifelse(margin_victory_heaped >= 0, 1, 0)
    )

# Plot the heaped running variable in the same style as the main RD plot
ggplot(
    df_heap,
    aes(
        x = margin_victory_heaped,
        y = gdp_growth,
        color = factor(high_educ_win_heaped)
    )
) +
    geom_point(alpha = 0.35) +
    geom_vline(xintercept = 0, linetype = "dashed") +
    theme_minimal() +
    labs(
        title = "Heaping Example: Election Margin vs GDP Growth",
        subtitle = "Vertical stripes show observations piling up at rounded values",
        x = "Recorded Margin of Victory (%)",
        y = "GDP Growth (%)",
        color = "Treatment"
    ) +
    theme(legend.position = "none")

2. Specification Checks

Ensure your results hold under different functional forms. Higher-order global polynomials can be misleading because they put too much weight on observations far from the cutoff and can introduce spurious wiggles near the threshold.

• Linear: \(Y = \alpha + \tau D + \beta_1 X + \epsilon\)
• Linear with interaction (different slopes on each side): \(Y = \alpha + \tau D + \beta_1 X + \beta_2 (D \cdot X) + \epsilon\)
• Quadratic: Add \(X^2\) terms.
• Cubic: Add \(X^3\) terms, but use this only as a sensitivity check, not as the main specification.

The reason practitioners usually prefer local linear fits is that they behave better near the cutoff and are less sensitive to distant observations than cubic or quartic polynomials.

3. Bandwidth Sensitivity

The “bandwidth” determines how many observations around the cutoff are used. A smaller bandwidth reduces bias because observations are more similar, but it also reduces power. A larger bandwidth improves precision but increases the risk that the control function is no longer approximating the counterfactual trend well.

The practical question is whether the estimate is stable over a reasonable window of cutoffs. Calonico, Cattaneo, and Farrell (2020) emphasize that stability should be judged relative to the scale of the running variable and the amount of data near the threshold. The RD bandwidth app is a useful tool for exploring those tradeoffs.

One way to assess robustness is to check whether the estimated effect remains stable across a range of bandwidths.

# Define a range of bandwidths to test (from 3 to 25)
bandwidths <- seq(3, 25, by = 1)

# Initialize a data frame to store results
results <- data.frame(
    bandwidth = bandwidths,
    estimate = NA,
    ci_lower = NA,
    ci_upper = NA
)

# Loop through bandwidths and estimate RDD using rdrobust
for (i in 1:length(bandwidths)) {
    bw <- bandwidths[i]
    # Run rdrobust with fixed bandwidth 'h'
    model <- rdrobust(y = df$gdp_growth, x = df$margin_victory, c = 0, h = bw, p = 1)

    # Extract coefficient and robust CI
    results$estimate[i] <- model$coef[3]
    results$ci_lower[i] <- model$ci[3] # Robust CI lower bound
    results$ci_upper[i] <- model$ci[6] # Robust CI upper bound
}

# Plot the sensitivity analysis
ggplot(results, aes(x = bandwidth, y = estimate)) +
    geom_point() +
    geom_errorbar(aes(ymin = ci_lower, ymax = ci_upper), width = 0.5) +
    geom_hline(yintercept = 2.5, linetype = "dashed", color = "red") + # True Effect
    geom_hline(yintercept = 0, linetype = "dotted", color = "black") +
    labs(
        title = "RDD Sensitivity to Bandwidth Selection",
        x = "Bandwidth (Margin of Victory %)",
        y = "Estimated Treatment Effect (with 95% CI)"
    ) +
    theme_minimal()

This plot shows the estimated treatment effect (dots) and 95% confidence intervals (bars) for different bandwidths.

• The red dashed line represents the true effect (2.5).
• If the estimates are broadly stable, do not flip sign, and remain within a substantively similar range, the result is more credible.
• If the estimate only appears under a very narrow or very wide window, that is a warning sign.
• Smaller bandwidths usually have wider confidence intervals because there are fewer observations near the cutoff.

4. Placebo Test: Other Outcomes

Placebo tests should distinguish between two cases:

• Predetermined covariates, such as pre-election demographics or baseline city size, should be balanced at the cutoff.
• Predetermined outcomes, such as lagged GDP growth or pre-election crime rates, should show no discontinuity.

If either one jumps at the cutoff, the design is likely not comparing like with like.

# Placebo test: lagged GDP growth should not jump at the cutoff
fit_placebo <- rdrobust(y = df$lagged_gdp_growth, x = df$margin_victory, c = 0)
summary(fit_placebo)

Call: rdrobust

Sharp RD estimates using local polynomial regression.

Number of Obs.                  500
BW type                       mserd
Kernel                   Triangular
VCE method                       NN

                               Left        Right
Number of Obs.                  265          235
Eff. Number of Obs.             103           83
Order est. (p)                    1            1
Order bias (q)                    2            2
BW est. (h)                  36.838       36.838
BW bias (b)                  57.752       57.752
rho (h/b)                     0.638        0.638
Unique Obs.                     265          235

=====================================================================
                   Point    Robust Inference
                Estimate         z     P>|z|      [ 95% C.I. ]       
---------------------------------------------------------------------
     RD Effect     0.152     0.303     0.762    [-1.536 , 2.097]     
=====================================================================

Interpretation: The estimated discontinuity in lagged GDP growth is close to zero and not statistically significant. This is reassuring: because lagged GDP growth was determined before the election outcome, any jump at the cutoff would indicate that the running variable itself (election margin) was correlated with pre-existing outcomes. That would violate the RD assumption that units near the cutoff are comparable. Here, the placebo test passes.

For close-elections applications, a good placebo outcome is a pre-election economic trend, while a good placebo covariate is something like pre-election city population or partisan composition.

For formal placebo testing across repeated cutoff assignments, Canay and Kamat (2018) provide permutation-style inference that can complement the standard placebo plots.

5. Placebo Test: Artificial Cutoffs

Test for discontinuities at fake thresholds where there should be no effect. In a close-election setting, that means checking margins such as +10% or -10%, not just the main 0% cutoff.

• If the estimate appears at many fake cutoffs, the real result may reflect functional form misspecification or random noise.
• If the jump only appears at the true cutoff and not nearby placebo thresholds, that is more reassuring.

# Placebo cutoff at +10%
rdplot(
    y = df$gdp_growth, x = df$margin_victory, c = 10,
    title = "Placebo Test: Cutoff at +10%",
    x.label = "Margin of Victory", y.label = "GDP Growth", p = 1
)

# Formal test at fake cutoff
summary(rdrobust(y = df$gdp_growth, x = df$margin_victory, c = 10))

Call: rdrobust

Sharp RD estimates using local polynomial regression.

Number of Obs.                  500
BW type                       mserd
Kernel                   Triangular
VCE method                       NN

                               Left        Right
Number of Obs.                  285          215
Eff. Number of Obs.              53           47
Order est. (p)                    1            1
Order bias (q)                    2            2
BW est. (h)                  21.824       21.824
BW bias (b)                  35.941       35.941
rho (h/b)                     0.607        0.607
Unique Obs.                     285          215

=====================================================================
                   Point    Robust Inference
                Estimate         z     P>|z|      [ 95% C.I. ]       
---------------------------------------------------------------------
     RD Effect    -0.182    -0.395     0.693    [-2.806 , 1.866]     
=====================================================================

6. Power Analysis

RDD often has limited power because identification relies on units very close to the cutoff. Even large datasets can have a small effective sample size if relatively few observations fall near the threshold.

If you find a null result, it is important to ask whether the design is underpowered rather than concluding that the effect is zero. Likewise, before starting the analysis, it is useful to check whether the design can detect a substantively meaningful effect.

We can use rdpower for power calculations and sample-size planning.

library(rdpower)
# Power calculation with plot
rdpower(data = cbind(df$gdp_growth, df$margin_victory), cutoff = 0, tau = 2.5, plot = TRUE)

Number of obs =       500
BW type       =       mserd
Kernel type   =       Triangular
VCE method    =       NN
Derivative    =       0
HA:       tau =       2.5


Cutoff c = 0          Left of c       Right of c      
Number of obs         265             235             
Eff. number of obs    86              64              
BW loc. poly.         30.286          30.286          
Order loc. poly.      1               1               
Sampling BW           30.286          30.286          
New sample            86              64              


=========================================================================================
Power against:           H0: tau =      0.2*tau =      0.5*tau =      0.8*tau =    tau =          
                         0              0.5            1.25           2            2.5            
-----------------------------------------------------------------------------------------
Robust bias-corrected    0.05           0.093          0.329          0.68         0.859          
=========================================================================================

If the target effect size yields adequate power, the design is better suited to detect that effect. If not, a null estimate may simply reflect limited precision.

One way to deal with low power is to widen the bandwidth, but that comes at the cost of including observations farther from the cutoff. A 1 percent window may be very credible but underpowered, while a 15 percent window may be more precise but less defensible because it stretches the local design.

Another way to improve precision is to use a lower-level outcome, such as zip-code-level economic indicators rather than city averages. That can reduce noise, but the treatment assignment still happens at the city-election level, so standard errors should still be clustered at the appropriate treatment level.

Building Base RD Intuition (Parametric Approach)

While it is most recommended to use a kernel or non-parametric estimator (like rdrobust) for RD, there are simpler ways to do so that can help build intuition. We can think of RDD as a regression problem where we carefully control for the running variable (\(X\)).

We will use feols from the fixest package to show how this works using linear regressions with increasing flexibility.

1. Linear Model (Same Slope) This assumes the relationship between the running variable and outcome is linear and has the same slope on both sides of the cutoff. It controls for \(X\) but forces the trend to be parallel. \[ Y = \alpha + \tau D + \beta X + \epsilon \]

library(fixest)
# Simple linear control
feols(gdp_growth ~ high_educ_win + margin_victory, data = df)

2. Linear Interaction Model (Different Slopes) This allows the slope of the running variable to be different for treated and control units (“linear from both sides”). This is the standard “local linear” intuition but applied to the whole window. \[ Y = \alpha + \tau D + \beta_1 X + \beta_2 (D \times X) + \epsilon \]

# Allowing different slopes (interaction)
feols(gdp_growth ~ high_educ_win * margin_victory, data = df)

3. Quadratic Model Allows for non-linear relationships (curvature) by adding a squared term. This helps if the underlying trend is curved. \[ Y = \alpha + \tau D + \beta_1 X + \beta_2 X^2 + \dots \]

# Quadratic polynomial (interacted)
feols(gdp_growth ~ high_educ_win * poly(margin_victory, 2), data = df)

4. Cubic Model Adds even more flexibility with a cubic term.

# Cubic polynomial (interacted)
feols(gdp_growth ~ high_educ_win * poly(margin_victory, 3), data = df)

While these parametric models are intuitive, high-order global polynomials (like cubic or quartic) can be misleading and suffer from bias at the cutoff. This is why local linear regression (used in rdrobust) is preferred.

Common Pitfalls in RDD

RDD is easy to misuse. Here we overview some common mistakes to avoid.

Discrete Running Variables

Sometimes the running variable is discrete rather than continuous. For example, age may be reported only in integers. In those cases, the support around the cutoff is sparse, so the usual smooth-continuity intuition needs to be interpreted more carefully. Kolesár and Rothe discuss how identification and inference change when the running variable is discrete, and the practical implication is that you should be cautious about over-interpreting very small jumps when there are only a few mass points near the cutoff.

Donut RDD

If there is heaping or obvious manipulation exactly at the cutoff, one option is a donut RD: drop observations in a small window around the threshold and estimate the effect using observations just outside the contaminated region. That approach is useful when the very closest observations are suspect, but it trades away some of the most informative data and should be justified carefully. Noack and Rothe (2023) discuss this strategy in settings with bunching or manipulation.

Clustering

If your data has a nested structure, cluster at the level where treatment is assigned, not necessarily at the level where the outcome is measured. In the mayor example, that usually means clustering at the city-election level even if the outcome is observed at a lower geographic unit.

rdrobust allows this via the cluster argument.

RDD with Sub-outcomes

Sometimes you might have data at a lower level than the assignment. For example, instead of city-level GDP, you might have neighborhood-level economic data. The assignment (mayor education) is at the city level. You can still run the RDD, but ensure your clustering accounts for the city-level assignment.

Panel Data and Fixed Effects

If you have panel data with repeated elections for the same city, you can sometimes include unit fixed effects to absorb time-invariant differences across cities. That can be helpful when the same unit appears multiple times, but it does not replace the RD identification argument at the cutoff.

In many applications, it is simpler to treat the data as repeated cross-sections and cluster standard errors at the city level. If you use panel structure, be explicit about the level at which treatment is assigned and the level at which errors may be correlated.

The rdrobust authors do not encourage fixed effects as a default RD strategy, but you can also implement an RD-style regression in fixest if your design needs panel controls.

feols(gdp_growth ~ high_education + election_margin, )

What if there is imbalance?

If baseline covariates are slightly imbalanced, you can include them as controls in the outcome equation, in a covariate-adjusted RDD. The important caveat is that covariates should help explain the outcome, not help to determine the bandwidth. In other words, covariates are a precision and robustness device.

rdrobust(y = Y, x = X, covs = cbind(cov1, cov2))

If the RD estimate is similar with and without covariates, that suggests the result is not driven by small residual imbalance. If the estimate changes substantially, you should investigate why the covariates matter so much and justify it. In a case of close elections between two candidate types, it could be that the type of the candidate is correlated withother covariate of the candidate that should be controlled. For instance, if high-education mayors are older on average, to estimate the effect of high-education mayors we need to control for the age.

What if there is fuzzy continuity? (Fuzzy RDD)

Sometimes the cutoff does not perfectly determine treatment assignment, but changes the probability of treatment.

• Sharp RDD: \(P(D=1 | X \ge c) = 1\) and \(P(D=1 | X < c) = 0\).
• Fuzzy RDD: \(P(D=1 | X \ge c) > P(D=1 | X < c)\), but not 0 and 1.
  • Example: Instead of a strict election rule, imagine a scholarship program where students above a test score cutoff are eligible, but not all take it, and some below might get exceptions.

In this case, the RDD estimator is an Instrumental Variable (IV) estimator.

• Instrument (\(Z\)): Being above the cutoff (\(Z = 1\) if \(X \ge c\)).
• Treatment (\(D\)): Actually receiving the scholarship.
• We use Two-Stage Least Squares (2SLS):
  1. First Stage: Regress treatment \(D\) on cutoff \(Z\) (and X).
  2. Reduced Form: Regress outcome \(Y\) on cutoff \(Z\) (and X).
  3. Fuzzy Effect = (Reduced Form Jump) / (First Stage Jump).

# Fuzzy RDD using rdrobust
# 'fuzzy' argument takes the actual treatment indicator D
fit_fuzzy <- rdrobust(y = df$outcome, x = df$score, c = 0, fuzzy = df$actual_treatment)
summary(fit_fuzzy)

This estimates the Local Average Treatment Effect (LATE) for compliers: students whose treatment status was determined by the cutoff rule.

Geographic Discontinuity

Instead of a vote margin, the “running variable” is distance to a border.

• Example: Comparing house prices across a school district boundary.
• \(X\) is distance (positive if inside, negative if outside).
• Assumption: Households just inside and just outside are similar. This is often violated if people “sort” (move) specifically for the school district.

Candidate Feature Discontinuity

Sometimes the “treatment” is bundled. A mayor with higher education might also naturally be older or have more reliable prior political experience.

• If high-education candidates are consistently 10 years older than low-education candidates at the cutoff, you cannot separate the effect of “education” from “age.”
• You are estimating the effect of the bundle of characteristics of the winning candidate.

Solution: Add predetermined characteristics as controls in the outcome equation. If age or experience are measured before the election and are not themselves affected by winning, they can improve precision and help separate education from correlated features.

Controls do not change the RD logic; they only adjust for residual imbalance in observed covariates. If the estimate disappears only after adding a particular control, that is a signal to think carefully about whether the control is a confounder, a proxy, or a post-treatment variable.

# Controlling for age and experience explicitly
# Assuming 'age' and 'experience' are columns in your dataframe
fit_controls <- rdrobust(
    y = df$gdp_growth, x = df$margin_victory, c = 0,
    covs = cbind(df$age, df$experience)
)
summary(fit_controls)

Difference in Discontinuities (Diff-in-Disc)

What if the effect of education depends on incentives?

• Imagine some cities pay mayors a high salary (based on a population threshold) and others pay less.
• You have two discontinuities:
  1. Close elections in “High Salary” cities.
  2. Close elections in “Low Salary” cities.
• You can estimate \(\tau_{high}\) and \(\tau_{low}\) and compare them: \(\tau_{diff} = \tau_{high} - \tau_{low}\).
• This allows you to answer: “Does the effect of a highly educated mayor on growth depend on whether they are financially incentivized?”

Simulation code

The code below defines the exact function used to generate rdd_data.csv. Copy it into an R script, then call

df <- simulate_rdd_data()

to reproduce the dataset.

#' Simulate data for a sharp regression discontinuity design
#'
#' Generates data mimicking a close-election setup for demonstrating sharp
#' regression discontinuity (RD) estimation. A running variable (margin of
#' victory for a hypothetical "high-education" candidate) determines
#' treatment deterministically at a cutoff of zero, and an outcome variable
#' (GDP growth) is constructed with both a smooth trend in the running
#' variable and a discontinuous jump at the cutoff. A predetermined lagged
#' outcome is also generated, intended for use in placebo/falsification
#' tests, and is constructed to have no jump at the cutoff by design.
#'
#' @param seed Integer. Random seed for reproducibility. Defaults to 123.
#' @param n Integer. Sample size. Defaults to 500.
#'
#' @return A data frame with columns \code{margin_victory} (numeric, the
#'   running variable, uniform on -100 to 100), \code{high_educ_win}
#'   (integer, 1 if \code{margin_victory >= 0} else 0, the treatment
#'   indicator), \code{gdp_growth} (numeric outcome, with a true jump of
#'   2.5 at the cutoff and a slope of 0.05 in the running variable), and
#'   \code{lagged_gdp_growth} (numeric, a predetermined outcome with the
#'   same slope but no discontinuity, for placebo testing).
#'
#' @examples
#' df <- simulate_rdd_data()
#'
#' @export
simulate_rdd_data <- function(seed = 123, n = 500) {
    set.seed(seed)
    library(magrittr)

    df <- data.frame(margin_victory = runif(n, -100, 100))

    df <- df %>%
        dplyr::mutate(high_educ_win = ifelse(margin_victory >= 0, 1, 0))

    df <- df %>%
        dplyr::mutate(
            gdp_growth = 5 + 0.05 * margin_victory + 2.5 * high_educ_win + rnorm(n, sd = 2.5),
            lagged_gdp_growth = 4.8 + 0.04 * margin_victory + rnorm(n, sd = 2.3)
        )

    df
}