Regression Discontinuity Design (RDD) Skill

This skill covers sharp and fuzzy RDD: identification assumptions, bandwidth selection, local polynomial estimation, validity tests, and reporting standards for academic papers.

Core Logic

RDD exploits a known threshold in a continuous "running variable" (X) that determines treatment assignment. Units just above and below the cutoff (c) are comparable on all dimensions except treatment.

Sharp RDD: Treatment perfectly determined by crossing cutoff

• T_i = 1 if X_i ≥ c, T_i = 0 if X_i < c
• Estimand: Average treatment effect at the cutoff (τ_SRD)

Fuzzy RDD: Crossing cutoff increases probability of treatment (like an instrument)

• Use when there's non-compliance around the cutoff
• Estimand: LATE at the cutoff (τ_FRD = reduced form / first stage)

RDD Assumptions

1. Continuity of conditional expectation: E[Y(0)|X] and E[Y(1)|X] are continuous at X = c
   • Means: units cannot precisely manipulate running variable to select into treatment
2. No other discontinuities: Nothing else changes discontinuously at the cutoff
3. Bandwidth continuity: Observations near cutoff are locally valid comparisons

Complete RDD Workflow

Step 1: Visualize the Discontinuity

Always plot the raw data with binned means before any regression.

# Python
import matplotlib.pyplot as plt
import numpy as np

# Bin the running variable
df['bin'] = pd.cut(df['running_var'], bins=50)
bin_means = df.groupby('bin')[['running_var', 'y']].mean().reset_index()

plt.figure(figsize=(10, 6))
plt.scatter(bin_means['running_var'], bin_means['y'], s=30, color='steelblue')
plt.axvline(x=cutoff, color='red', linestyle='--', label='Cutoff')
plt.xlabel('Running Variable'); plt.ylabel('Outcome')
plt.title('RDD: Binned Scatter Plot')
plt.legend(); plt.show()

# R (rdplot from rdrobust)
library(rdrobust)
rdplot(y = df$y, x = df$running_var, c = cutoff,
       title = "RDD Binned Scatter", x.label = "Running Variable",
       y.label = "Outcome")

rdplot y running_var, c(cutoff) graph_options(title("RDD Visualization"))

Step 2: Bandwidth Selection

Default: Use Imbens-Kalyanaraman (IK) or Calonico-Cattaneo-Titiunik (CCT) optimal bandwidth.

from rdrobust import rdrobust
result = rdrobust(df['y'], df['running_var'], c=cutoff)
print(result.summary())

rdbwselect(y = df$y, x = df$running_var, c = cutoff)

rdbwselect y running_var, c(cutoff) all

Step 3: Main RDD Estimate

# Python (rdrobust) — triangular kernel, local linear
result = rdrobust(y=df['y'], x=df['running_var'], c=cutoff,
                  kernel='triangular', p=1)
print(result.summary())

# R
main_rdd <- rdrobust(y = df$y, x = df$running_var, c = cutoff,
                     kernel = "triangular", p = 1)
summary(main_rdd)

rdrobust y running_var, c(cutoff) kernel(triangular) p(1)

Step 4: Validity Tests (All Required)

4a. Density/Manipulation Test (McCrary Test)

H₀: No discontinuity in density of running variable at cutoff

from rdrobust import rddensity
density_test = rddensity(df['running_var'], c=cutoff)
print(density_test.summary())

library(rddensity)
rdd_density <- rddensity(df$running_var, c = cutoff)
summary(rdd_density)
rdplotdensity(rdd_density, df$running_var)

rddensity running_var, c(cutoff)

Interpretation: p > 0.05 → no bunching; manipulation unlikely ✓

4b. Covariate Balance (Placebo Outcome Tests)

Run RDD on pre-determined covariates — should find no discontinuity.

for (cov in c("age", "income_pre", "gender")) {
  res <- rdrobust(y = df[[cov]], x = df$running_var, c = cutoff)
  cat(cov, ": coef =", res$coef[1], ", p =", res$pv[3], "\n")
}

4c. Placebo Cutoff Tests

Run RDD at fake cutoffs above and below actual cutoff — should find no effects.

for (fake_c in c(cutoff - 5, cutoff + 5)) {
  df_sub <- df[df$running_var < cutoff, ]  # Use only control side
  res <- rdrobust(df_sub$y, df_sub$running_var, c = fake_c)
  cat("Placebo c =", fake_c, ": coef =", res$coef[1], "\n")
}

4d. Bandwidth Sensitivity

Report estimates at 50%, 75%, 125%, 150% of optimal bandwidth.

bw_opt <- rdbwselect(df$y, df$running_var, c = cutoff)$bws[1,1]
for (mult in c(0.5, 0.75, 1, 1.25, 1.5)) {
  res <- rdrobust(df$y, df$running_var, c = cutoff, h = bw_opt * mult)
  cat("BW =", round(bw_opt*mult,2), ": coef =", round(res$coef[1],3),
      ", p =", round(res$pv[3],3), "\n")
}

Fuzzy RDD

# R — fuzzy RDD (uses crossing as instrument for actual treatment)
fuzzy_rdd <- rdrobust(y = df$y, x = df$running_var, c = cutoff,
                      fuzzy = df$actual_treatment)
summary(fuzzy_rdd)

rdrobust y running_var, c(cutoff) fuzzy(actual_treatment)

Reporting Standards

Report in this order:

1. Binned scatter plot showing discontinuity visually
2. Main estimate with optimal CCT bandwidth, triangular kernel, local linear
3. Sensitivity table: estimates across bandwidth multiples and polynomial orders
4. Validity tests: density test (McCrary), covariate balance, placebo cutoffs
5. Sample size: N total, N within bandwidth (left and right)

Key sentence template for papers:

"We estimate the RDD using a local linear regression with a triangular kernel and the CCT optimal bandwidth (h = [X]). The point estimate at the cutoff is [β] (SE = [se], p = [p])."

See references/rdd-reference.md for geographic RDD, kink designs, donut-hole robustness, discrete running variable RDD, and multi-cutoff/multi-score RDD.

Common Pitfalls

• Using global polynomial regression: High-order global polynomials (e.g., 5th degree) overfit and produce misleading results — always use local linear or local quadratic
• Not showing the binned scatter plot: The visual discontinuity is crucial for credibility — always include it
• Ignoring manipulation: A failed McCrary test means your RDD is fundamentally compromised — address the sorting concern
• Reporting only one bandwidth: Show sensitivity across 50%–150% of optimal bandwidth to demonstrate robustness
• Using RDD far from cutoff: RDD estimates are valid only at the cutoff — do not extrapolate to units far from the threshold