Lab: Regression Discontinuity Design from Scratch

Overview

Regression Discontinuity Design (RDD) exploits situations where treatment is assigned based on whether a running variable (also called a forcing variable or score) crosses a known cutoff. Near the cutoff, treatment assignment is essentially random, giving us a credible causal estimate.

What you will learn:

• How to set up and visualize an RD design
• How to estimate the treatment effect using local linear regression
• How to choose bandwidth and assess sensitivity
• How to test the no-manipulation assumption (McCrary density test)
• How to run validity checks (covariate balance at the cutoff)

Prerequisites: OLS regression (see the OLS lab).

Step 1: The Setting

Imagine a university that places students on academic probation if their GPA falls below 2.0. We want to estimate whether academic probation affects graduation rates. The running variable is GPA, the cutoff is 2.0, and the outcome is whether the student eventually graduates.

Students just below 2.0 are placed on probation; students just above 2.0 are not. The key insight: students at 1.99 GPA and 2.01 GPA are expected to be very similar on observed and unobserved characteristics — the primary difference is probation status.

Step 2: Simulate the Data

1library(rdrobust)
2library(rddensity)
3library(ggplot2)
4
5set.seed(42)
6n <- 5000
7
8# Running variable: GPA centered at 2.0
9gpa_raw <- pmin(pmax(rnorm(n, 2.5, 0.7), 0), 4.0)
10X <- gpa_raw - 2.0
11
12# Treatment: probation if GPA < 2.0 (X < 0)
13D <- as.integer(X < 0)
14
15# Potential outcomes
16tau <- 0.08
17Y0 <- 0.5 + 0.15 * X + 0.02 * X^2 + rnorm(n, 0, 0.15)
18Y1 <- Y0 + tau
19Y <- D * Y1 + (1 - D) * Y0
20Y <- pmin(pmax(Y, 0), 1)
21
22df <- data.frame(gpa = gpa_raw, X = X, probation = D, graduated = Y)
23
24cat("Sample size:", n, "\n")
25cat("On probation:", sum(D), "(", round(mean(D)*100,1), "%)\n")
26cat("Mean graduation:", round(mean(Y), 3), "\n")

Expected output:

Summary statistics:

Step 3: Visualize the Discontinuity

The first and most important step in any RD analysis is to plot the data. Bin the running variable and plot mean outcomes within each bin.

# RD plot using rdrobust package
rdplot(y = df$graduated, x = df$X, c = 0,
     title = "RD Plot: Academic Probation and Graduation",
     x.label = "GPA - 2.0 (Running Variable)",
     y.label = "Graduation Rate")

Step 4: Estimate the Treatment Effect

The standard RD estimator uses local linear regression: fit a separate linear regression on each side of the cutoff, using only observations within a bandwidth $h$ of the cutoff.

1# Using rdrobust (the standard approach)
2rd_result <- rdrobust(y = df$graduated, x = df$X, c = 0)
3summary(rd_result)
4
5# The default uses MSE-optimal bandwidth selection
6# and bias-corrected confidence intervals

Expected output: Local linear RD estimate (manual, bandwidth = 0.5)

The rdrobust estimate with data-driven MSE-optimal bandwidth will produce a point estimate near 0.08, with a bandwidth typically between 0.3 and 0.8 GPA points and bias-corrected confidence intervals that cover the true value.

Step 5: Bandwidth Sensitivity

The choice of bandwidth $h$ involves a bias-variance tradeoff. A smaller bandwidth means less bias (more "local" comparison) but fewer observations and more variance. A larger bandwidth means more data but potentially more bias.

1# Bandwidth sensitivity
2bandwidths <- c(0.2, 0.3, 0.4, 0.5, 0.7, 1.0, 1.5)
3
4for (h in bandwidths) {
5rd <- rdrobust(y = df$graduated, x = df$X, c = 0, h = h)
6cat(sprintf("h = %.1f: estimate = %.4f, SE = %.4f, N = %d\n",
7            h, rd$coef[1], rd$se[3], rd$N_h[1] + rd$N_h[2]))
8}
9cat("\nTrue effect: 0.08\n")
10cat("Estimate should be stable across bandwidths.\n")

Expected output: Bandwidth sensitivity

The true treatment effect is 0.08. Estimates are stable across bandwidths from 0.2 to 1.0, ranging between approximately 0.07 and 0.08. At larger bandwidths (1.0+), estimates drift slightly downward as the linear specification picks up curvature from the quadratic term in the DGP. This stability is reassuring and characteristic of a well-behaved RD design.

Step 6: Test for Manipulation (McCrary Test)

The key RD assumption is that units cannot precisely manipulate the running variable to sort above or below the cutoff. If students can manipulate their GPA to avoid probation, there will be an unusual density of students just above 2.0 and a gap just below 2.0.

1# McCrary-style density test
2library(rddensity)
3density_test <- rddensity(X = df$X, c = 0)
4summary(density_test)
5
6# Visual
7rdplotdensity(density_test, df$X,
8            title = "Density Test at Cutoff",
9            xlabel = "GPA - 2.0")

Expected output: McCrary density test

Since the data is simulated from a smooth normal distribution with no manipulation, the p-value should be well above 0.05, confirming that the density of the running variable is continuous through the cutoff.

Step 7: Covariate Balance at the Cutoff

If the RD design is valid, pre-determined covariates should be balanced at the cutoff. We can test this by running the RD regression with covariates as the outcome.

1# Add covariates
2df$female <- rbinom(n, 1, 0.5)
3df$age <- 18 + rpois(n, 2)
4df$sat_score <- 1000 + 100 * (df$gpa - 2.5) + rnorm(n, 0, 80)
5
6# Balance tests
7covariates <- c("female", "age", "sat_score")
8for (cov in covariates) {
9rd <- rdrobust(y = df[[cov]], x = df$X, c = 0)
10cat(sprintf("%s: coef = %.3f, p = %.3f\n",
11            cov, rd$coef[1], rd$pv[3]))
12}
13cat("\nAll p-values should be large (no discontinuities)\n")

Expected output: Covariate balance tests at cutoff (bandwidth = 0.5)

All p-values are large (well above 0.05), indicating no statistically significant discontinuities in pre-determined covariates at the cutoff. This finding is expected: female and age are generated independently of GPA, and sat_score is correlated with GPA but the local linear specification absorbs the smooth relationship, leaving no jump at the cutoff. These results support the validity of the RD design.

Step 8: Exercises

1. Fuzzy RD. Modify the simulation so that only 70% of students below 2.0 actually receive probation (imperfect compliance). Estimate the treatment effect using fuzzy RD (essentially IV at the cutoff). Compare with the sharp RD estimate.

2. Donut hole RD. Drop observations very close to the cutoff (e.g., within 0.05 of 2.0) and re-estimate. If the estimate changes substantially, this shift suggests manipulation right at the cutoff.

3. Placebo cutoffs. Estimate the "treatment effect" at fake cutoffs (e.g., GPA = 2.5, GPA = 3.0). You should find no discontinuity at these placebo cutoffs.

4. Quadratic specification. Instead of local linear, try local quadratic regression. Compare the estimates. The results should be similar if the bandwidth is not too large.

Summary

In this lab you learned:

• RD exploits a known cutoff in the running variable to estimate causal effects for units near the threshold
• The RD plot is the most important piece of evidence — include it in every RD analysis
• Local linear regression with optimal bandwidth (via rdrobust) is the standard estimation approach
• Check bandwidth sensitivity by reporting estimates across a range of bandwidths
• Test the no-manipulation assumption using the McCrary density test
• Verify covariate balance at the cutoff as additional evidence of design validity
• The RD estimate is local: it applies to units near the cutoff and may not generalize to other parts of the running variable distribution