Regression Discontinuity Design

Regression discontinuity (RD) exploits a threshold rule that assigns treatment based on whether a running variable crosses a cutoff. Units just above and just below the cutoff are assumed to be comparable.

Sharp RD

Treatment is deterministically assigned: everyone above the cutoff is treated, everyone below is not.

Fuzzy RD

Treatment assignment is probabilistic but has a jump at the cutoff. This is analyzed using IV-like local estimation.

Key Assumption

| Violation | Consequence |

|-----------|------------|

| Manipulation of running variable | Bias — people sort around cutoff |

| Discrete running variable | Binning required; may introduce bias |

| Covariate imbalance at cutoff | Suggests manipulation or confounding |

Local Estimation

In practice, estimate local polynomial regressions on each side of the cutoff.

Bandwidth Selection

The bandwidth determines the window around the cutoff used for estimation.

Covariate Balance Check

Before interpreting results, check that baseline covariates are continuous at the cutoff:

If covariates show discontinuities at the cutoff, the identifying assumption may be violated.

McCrary Density Test

Tests for manipulation of the running variable at the cutoff. If people can precisely control their running variable, they may sort around the cutoff.

Python Implementation

import numpy as np

import pandas as pd

import matplotlib.pyplot as plt

from rdrobust import rdrobust

from rdrobust import rddensity



np.random.seed(42)



# Simulate sharp RD data

n = 1000

X = np.random.uniform(-1, 1, n)  # Running variable

T = (X >= 0).astype(int)  # Treatment

Y = 2.0 * T + 3.0 * X + np.random.randn(n) * 0.5



# Main RD estimate

result = rdrobust(Y, X, c=0)

print("RD Estimate:")

print(result)



# Covariate balance check

Z = np.random.randn(n)  # Covariate

print("\nCovariate balance at cutoff:")

rd_z = rdrobust(Z, X, c=0)

print(rd_z)



# McCrary density test

density = rddensity(X, c=0)

print("\nMcCrary density test:")

print(density)



# Plot

fig, axes = plt.subplots(1, 2, figsize=(12, 5))



# Outcome

axes[0].scatter(X, Y, alpha=0.3, s=10)

axes[0].axvline(x=0, color='red', linestyle='--')

axes[0].set_title('Outcome vs Running Variable')

axes[0].set_xlabel('Running Variable')

axes[0].set_ylabel('Outcome')



# Density

axes[1].hist(X, bins=50, edgecolor='black')

axes[1].axvline(x=0, color='red', linestyle='--')

axes[1].set_title('Density of Running Variable')



plt.tight_layout()

plt.show()

Worked Example

Key Takeaways

Related Topics

• See Instrumental Variables for another quasi-experimental method

• See Difference-in-Differences for policy evaluation

• See Causal Inference for the potential outcomes framework