11.6 Chapter Summary

"The beauty of regression discontinuity is that it provides a transparent way to learn about causal effects."— Joshua Angrist & Jörn-Steffen Pischke, Mostly Harmless Econometrics

Master RDD to become a causal inference expert

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Core Concepts Review

The Essence of RDD

Regression Discontinuity Design is a quasi-experimental method that exploits the discontinuity in treatment assignment rules at a threshold to identify causal effects.

Core idea:

"In the neighborhood of the cutoff, RDD is as good as a randomized experiment. The discontinuity is your friend." — Joshua Angrist

Key elements:

1. Running Variable : Continuous variable determining treatment assignment
2. Cutoff : Threshold where treatment assignment changes
3. Treatment : Changes discontinuously at cutoff
4. Outcome : Variable we care about

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Sharp RDD vs Fuzzy RDD

Sharp RDD

Definition: Treatment status completely determined by cutoff

Identification:

Causal interpretation: Local average treatment effect at cutoff (LATE)

Classic cases:

• Lee (2008): Electoral vote share 50%
• Carpenter & Dobkin (2009): 21-year drinking age

Fuzzy RDD

Definition: Treatment probability jumps at cutoff, but not from 0 to 1

Identification (Wald estimator):

Essence: Instrumental variable (IV) estimation, cutoff as instrument

Causal interpretation: LATE for compliers

Classic case:

• Angrist & Lavy (1999): Class size (Maimonides' Rule)

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Core Assumptions

1. Continuity Assumption ⭐

Assumption: Potential outcome functions continuous at cutoff

In plain language: Without treatment, outcome doesn't jump at cutoff

Not directly testable, but can be verified indirectly through:

• Covariate balance tests
• Density test (McCrary Test)
• Placebo tests

2. No Precise Manipulation

Assumption: Individuals cannot precisely manipulate running variable to just cross cutoff

Test: McCrary density test

• : Density continuous at cutoff
• Reject → Possible manipulation

3. Local Exclusion

Assumption: Running variable affects outcome only through treatment (near cutoff)

Practice: Choose "exogenous" running variables (age, lottery numbers, exam scores, etc.)

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️ Empirical Implementation Steps

1. Data Preparation

import numpy as np
import pandas as pd
from rdrobust import rdrobust, rdplot, rdbwselect

# Center running variable
df['X_centered'] = df['X'] - cutoff
df['D'] = (df['X'] >= cutoff).astype(int)

2. Visualization (RDD Plot)

# Using rdplot
rdplot(y=df['Y'], x=df['X'], c=cutoff,
       title='RDD Plot',
       x_label='Running Variable',
       y_label='Outcome')

3. Main Effect Estimation

# Using rdrobust (CCT optimal bandwidth + robust inference)
result = rdrobust(y=df['Y'], x=df['X'], c=cutoff)
print(result)

4. Validity Tests

# Covariate balance
for cov in covariates:
    balance_test = rdrobust(y=df[cov], x=df['X'], c=cutoff)
    print(f"{cov}: p={balance_test.pval[0]:.4f}")

# Density test
from rddensity import rddensity
density_test = rddensity(X=df['X'], c=cutoff)
print(f"McCrary Test p-value: {density_test.pval[0]:.4f}")

# Placebo tests (false cutoffs)
placebo_cutoffs = [cutoff - 10, cutoff + 10]
for pc in placebo_cutoffs:
    placebo_result = rdrobust(y=df['Y'], x=df['X'], c=pc)
    print(f"Placebo {pc}: p={placebo_result.pval[0]:.4f}")

5. Robustness Tests

# Bandwidth sensitivity
bandwidths = [0.5*h_optimal, 0.75*h_optimal, h_optimal, 1.25*h_optimal]
for h in bandwidths:
    result_h = rdrobust(y=df['Y'], x=df['X'], c=cutoff, h=h)
    print(f"h={h:.2f}: Effect={result_h.coef[0]:.4f}")

# Donut-hole
df_donut = df[np.abs(df['X_centered']) >= 1]  # Exclude |X-c| < 1
result_donut = rdrobust(y=df_donut['Y'], x=df_donut['X'], c=cutoff)

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️ Common Pitfalls and Cautions

Pitfall 1: High-Order Polynomials ()

Problem: Gelman & Imbens (2019) warn against using high-order polynomials

• Prone to overfitting
• Unreliable confidence intervals

Recommendation:

• Use local linear ()
• At most local quadratic ()
• Avoid

Pitfall 2: Ignoring Clustering

Problem: If data has clustered structure (schools, regions), standard errors underestimated

Solution: Use clustered standard errors

result = rdrobust(y=df['Y'], x=df['X'], c=cutoff,
                 cluster=df['cluster_id'])

Pitfall 3: Not Reporting Multiple Cutoffs

Problem: If multiple cutoffs exist (e.g., 40, 80, 120), reporting only one creates selection bias

Solution:

• Report results for all cutoffs
• Or pool them (pooled RDD)

Pitfall 4: Extrapolating Beyond the Cutoff

Problem: RDD estimates LATE at cutoff, not overall ATE

Lesson:

• Clearly state estimating local effect
• Don't over-extrapolate to entire distribution

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Literature Recommendations

Essential Reviews

1. Lee, D. S., & Lemieux, T. (2010). "Regression Discontinuity Designs in Economics." Journal of Economic Literature, 48(2), 281-355.

   • The bible of RDD, must-read!

2. Imbens, G. W., & Lemieux, T. (2008). "Regression discontinuity designs: A guide to practice." Journal of Econometrics, 142(2), 615-635.

Methodological Breakthroughs

3. Calonico, S., Cattaneo, M. D., & Titiunik, R. (2014). "Robust Nonparametric Confidence Intervals for Regression-Discontinuity Designs." Econometrica, 82(6), 2295-2326.

   • CCT robust inference

4. Gelman, A., & Imbens, G. (2019). "Why High-Order Polynomials Should Not Be Used in Regression Discontinuity Designs." JBES, 37(3), 447-456.

   • Warning against high-order polynomials

Classic Applications

5. Thistlethwaite, D. L., & Campbell, D. T. (1960). "Regression-discontinuity analysis." J. Educational Psychology, 51(6), 309.

   • Birth of RDD

6. Angrist, J. D., & Lavy, V. (1999). "Using Maimonides' Rule to Estimate the Effect of Class Size on Scholastic Achievement." QJE, 114(2), 533-575.

7. Lee, D. S. (2008). "Randomized Experiments from Non-random Selection in U.S. House Elections." J. Econometrics, 142(2), 675-697.

8. Carpenter, C., & Dobkin, C. (2009). "The Effect of Alcohol Consumption on Mortality: Regression Discontinuity Evidence from the Minimum Drinking Age." AEJ: Applied, 1(1), 164-182.

Recommended Textbooks

• Angrist & Pischke (2009). Mostly Harmless Econometrics, Chapter 6
• Cunningham (2021). Causal Inference: The Mixtape, Chapter 6
• Huntington-Klein (2022). The Effect, Chapter 20

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Python Toolkit

Core Packages

# Installation
pip install rdrobust rddensity numpy pandas matplotlib seaborn statsmodels

# Import
from rdrobust import rdrobust, rdplot, rdbwselect
from rddensity import rddensity
import statsmodels.formula.api as smf
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

Quick Template

# 1. Basic RDD
result = rdrobust(y=Y, x=X, c=cutoff)
print(result)

# 2. Plotting
rdplot(y=Y, x=X, c=cutoff)

# 3. Fuzzy RDD
result_fuzzy = rdrobust(y=Y, x=X, c=cutoff, fuzzy=D)

# 4. Bandwidth selection
bw = rdbwselect(y=Y, x=X, c=cutoff)
print(f"Optimal bandwidth: {bw.bws[0]}")

# 5. Density test
density_test = rddensity(X=X, c=cutoff)
print(f"p-value: {density_test.pval[0]}")

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Practice Exercises

Exercise 1: Conceptual Understanding

1. Explain: Why is RDD called a "local RCT"?
2. Distinguish: What is the essential difference between Sharp RDD and Fuzzy RDD?
3. Assumption: Why can't the continuity assumption be directly tested? What can we do?

Exercise 2: Data Analysis

Data generation:

np.random.seed(123)
n = 1000
X = np.random.uniform(-50, 50, n)
D = (X >= 0).astype(int)
Y = 50 + 0.5*X + 0.01*X**2 + 10*D + np.random.normal(0, 5, n)

Tasks:

1. Draw RDD plot
2. Estimate treatment effect (using rdrobust)
3. Conduct covariate balance tests (generate your own covariates)
4. Conduct density test
5. Conduct bandwidth sensitivity analysis

Exercise 3: Case Replication

Choose one of the following classic papers and attempt to replicate main results:

• Angrist & Lavy (1999)
• Lee (2008)
• Carpenter & Dobkin (2009)

Hint: Many papers' data available on author websites or journal websites.

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From RDD to Broader Causal Inference

RDD's Position in the Causal Inference Toolkit

Method	Data Requirement	Core Assumption	External Validity	Typical Applications
RCT	Experimental data	Random assignment	May be lower	Medicine, development economics
RDD	Cutoff rule	Continuity assumption	Local effect	Education, politics, public health
DID	Panel data	Parallel trends	May be broader	Policy evaluation
IV	Instrumental variable	Exclusion, relevance	Compliers	Returns to education, health

When to Use RDD?

Ideal scenarios:

1. Clear cutoff rule exists (policy, law, institution)
2. Cutoff is exogenous (non-manipulable)
3. Sufficient observations near cutoff
4. Care about local effect (at cutoff)

Not suitable scenarios:

1. Cutoff fuzzy or subjective
2. Severe manipulation (election fraud, exam cheating)
3. Too few observations near cutoff
4. Need overall ATE

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Next Steps in Learning

Advanced Topics

1. Multidimensional RDD: Multiple running variables
2. Dynamic RDD: Treatment effects varying over time
3. Kink RDD: Slope change at cutoff (not level jump)
4. Geographic RDD: Geographic boundaries as cutoffs
5. Machine Learning + RDD: Using ML to estimate conditional expectation functions

Related Methods

• Synthetic Control: Constructing counterfactuals
• Matching: Finding comparable control groups
• Event Study: Dynamic effects

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Final Words

Josh Angrist's advice:

"Econometrics is about using the right tool for the job. RDD is a powerful tool when you have a cutoff, but it's not a panacea. Always think carefully about identification."

The essence of RDD:

1. Simple yet powerful: Intuitive concept, credible identification
2. Local yet precise: Sacrifice external validity for internal validity
3. Transparent and testable: Assumptions can be verified indirectly

Remember:

• Seek clear cutoff rules
• Rigorously test continuity assumption
• Report robustness analyses
• Interpret carefully (LATE, not ATE)

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Conclusion

Congratulations on completing Module 11! You have now mastered:

• Core ideas and identification logic of RDD
• Differences between Sharp and Fuzzy RDD
• Validity and robustness testing
• Replication of classic cases
• Python implementation best practices

You now possess:

• Ability to identify RDD research opportunities
• Skills to rigorously implement RDD analysis
• Foundation to publish RDD research in top journals

Next steps:

• Read more classic RDD papers
• Seek cutoff rules in your research
• Practice, practice, practice!

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RDD: Local randomization, a powerful tool for causal inference!

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Chapter Complete

Thank you for studying Module 11!

For questions or suggestions, please feel free to discuss.

Happy Coding & Happy Researching!