Key Concepts of Regression Discontinuity Designs

Why This Matters

Regression Discontinuity Design sits at the heart of modern causal inference because it solves a fundamental problem: how do we estimate causal effects when we can't randomly assign treatment? RDD exploits arbitrary cutoffs—test score thresholds, age limits, income eligibility lines—to create conditions that approximate a randomized experiment. You're being tested on your ability to recognize when RDD is appropriate, understand what makes it credible, and identify threats to its validity.

The concepts here connect directly to broader themes in causal inference: identification strategies, local versus average treatment effects, and the bias-variance tradeoff in estimation. Don't just memorize that "bandwidth matters" or "manipulation is bad"—understand why observations near a cutoff serve as valid counterfactuals and what assumptions must hold for that logic to work. These distinctions will drive your FRQ responses and help you critically evaluate research designs.

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The Core Logic: Exploiting Discontinuities

RDD works because individuals just above and below a cutoff are essentially randomly assigned to treatment—at least locally. The key insight is that if the cutoff is arbitrary, people on either side should be comparable in all ways except their treatment status.

Definition and Basic Concept of RDD

• Quasi-experimental design exploiting a cutoff—the running variable (also called the forcing variable) determines treatment assignment at a specific threshold
• Local randomization assumption means individuals just above and below the cutoff are statistically identical, differing only in treatment receipt
• Identifies causal effects without randomization, making RDD one of the most credible observational methods when its assumptions hold

Sharp vs. Fuzzy RDD

• Sharp RDD has deterministic assignment—crossing the threshold guarantees treatment (e.g., everyone scoring ≥70 gets the scholarship)
• Fuzzy RDD involves probabilistic assignment where the cutoff changes the probability of treatment but doesn't determine it perfectly (think of it as an intention-to-treat scenario)
• Fuzzy designs require IV-style estimation, using the cutoff as an instrument; this estimates a Local Average Treatment Effect (LATE) for compliers

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Validity Requirements: What Must Hold

The credibility of any RDD hinges on whether the discontinuity in treatment is the only thing changing at the cutoff. These assumptions determine whether your causal claims are defensible.

Assumptions and Requirements for Valid RDD

• Continuity assumption requires that potential outcomes are smooth functions of the running variable—no other discontinuities exist at the cutoff
• No manipulation means individuals cannot precisely control their running variable to sort above or below the threshold
• Running variable must be continuous and treatment assignment must depend solely on its relationship to the cutoff

Testing for Manipulation of the Running Variable

• McCrary density test examines whether observations "bunch" suspiciously at the cutoff—a spike suggests people are gaming the system
• Covariate balance checks verify that predetermined characteristics don't jump at the threshold (if they do, something other than treatment is changing)
• Manipulation invalidates the design entirely because it means people on either side of the cutoff are systematically different

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Estimation Choices: The Bias-Variance Tradeoff

How you estimate the treatment effect matters enormously. The core tension: using more data reduces variance but risks bias from observations far from the cutoff.

Bandwidth Selection and Local Linear Regression

• Bandwidth defines the analysis window—only observations within this range of the cutoff contribute to estimation
• Narrower bandwidths reduce bias by focusing on observations most similar to each other, but increase variance due to smaller sample sizes
• Optimal bandwidth selection methods (like Imbens-Kalyanaraman or Calonico-Cattaneo-Titiunik) balance this tradeoff systematically

Graphical Analysis and Visualization in RDD

• Binned scatter plots show average outcomes at different values of the running variable—a visible "jump" at the cutoff signals a treatment effect
• Fitted regression lines on either side of the cutoff help visualize the estimated discontinuity and functional form assumptions
• Graphs serve as transparency devices, allowing readers to assess whether the effect is real or an artifact of modeling choices

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Robustness and Limitations: Stress-Testing Your Results

No single RDD estimate should be taken at face value. Credible research demonstrates that findings survive alternative specifications and acknowledges inherent limitations.

Sensitivity Analysis and Robustness Checks

• Vary the bandwidth to show results aren't driven by a single arbitrary choice—effects should be stable across reasonable ranges
• Test different functional forms (linear, quadratic, polynomial) to ensure the discontinuity isn't an artifact of model specification
• Placebo cutoffs at points where no treatment effect should exist help confirm that the real cutoff is special

Limitations and External Validity of RDD

• Local Average Treatment Effect (LATE) means RDD only identifies effects at the cutoff—not for the broader population
• Limited generalizability arises because people near thresholds may differ systematically from those far away
• Single-cutoff dependence means the entire design rests on one discontinuity; if that cutoff is problematic, the whole analysis fails

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Applications and Comparisons: RDD in Context

Understanding where RDD fits in the causal inference toolkit helps you recognize when it's the right method—and when alternatives might be stronger.

Applications and Examples of RDD in Various Fields

• Education research uses test score cutoffs for scholarships, gifted programs, or remediation (Angrist & Lavy's class size study is a classic)
• Policy evaluation exploits eligibility thresholds for social programs—age cutoffs for Medicare, income limits for benefits
• Health economics leverages age-based treatment rules or diagnostic thresholds to estimate intervention effects

Comparison of RDD with Other Causal Inference Methods

• Unlike RCTs, RDD doesn't require randomization but achieves credibility through local quasi-randomization at the cutoff
• Differs from matching methods because RDD exploits a known assignment mechanism rather than trying to balance observed covariates
• Related to instrumental variables—fuzzy RDD is an IV design where the cutoff instruments for treatment; both identify LATE for compliers

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Quick Reference Table

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Self-Check Questions

1. What distinguishes sharp RDD from fuzzy RDD, and how does this distinction affect what parameter you're estimating?

2. A researcher finds that predetermined covariates show a discontinuity at the cutoff. What does this suggest about the validity of the RDD, and what assumption is likely violated?

3. Compare and contrast bandwidth selection in RDD with the bias-variance tradeoff: why might a researcher report results across multiple bandwidths rather than choosing a single "optimal" one?

4. If an FRQ asks you to evaluate a study using RDD to estimate the effect of a scholarship on college completion, what three validity checks would you look for in the research design?

5. Why does RDD estimate a Local Average Treatment Effect rather than an Average Treatment Effect, and what does this imply for generalizing findings to other populations?