Causal Inference with Regression Discontinuity

Causal inference asks what would have happened differently if a treatment had not been applied. When randomized controlled trials are unethical or impossible, researchers turn to clever quasi-experimental designs. Regression Discontinuity Design (RDD) is one of the most credible, exploiting arbitrary rules in policy or nature to estimate causal effects. By comparing units just above and just below a strict cutoff, you can isolate the impact of a program or intervention as if by a localized experiment.

The Core Logic of a Discontinuity

At its heart, RDD leverages a precise, formulaic rule for treatment assignment. Imagine a policy where students who score 80% or higher on an entrance exam receive a scholarship. The running variable (or score) is the pre-determined, continuous measure used for assignment—here, the exam score. The threshold (or cutoff) is the specific value (80%) that determines eligibility.

The central identifying assumption is that units (students) just below and just above the cutoff are virtually identical in all other observed and unobserved characteristics. Any sudden jump in an outcome (e.g., college graduation rates) at precisely the 80% threshold can then be plausibly attributed to the scholarship, not to underlying differences between students. This is because a student who scores 79.9% is likely very similar to one who scores 80.1%, except for the scholarship award.

Sharp vs. Fuzzy RDD: Compliance with the Rule

RDD comes in two primary flavors, distinguished by how strictly the assignment rule is followed. In a Sharp RDD, treatment assignment is a deterministic function of the running variable. If you meet the cutoff, you get the treatment; if you don’t, you don’t. Our scholarship example is sharp: every student with a score ≥80 gets the scholarship, and no student below 80 does. The probability of treatment jumps from 0 to 1 at the threshold.

In a Fuzzy RDD, the rule is imperfectly followed. While the probability of receiving treatment jumps at the cutoff, it is not a perfect 0-to-1 shift. Perhaps some students above 80% decline the scholarship, or some below 80% receive one through a different channel. This non-compliance mirrors the structure of an instrumental variables setup. The causal effect is estimated for compliers—those units whose treatment status is actually changed by the rule—using the discontinuity as an instrument. The estimator is the ratio of the jump in the outcome to the jump in the treatment probability at the cutoff.

Estimation: Local Polynomial Regression and Bandwidth

You cannot simply compare the means of all treated and untreated units, as those far from the cutoff are likely very different. RDD relies on local polynomial estimation, typically linear or quadratic, fitted separately on either side of the cutoff. The goal is to extrapolate the trend of the outcome near, but not across, the threshold to estimate what would have happened to the treated group had they not received treatment.

The choice of bandwidth—how far from the cutoff to include data—is a critical trade-off. A narrower bandwidth uses data closer to the cutoff, where the "as-good-as-random" assumption is strongest, but it yields fewer data points and higher variance. A wider bandwidth increases precision but risks bias by including units that are less comparable. Modern methods, like cross-validation or plug-in estimators, select a bandwidth to minimize mean squared error. The key is to check that the estimated effect is not overly sensitive to reasonable bandwidth choices.

Testing the Design: Manipulation and Validity

The validity of an RDD rests on the assumption that units cannot precisely manipulate their running variable to land on a specific side of the cutoff. If they could, then those just above and just below are no longer comparable—they differ in their motivation or ability to manipulate. The McCrary density test is a standard check for this. It examines whether the density (number of observations) of the running variable is continuous at the threshold. A significant drop or spike suggests individuals are sorting around the cutoff, threatening the design's integrity.

Other essential validity checks include testing for continuity in pre-determined covariates. If the units are truly comparable at the threshold, then characteristics like age, prior income, or gender should not jump at the cutoff. Plotting these covariates against the running variable and testing for a discontinuity is a routine diagnostic.

Applications in Policy and Natural Experiments

RDD is a workhorse in applied economics and policy evaluation. Classic examples include estimating the effect of: financial aid on college persistence (using GPA cutoffs), class size on student achievement (using enrollment-based rules), or medical treatments on health outcomes (using age or test score eligibility thresholds). In natural experiments, researchers seek out real-world scenarios where similar arbitrary rules create a discontinuity, such as laws that take effect on a specific date or funding formulas based on population thresholds.

The design's strength is its transparent, visual intuition. A compelling RDD result is often shown in a scatterplot with the outcome on the Y-axis and the running variable on the X-axis, with fitted regression lines on either side of a clear vertical cutoff line. The gap between the lines at the point of discontinuity is the estimated causal effect.

Common Pitfalls

1. Ignoring Bandwidth Sensitivity: Reporting an effect from a single, arbitrarily chosen bandwidth is misleading. Always present a sensitivity analysis, such as a plot showing how the point estimate and its confidence interval change across a range of plausible bandwidths. The effect should be reasonably stable.
2. Overlooking Functional Form Mis-specification: Using the wrong polynomial order (e.g., a linear fit when the relationship is clearly curved) can create a false discontinuity or mask a real one. Always compare linear, quadratic, and perhaps cubic fits, and use placebo tests by checking for discontinuities at false cutoffs where no treatment changes.
3. Misinterpreting a Fuzzy RDD Estimate: The estimated Local Average Treatment Effect (LATE) in a fuzzy RDD applies only to compilers—those whose treatment status was changed by the rule. It is not the average effect for the entire population. Clearly define and describe this subgroup in your interpretation.
4. Relying Solely on the McCrary Test: While important, the McCrary test has limited power. The absence of a density jump does not prove no manipulation. Use it alongside other tests, like covariate balance checks, and consider the institutional context: could people theoretically manipulate their score?

Summary

• Regression Discontinuity Design (RDD) isolates causal effects by comparing units just above and just below an arbitrary cutoff in a running variable that determines treatment eligibility.
• In a Sharp RDD, treatment assignment is deterministic at the cutoff; in a Fuzzy RDD, the probability of treatment jumps discontinuously, requiring an instrumental variables approach to estimate the effect for compilers.
• Estimation is performed using local polynomial regression within an optimal bandwidth around the cutoff, balancing the bias-variance trade-off.
• Critical validation includes testing for manipulation of the running variable using the McCrary density test and confirming that pre-determined covariates are continuous at the threshold.
• RDD is a powerful tool for evaluating policies and natural experiments, providing highly credible causal estimates when randomization is not feasible, but its validity hinges on the untestable assumption of no precise manipulation at the cutoff.