Regression Discontinuity

Regression discontinuity (RD) is a quasi experimental research design that estimates causal effects by exploiting a known threshold or cutoff on a continuous assignment variable that determines treatment eligibility. By comparing outcomes for individuals just above and just below the cutoff, where assignment is effectively as if random, the design produces credible causal estimates without randomization.

In many real world settings, treatments, programs, or policies are assigned based on whether an individual's score on a continuous measure exceeds a cutoff: students receive remedial education based on test scores, patients receive treatment based on clinical risk scores, and welfare benefits are determined by income thresholds. Regression discontinuity exploits these assignment rules to produce causal estimates that are considered among the most credible in all of quasi experimental research. Economists and policy researchers have used RD designs to evaluate educational programs, healthcare policies, environmental regulations, and electoral outcomes, producing evidence that is often cited alongside randomized trial results.

The logic of the RD design relies on the assumption that individuals just above and just below the cutoff are similar on all characteristics except for their treatment status. If the cutoff for receiving a scholarship is a test score of 80, then students scoring 79 and 81 are, in expectation, nearly identical in ability, motivation, and background. The only systematic difference is that the 81 scorers received the scholarship and the 79 scorers did not. Any discontinuity (jump) in the outcome at the cutoff can therefore be attributed to the treatment.

In a sharp RD design, treatment assignment is deterministic at the cutoff: everyone on one side of the cutoff receives treatment and no one on the other side does. The causal effect is estimated by comparing the regression functions of the outcome on the assignment variable from each side of the cutoff. Local polynomial regression or parametric models are fitted to data near the threshold, and the difference in the predicted outcomes at the cutoff is the estimated treatment effect.

In a fuzzy RD design, the cutoff increases the probability of treatment but does not determine it with certainty (some individuals above the cutoff do not receive treatment, and some below do). The fuzzy RD is analyzed as an instrumental variables problem, using the cutoff as an instrument for actual treatment receipt, and the estimated effect is a local average treatment effect for those whose treatment status is changed by the cutoff.

Key assumptions include continuity of potential outcomes at the cutoff (no other factor changes discontinuously at the threshold) and no manipulation of the assignment variable (individuals cannot precisely control whether they fall above or below the cutoff). The McCrary density test checks for bunching of observations at the cutoff, which would suggest manipulation.

Use regression discontinuity when a treatment, program, or policy is assigned based on a cutoff on a continuous variable and when the research question is whether that treatment has a causal effect on outcomes. The design is most credible when the cutoff is sharp, the assignment variable cannot be manipulated, and there are sufficient observations near the threshold. Note that RD estimates are local, meaning they apply to individuals near the cutoff and may not generalize to the broader population. When no meaningful cutoff exists, RD is not applicable, and when the assignment variable takes only a few distinct values (for example, a categorical variable), standard RD methods do not apply.

A common error is specifying the wrong functional form for the relationship between the assignment variable and the outcome, which can create spurious apparent discontinuities or mask real ones. Another mistake is using observations far from the cutoff, which introduces extrapolation bias. The bandwidth (how much data around the cutoff to include) should be selected using data driven methods such as those proposed by Imbens and Kalyanaraman, or by Calonico, Cattaneo and Titiunik, whose robust bias corrected confidence intervals are now widely used. Researchers sometimes fail to test for manipulation of the assignment variable, undermining the key assumption that individuals near the cutoff are comparable. Finally, generalizing RD estimates beyond the neighborhood of the cutoff requires additional assumptions that should be made explicit.

A state provides free health insurance to households earning below 200% of the federal poverty level. Researchers examine emergency department visit rates for households earning between 180% and 220% of the poverty level. Insurance coverage jumps from 35% just above the cutoff to 78% just below it, so the cutoff raises the probability of coverage without determining it. Using a fuzzy RD design, the researchers estimate that gaining health insurance reduces emergency department visits by 0.14 visits per person per year (95% CI: 0.06 to 0.22) among households near the eligibility threshold whose coverage depended on eligibility. The McCrary test shows no evidence of income manipulation at the cutoff.