A regression discontinuity design (RDD) is an example of a quasi-experimental design in which the probability of receiving a treatment is a discontinuous functions of one or more underlying variables. Comparing observations near the discontinuity allows for the estimation of a local causal effect of the treatment on an outcome.
In a regression discontinuity design, treatment D get assigned on the basis of some continuous covariate or covariates crossing an explicit threshold. This often happens in circumstances where a treatment is triggered by an administrative or organizational rule. One example is the effect of scholarship receipt on college enrollment. If students become eligible when their test score or grade point average exceed some value, one can compare the difference in enrollment rates for students who are just below the cutoff to students who are just over it. The variable defining the discontinuity must not be easily changed by the agent (or anyone else) to obtain or avoid the treatment.
The basic formula is a ratio of two differences in means: $$\Delta_{RD} = \frac{\bar Y_+-\bar Y_-}{\bar D_+-\bar D_-},$$
where $Y$ is the outcome of interest, $D$ is the binary treatment indicator, and $+$ and $-$ subscripts indicate the position relative to the threshold.
There are two types of RDDs:
- Sharp RDD: probability of treatment moves from one to zero at the threshold value, so the denominator above is 1.
- Fuzzy RDD: probability of treatment is discontinuous at the threshold value, so the denominator is between one and zero.
Fuzzy RDD can be thought as a special case of instrumental variables and the Wald estimator. Sharp RDD can be considered a special case of matching.
