27.1 Conceptual Framework
Regression discontinuity is best understood as a localized experiment at the threshold:
Internal Validity: Strong within a narrow bandwidth around the cutoff.
External Validity: Limited—results may not generalize beyond the bandwidth.
Comparison to Randomized Experiments: Empirical evidence suggests that RD and randomized controlled trials yield similar estimates ((Chaplin et al. 2018), Mathematica).
RD is connected to other causal inference methods:
Randomized Experiment: It’s local randomization.
Instrumental Variables: RD can be viewed as a structural IV model (J. D. Angrist and Lavy 1999).
Matching Methods: RD is a special case where matching occurs at a single threshold (J. J. Heckman, LaLonde, and Smith 1999).
Interrupted Time Series (ITS): RD is preferable when the running variable is finely measured. However, with highly discrete time data (e.g., quarterly or annual), ITS may be more appropriate. Hence, RD is always better than ITS if data are infinite (or substantially large).
27.1.1 Types of Regression Discontinuity Designs
- Sharp RD: Treatment probability jumps from 0 to 1 at the cutoff.
- Fuzzy RD: Treatment probability changes discontinuously but does not reach 1.
- Kink RD: Discontinuity occurs in the slope rather than the level of the running variable (Nielsen, Sørensen, and Taber 2010) (see applications in Böckerman, Kanninen, and Suoniemi (2018) and theoretical foundations in Card et al. (2015)).
- Regression Discontinuity in Time (i.e., Interrupted Time Series): The running variable is time.
Additional variations:
Multiple Cutoffs: Different thresholds across subgroups.
Multiple Scores: More than one running variable.
Geographic RD: Cutoff is spatially defined.
Dynamic Treatments: Treatment effects evolve over time.
Continuous Treatments: Instead of binary treatment, intensity varies.
27.1.2 Assumptions for RD Validity
Independent Assignment: The treatment is assigned solely based on the running variable.
Continuity of Conditional Expectations: The expected outcomes without treatment are continuous at the cutoff:
E[Y(0)|X=x] and E[Y(1)|X=x] are continuous at x=c.
Exogeneity of the Cutoff: The cutoff should not be manipulable. No confounding interventions at the cutoff.
No Discontinuity in Confounding Variables: Other covariates should be smooth at the threshold. A common test is to check for jumps in covariates unrelated to treatment.
27.1.3 Threats to RD Validity
| Issue | Description | Solution |
|---|---|---|
| Violation of Continuity in Covariates | If other variables besides treatment exhibit a discontinuity at the cutoff, the estimated effect may be biased. | Conduct balance tests on pre-treatment covariates. |
| Multiple Discontinuities | When multiple threshold effects exist, identification becomes more challenging. | Use robustness checks with alternative model specifications. |
| Manipulation of the Running Variable | Subjects may manipulate Xi to qualify for treatment (e.g., strategic behavior in test scores). | Implement McCrary’s density test to check for discontinuities in the distribution of Xi. |