27.1 Conceptual Framework

Regression discontinuity is best understood as a localized experiment at the threshold:

RD is connected to other causal inference methods:


27.1.1 Types of Regression Discontinuity Designs

  1. Sharp RD: Treatment probability jumps from 0 to 1 at the cutoff.
  2. Fuzzy RD: Treatment probability changes discontinuously but does not reach 1.
  3. 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)).
  4. 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

  1. Independent Assignment: The treatment is assigned solely based on the running variable.

  2. 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.

  3. Exogeneity of the Cutoff: The cutoff should not be manipulable.

  4. 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

27.1.4 Violation of Continuity in Covariates

If other variables besides treatment exhibit a discontinuity at the cutoff, the estimated effect may be biased.

Solution: Conduct balance tests on pre-treatment covariates.

27.1.5 Multiple Discontinuities

When multiple threshold effects exist, identification becomes more challenging.

Solution: Use robustness checks with alternative model specifications.

27.1.6 Manipulation of the Running Variable

Subjects may manipulate Xi to qualify for treatment (e.g., strategic behavior in test scores).

Solution: Implement McCrary’s density test to check for discontinuities in the distribution of Xi.


References

Angrist, Joshua D, and Victor Lavy. 1999. “Using Maimonides’ Rule to Estimate the Effect of Class Size on Scholastic Achievement.” The Quarterly Journal of Economics 114 (2): 533–75.
Böckerman, Petri, Ohto Kanninen, and Ilpo Suoniemi. 2018. “A Kink That Makes You Sick: The Effect of Sick Pay on Absence.” Journal of Applied Econometrics 33 (4): 568–79.
Card, David, David S Lee, Zhuan Pei, and Andrea Weber. 2015. “Inference on Causal Effects in a Generalized Regression Kink Design.” Econometrica 83 (6): 2453–83.
Chaplin, Duncan D, Thomas D Cook, Jelena Zurovac, Jared S Coopersmith, Mariel M Finucane, Lauren N Vollmer, and Rebecca E Morris. 2018. “The Internal and External Validity of the Regression Discontinuity Design: A Meta-Analysis of 15 Within-Study Comparisons.” Journal of Policy Analysis and Management 37 (2): 403–29.
Heckman, James J, Robert J LaLonde, and Jeffrey A Smith. 1999. “The Economics and Econometrics of Active Labor Market Programs.” In Handbook of Labor Economics, 3:1865–2097. Elsevier.
Nielsen, Helena Skyt, Torben Sørensen, and Christopher Taber. 2010. “Estimating the Effect of Student Aid on College Enrollment: Evidence from a Government Grant Policy Reform.” American Economic Journal: Economic Policy 2 (2): 185–215.