27.2 Types of Regression Discontinuity Designs

RD is not a one-size-fits-all approach. While the classic setup involves a binary treatment determined by a threshold-crossing rule, numerous extensions and variants exist, each tailored to different research contexts. Understanding these variations is essential for choosing the appropriate design and interpreting results correctly.

1. Sharp RD:The simplest and most intuitive form of RD:
   • Definition: Treatment assignment jumps deterministically from 0 to 1 at the cutoff.
   • Implication: Individuals below the threshold are never treated; those above always are.
   • Identification: The treatment effect is identified as the jump in the outcome at the cutoff.
   • Example: A scholarship awarded strictly to students with test scores ≥ 85.
2. Fuzzy RD:A generalization of the sharp design:
   • Definition: The probability of treatment increases discontinuously at the cutoff, but does not jump from 0 to 1.
   • Implication: Some individuals below the cutoff may receive the treatment, and some above may not.
   • Identification: Requires instrumental variable methods to estimate the local average treatment effects (LATE) for compliers.
   • Example: A healthcare subsidy that is more likely, but not guaranteed, to be received by patients above a certain income level.
3. Kink RD:A subtler variant based on a change in slope:
   • Definition: The first derivative (slope) of the outcome changes at the cutoff, rather than its level.
   • Identification: The treatment effect is identified from a discontinuity in the marginal effect of the running variable.
   • Example: Tax incentives where the marginal benefit changes at a threshold income level.
   • References: See theoretical foundations in (Card et al. 2015) and empirical applications in (Böckerman, Kanninen, and Suoniemi 2018; Nielsen, Sørensen, and Taber 2010).
4. Regression Discontinuity in Time (RDiT): Also known as a special case of Interrupted Time Series:
   1. Definition: The running variable is time, and the cutoff corresponds to a policy implementation date or event.
   2. Application: Evaluates the immediate impact of an intervention that begins at a known point in time.
   3. Caveat: Since time trends are often confounded by other events, careful modeling of pre- and post-trends is essential.

Note: RDiT is often less reliable than traditional RD due to potential violations of the continuity assumption, especially when time is measured in coarse intervals (e.g., quarterly or yearly).

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RD designs have been extended in several directions to accommodate real-world complexities:

• Multiple Cutoffs:
  • Use case: Different treatment thresholds apply to different subgroups (e.g., age groups, regions).
  • Challenge: Requires careful pooling and stratification to maintain comparability.
• Multiple Scores (Multiple Running Variables):
  • Use case: Treatment depends on more than one score (e.g., income and age).
  • Approach: Involves multivariate threshold rules or interaction effects.
• Geographic RD (Spatial RD):
  • Definition: The cutoff is defined in space, such as administrative borders or policy boundaries.
  • Example: A tax policy that applies only to businesses within a specific jurisdiction.
• Dynamic Treatments:
  • Definition: Treatment effects may evolve over time after the initial intervention.
  • Consideration: Requires longitudinal data and appropriate modeling of lagged effects.
• Continuous Treatment Intensity (Dose-Response RD):
  • Definition: Instead of a binary treatment, units receive varying degrees of treatment intensity based on the running variable.
  • Example: Advertising exposure that increases progressively with customer engagement scores.

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27.2.1 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:

   $$E[Y(0)|X=x] \text{ and } E[Y(1)|X=x] \text{ are continuous at } x = c.$$

3. Exogeneity of the Cutoff: The cutoff should not be manipulable. No confounding interventions at the cutoff.

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.

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 $X_i$ to qualify for treatment (e.g., strategic behavior in test scores).	Implement McCrary’s density test to check for discontinuities in the distribution of $X_i$.

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References

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.

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.