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).

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.

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 \(X_i\).

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.