7.4 Regression discontinuity analysis

Regression discontinuity analysis is a powerful tool for estimating causal effects when treatment depends on a cutoff. It assumes units just above and below the threshold are comparable, allowing for causal inference within a narrow bandwidth around the cutoff.

This method is particularly useful in policy evaluation, education, and healthcare settings. It requires careful consideration of assumptions, appropriate estimation techniques, and thorough robustness checks to ensure valid and reliable results.

Regression Discontinuity Analysis: Principles and Assumptions

Fundamental Concepts and Design

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• Regression discontinuity (RD) analysis employs quasi-experimental design to estimate causal effects when treatment assignment depends on a cutoff on a continuous variable
• Key assumption posits units just above and below the cutoff are comparable, differing only in treatment status
• RD designs categorized as sharp (perfect compliance with cutoff rule) or fuzzy (imperfect compliance), requiring distinct estimation approaches
• Continuity assumption states the relationship between outcome and assignment variable would be continuous at the cutoff without treatment
• "No manipulation" assumption prevents individuals from precisely manipulating their position relative to the cutoff
• Local randomization assumed near the cutoff allows for causal inference within a narrow bandwidth around the threshold

Statistical Assumptions and Implications

• Continuity of potential outcomes function at the cutoff underpins causal inference in RD designs
• Smoothness assumption requires the conditional expectation of potential outcomes to be continuous in the running variable
• Local conditional independence assumption states treatment assignment is as good as random within a small neighborhood of the cutoff
• Monotonicity assumption in fuzzy RD designs requires the probability of treatment to be non-decreasing at the cutoff
• Stable Unit Treatment Value Assumption (SUTVA) precludes interference between units and ensures consistency of potential outcomes
• Exclusion restriction in fuzzy RD designs assumes the cutoff affects outcomes only through its impact on treatment probability

Suitable Contexts for Regression Discontinuity Designs

Policy Evaluation and Educational Settings

• RD designs suit policy evaluations where eligibility hinges on specific thresholds (test scores for program admission, age limits for benefits)
• Educational settings provide fertile ground for RD analysis due to test scores or grade levels determining intervention placement
• Examples include evaluating the impact of:
  • Remedial education programs based on standardized test scores
  • Gifted and talented programs using IQ thresholds
  • Financial aid eligibility based on family income cutoffs
• RD proves valuable in assessing the effectiveness of educational policies (class size reduction, teacher quality initiatives)

Economic and Healthcare Applications

• Economic policies with eligibility criteria based on continuous variables align well with RD analysis (income thresholds for subsidies, tax brackets)
• Healthcare interventions utilizing clinical measures or risk scores for treatment decisions suit RD designs
• Examples encompass:
  • Evaluating the impact of minimum wage laws on employment levels
  • Assessing the effects of environmental regulations based on pollution thresholds
  • Analyzing the efficacy of preventive healthcare measures based on risk scores
• RD facilitates evaluation of retirement policies, unemployment benefits, and social welfare programs with clear eligibility cutoffs

Considerations and Limitations

• RD requires a sufficiently large sample size near the cutoff to ensure reliable treatment effect estimates
• The method proves less suitable when the assignment variable can be easily manipulated or when the cutoff lacks strict enforcement
• RD designs necessitate a clear, known cutoff determining treatment assignment based on a continuous running variable
• Contextual knowledge remains crucial to assess the plausibility of RD assumptions in specific applications
• Potential limitations include reduced external validity due to the local nature of the treatment effect estimate

Local Average Treatment Effect (LATE) Estimation and Interpretation

Estimation Techniques and Considerations

• Local Average Treatment Effect (LATE) in RD represents the causal impact of treatment for units near the cutoff threshold
• LATE estimation involves comparing outcomes just above and below the cutoff, typically using local linear regression or non-parametric methods
• Bandwidth selection balances the trade-off between bias (using data far from the cutoff) and variance (using too little data)
• Common bandwidth selection methods include:
  • Cross-validation techniques
  • Optimal bandwidth selectors (Imbens-Kalyanaraman, Calonico-Cattaneo-Titiunik)
• Sharp RD designs allow direct interpretation of LATE as the treatment effect at the cutoff
• Fuzzy RD designs employ instrumental variables methods, using the cutoff as an instrument for treatment assignment

Graphical Analysis and Interpretation

• Graphical analysis proves essential for visualizing the discontinuity and interpreting the LATE
• Key visualization techniques include:
  • Scatter plots of individual data points
  • Binned means to smooth out noise in the data
  • Local linear regression fits on either side of the cutoff
• Interpreting LATE requires acknowledging its local nature, cautioning against broad generalizations to units far from the cutoff
• Examples of LATE interpretation:
  • Effect of financial aid on college enrollment for students near the eligibility threshold
  • Impact of a pollution regulation on health outcomes for areas just above the regulatory limit

Robustness and Validity of Regression Discontinuity Estimates

Sensitivity Analysis and Validation Techniques

• Sensitivity analysis to bandwidth choice assesses the robustness of RD estimates
• Placebo tests using alternative cutoffs or outcomes unrelated to the treatment help validate the RD design
• Density tests of the running variable around the cutoff detect potential manipulation of the assignment variable
• Covariate balance tests near the cutoff provide evidence for the comparability of units just above and below the threshold
• Assessing the continuity of pre-treatment variables at the cutoff validates RD assumptions
• Exploring alternative functional forms for the relationship between outcome and running variable tests result sensitivity to model specification

Statistical Power and Precision

• Power analysis determines the ability to detect meaningful treatment effects given the sample size and variability in the data
• Precision calculations help researchers understand the reliability of RD estimates, especially given the local nature of the treatment effect
• Factors affecting statistical power in RD designs include:
  • Sample size near the cutoff
  • Effect size of the intervention
  • Variance of the outcome variable
• Researchers often use simulation studies to assess power and precision in RD designs
• Reporting confidence intervals alongside point estimates provides a measure of estimate precision