Regression Discontinuity

Regression Discontinuity (RD) is a quasi-experimental research design used for causal inference when random assignment is not feasible. It exploits a predetermined cutoff or threshold in an assignment variable (also called the running or forcing variable) to identify the causal effect of a treatment or intervention. Units just above and just below the cutoff are assumed to be comparable, except for the treatment status, allowing researchers to estimate the local average treatment effect at the cutoff.

Core Concept and Identification Strategy

The fundamental idea behind Regression Discontinuity is that treatment assignment changes discontinuously at a known cutoff point of a continuous assignment variable. For example, a scholarship might be awarded only to students whose test scores exceed a particular threshold. Students who score just below and just above this threshold are likely similar in all respects except for receiving the scholarship, allowing for a credible comparison.

The causal effect is identified by comparing the expected outcomes immediately to the right and to the left of the cutoff. Formally, if the assignment variable is denoted by X and the cutoff by c, then treatment T is assigned if and only if X ≥ c. The parameter of interest is the difference in expected outcomes:

where Y is the outcome variable. This difference estimates the average causal effect of the treatment for units at the cutoff.

Types of Regression Discontinuity Designs

Sharp Regression Discontinuity

In a sharp RD design, the treatment assignment is perfectly determined by the cutoff. That is, every unit with an assignment variable value above the cutoff receives the treatment, and every unit below does not:

This clear rule simplifies the estimation and interpretation of the treatment effect.

Fuzzy Regression Discontinuity

In a fuzzy RD design, the probability of treatment changes discontinuously at the cutoff but not deterministically. The treatment assignment is not perfectly enforced, and some units may not comply with the cutoff rule. Formally, the probability of receiving treatment jumps at the cutoff but does not change from 0 to 1:

In this setting, the RD design is analogous to an instrumental variables estimator where the cutoff indicator acts as an instrument for treatment receipt.

Estimation and Implementation

Functional Form and Bandwidth Selection

RD estimation typically involves fitting regression functions on either side of the cutoff and comparing their limits at the threshold. Researchers must choose the functional form (e.g., linear, polynomial) and the bandwidth, which determines the range of observations near the cutoff used in estimation.

Smaller bandwidths reduce bias by focusing on observations closer to the cutoff but increase variance due to fewer data points. Conversely, larger bandwidths lower variance but may introduce bias by including observations less comparable to those at the threshold.

Common estimation techniques include local linear regression with uniform or triangular kernel weights, which assign greater weight to observations closer to the cutoff.

Estimation Equation in Sharp RD

A typical local linear regression for a sharp RD can be expressed as:

where:

• Y_i is the outcome for unit i
• D_i is the treatment indicator, equal to 1 if X_i ≥ c and 0 otherwise
• X_i - c is the running variable centered at the cutoff
• \tau captures the treatment effect at the cutoff
• \epsilon_i is the error term

The coefficient \tau is the primary parameter of interest, measuring the jump in the outcome at the cutoff.

Estimation in Fuzzy RD

For fuzzy RD, a two-stage least squares (2SLS) approach is used:

• First stage: Regress treatment receipt on the cutoff indicator and running variable
• Second stage: Regress the outcome on the predicted treatment status from the first stage

The cutoff indicator serves as an instrument for treatment receipt, and the estimated local average treatment effect is the ratio of the discontinuity in the outcome to the discontinuity in the treatment probability.

Assumptions and Validity

Continuity Assumption

The key identifying assumption is that all other factors affecting the outcome are continuous at the cutoff. This ensures that any discontinuity in the outcome can be attributed solely to the treatment effect. Formally, potential outcomes without treatment must be continuous at the threshold.

No Manipulation of the Running Variable

It is assumed that agents cannot precisely manipulate their assignment variable to sort just above or below the cutoff. If manipulation occurs, the comparison groups are no longer comparable, invalidating the RD design.

This assumption can be tested by examining the density of the running variable near the cutoff to detect any discontinuities, often using the McCrary density test.

Covariate Balance

Covariates should not show discontinuities at the cutoff, reinforcing that the only difference between groups at the threshold is the treatment status.

Advantages and Limitations

Advantages

• RD designs produce credible causal estimates with minimal modeling assumptions, often considered close to randomized experiments.
• They allow estimation of treatment effects in observational settings where randomization is infeasible.
• The local nature of the estimate provides a clear interpretation of the treatment effect at the cutoff.

Limitations

• The treatment effect is local to the cutoff and may not generalize to the entire population.
• Requires a sufficiently large number of observations near the cutoff to produce precise estimates.
• Sensitive to functional form and bandwidth choices; improper selection can bias results.
• Assumption of no manipulation may be violated in some contexts.

Extensions and Applications

Multivariate and Geographic RD

Extensions include designs where treatment assignment depends on multiple running variables or geographic boundaries, allowing for richer empirical applications.

RD with Multiple Cutoffs

Some settings involve multiple cutoffs, enabling comparisons of heterogeneous treatment effects across different thresholds.

Use in Policy Evaluation and Economics

Regression Discontinuity is widely used in economics for evaluating the impact of policies such as educational programs, social welfare eligibility, taxation thresholds, and more. By exploiting natural or administrative cutoff points, RD designs provide robust estimates of program effectiveness.

Practical Implementation Steps

1. Plot the data: Visual inspection of the outcome variable against the running variable helps detect discontinuities and guides bandwidth choice.
2. Check assumptions: Test for manipulation of the running variable and verify covariate continuity at the cutoff.
3. Choose bandwidth and functional form: Use cross-validation or data-driven methods to select an appropriate bandwidth and polynomial order.
4. Estimate the RD model: Apply local linear or polynomial regression techniques for sharp RD, or 2SLS for fuzzy RD.
5. Conduct robustness checks: Vary bandwidths, polynomial orders, and include placebo tests to confirm the stability of the results.
6. Interpret results cautiously: Emphasize the local nature of the estimated effect and discuss external validity considerations.

Summary of Key Elements

Regression Discontinuity designs provide a powerful framework for estimating causal effects in observational settings with a well-defined assignment rule based on a cutoff. Their rigorous assumptions and clear interpretation make them a cornerstone method in empirical economics and managerial economics research.