21  D6. Regression Discontinuity Design

21.1 About

Topics covered:

  • Sharp RDD: local average treatment effect at the threshold
  • Fuzzy RDD: IV interpretation when treatment assignment is probabilistic
  • Local linear regression estimation
  • Data-driven bandwidth selection (Calonico-Cattaneo-Titiunik)
  • Validity checks: McCrary density test, covariate smoothness, placebo cutoffs

21.2 Lecture Notes

 

TipComing soon

Detailed notes on RDD are under preparation.

21.3 Overview

Regression Discontinuity Design (RDD) exploits a discontinuous jump in the probability of treatment at a known threshold \(c\) of a continuous running variable \(X_i\):

\[D_i = \mathbf{1}[X_i \geq c]\]

Key idea: Units just below and just above the cutoff are similar on average. Any discontinuous jump in the outcome \(Y_i\) at \(c\) can be attributed to the treatment.

21.4 Sharp RDD

In the sharp RDD, treatment is a deterministic function of \(X_i\): \(D_i = \mathbf{1}[X_i \geq c]\).

The estimand is the Local Average Treatment Effect at the threshold: \[\tau_{RDD} = \lim_{x \downarrow c} E[Y_i \mid X_i=x] - \lim_{x \uparrow c} E[Y_i \mid X_i=x]\]

Identification assumption: \(E[Y_i(0)\mid X_i=x]\) and \(E[Y_i(1)\mid X_i=x]\) are continuous in \(x\) at \(c\).

21.5 Fuzzy RDD

In the fuzzy RDD, crossing the threshold changes the probability of treatment but does not determine it deterministically. The RDD estimator is an IV estimator using \(\mathbf{1}[X_i \geq c]\) as an instrument for \(D_i\):

\[\tau_{Fuzzy} = \frac{\lim_{x\downarrow c}E[Y_i\mid X_i=x] - \lim_{x\uparrow c}E[Y_i\mid X_i=x]}{\lim_{x\downarrow c}P(D_i=1\mid X_i=x) - \lim_{x\uparrow c}P(D_i=1\mid X_i=x)}\]

21.6 Estimation

Local linear regression

Estimate separate linear regressions on each side of the cutoff within a bandwidth \(h\):

\[Y_i = \alpha + \tau D_i + \beta_1(X_i - c) + \beta_2 D_i(X_i-c) + \varepsilon_i \quad \text{for } |X_i - c| \leq h\]

Local linear regression (weighted by a kernel) is preferred over polynomial regression for its better boundary properties.

Bandwidth selection

Optimal bandwidth trades off bias (from linearity approximation) and variance (from fewer observations). Calonico, Cattaneo & Titiunik (2014) provide data-driven bandwidth selection with bias-corrected inference.

21.7 Validity Checks

  • McCrary density test: test for manipulation of the running variable at \(c\) (no density jump should be present if assignment is as-good-as-random near \(c\))
  • Pre-treatment covariate smoothness: check that baseline covariates are smooth through \(c\)
  • Placebo cutoffs: test for jumps at non-threshold values

21.8 References

Imbens, G. W. and Lemieux, T. (2008). “Regression Discontinuity Designs: A Guide to Practice.” Journal of Econometrics, 142(2), 615–635.

Calonico, S., Cattaneo, M. D. and Titiunik, R. (2014). “Robust Nonparametric Confidence Intervals for Regression-Discontinuity Designs.” Econometrica, 82(6), 2295–2326.