Regression discontinuity
By the end of this chapter, you will be able to:
- Distinguish sharp from fuzzy regression-discontinuity designs and state the identifying assumptions for each.
- Estimate the treatment effect at the cutoff using local linear regression with triangular kernel weights.
- Apply the Calonico-Cattaneo-Titiunik (CCT) bias-corrected confidence intervals.
- Test the validity of RD via McCrary’s density discontinuity test and covariate smoothness checks.
- Interpret the RD estimand as a local average treatment effect at the cutoff.
Three 75–90 min lectures.
Lecture 1. Setup, sharp RD, identification at the cutoff, local linear estimator (75 min). Lecture 2. Bandwidth selection, CCT bias correction, fuzzy RD and 2SLS at the cutoff (90 min). Lecture 3. Validity diagnostics: McCrary density test, placebo cutoffs, covariate smoothness. Hands-on with rdrobust (75 min).
1 The setup
A regression-discontinuity design exploits a rule that assigns treatment based on whether a continuous running variable \(X\) crosses a threshold \(c\). Receiving a scholarship iff a test score exceeds 85. Legal drinking iff age exceeds 21. Enrollment in Medicare iff age exceeds 65.
In the sharp RD case, treatment is a deterministic function of the running variable:
\[ D_i = \mathbf{1}\{X_i \geq c\}. \]
In the fuzzy RD case, crossing the threshold changes the probability of treatment but does not deterministically assign it.
2 Sharp RD
2.1 Identification at the cutoff
Assume the conditional expectation functions \(\mu_d(x) = \mathbb{E}[Y(d) \mid X = x]\) are continuous at \(x = c\). Define the sharp RD estimand:
\[ \tau_{\text{RD}} := \mathbb{E}[Y(1) - Y(0) \mid X = c] = \lim_{x \downarrow c} \mathbb{E}[Y \mid X = x] - \lim_{x \uparrow c} \mathbb{E}[Y \mid X = x]. \tag{1}\]
Continuity of \(\mu_d\) ensures that any discontinuity in \(\mathbb{E}[Y \mid X]\) at \(c\) is attributable to the treatment, not to a jump in potential outcomes. This is the identifying assumption.
2.2 Local linear regression
Estimate \(\mu_d\) on each side of the cutoff using local linear regression with a kernel. For bandwidth \(h\), fit
\[ \hat\mu_1(c) = \arg\min_{\alpha_1, \beta_1} \sum_{i: X_i \geq c} K\left(\frac{X_i - c}{h}\right) (Y_i - \alpha_1 - \beta_1 (X_i - c))^2, \]
and similarly for \(\hat\mu_0(c)\). The RD estimator is \(\hat\tau_{\text{RD}} = \hat\mu_1(c) - \hat\mu_0(c)\).
Typical kernel is triangular: \(K(u) = (1 - |u|) \mathbf{1}\{|u| \leq 1\}\). The local linear specification is preferred over local constant because it has smaller bias at the boundary (Fan-Gijbels 1996).
3 Bandwidth selection and CCT bias correction
Choosing \(h\) involves a bias-variance tradeoff: small \(h\) means less bias but higher variance. Imbens and Kalyanaraman (2012) [@imbens2012optimal] proposed an MSE-optimal bandwidth. Calonico, Cattaneo, and Titiunik (2014) [@calonico2014robust] refined the approach with robust bias-corrected confidence intervals.
Under smoothness of \(\mu_0, \mu_1\) at the cutoff and with an MSE-optimal bandwidth \(h_{\text{MSE}}\), the conventional RD estimator has asymptotic bias of order \(h^2\). CCT construct a bias-estimate \(\hat B\) from a second derivative of \(\mu\) and form the bias-corrected estimator \(\hat\tau^{\text{BC}} = \hat\tau_{\text{RD}} - \hat B\), with robust standard errors.
In applied work, use rdrobust (R or Python). It handles bandwidth selection, bias correction, and robust inference automatically.
4 Fuzzy RD
In fuzzy RD, crossing \(c\) shifts treatment probability by less than 1. The ratio of the outcome jump to the treatment jump gives the fuzzy RD estimand:
\[ \tau_{\text{fuzzy}} = \frac{\lim_{x \downarrow c} \mathbb{E}[Y \mid X=x] - \lim_{x \uparrow c} \mathbb{E}[Y \mid X=x]}{\lim_{x \downarrow c} \mathbb{E}[D \mid X=x] - \lim_{x \uparrow c} \mathbb{E}[D \mid X=x]}. \]
This is a 2SLS at the cutoff with the threshold indicator \(\mathbf{1}\{X \geq c\}\) as an instrument for \(D\). Under the LATE assumptions from Chapter 6, \(\tau_{\text{fuzzy}}\) is the LATE for compliers at the cutoff.
5 Validity diagnostics
5.1 McCrary density test
If units can manipulate \(X\), e.g., to score just above a threshold for a scholarship, the identifying assumption of continuity fails. McCrary (2008) [@mccrary2008manipulation] proposed a test for discontinuity in the density of the running variable at the cutoff. Significant discontinuity suggests manipulation and invalidates RD.
5.2 Covariate smoothness
Predetermined covariates should be continuous at the cutoff. A discontinuity in baseline covariates indicates a confounded assignment and undermines the identifying assumption. Re-run the RD with each covariate as an outcome; large effects are diagnostic.
5.3 Placebo cutoffs
Apply the RD procedure at fake cutoffs \(c' \neq c\). Effects there should be near zero.
6 Synthetic example
import numpy as np
import matplotlib.pyplot as plt
rng = np.random.default_rng(0)
N = 2000
X = rng.uniform(-1, 1, N)
c = 0.0
D = (X >= c).astype(int)
# True treatment effect at cutoff = 1.0
Y0 = 0.5 * X + X ** 2 + rng.normal(0, 0.3, N)
Y1 = Y0 + 1.0 + 0.3 * X # heterogeneous around cutoff
Y = D * Y1 + (1 - D) * Y0
# Local linear on each side with bandwidth h = 0.3
h = 0.3
def local_linear(X, Y, x0, h):
mask = np.abs(X - x0) <= h
w = 1 - np.abs(X[mask] - x0) / h
Xw = np.column_stack([np.ones(mask.sum()), X[mask] - x0])
W = np.diag(w)
beta = np.linalg.inv(Xw.T @ W @ Xw) @ Xw.T @ W @ Y[mask]
return beta[0]
mu0 = local_linear(X[X < c], Y[X < c], c - 1e-6, h)
mu1 = local_linear(X[X >= c], Y[X >= c], c + 1e-6, h)
print(f"Estimated RD effect: {mu1 - mu0:.3f} (true = 1.0)")7 Bibliographic notes
Thistlethwaite and Campbell (1960) invented the RD design in an educational scholarship study. Hahn, Todd, and van der Klaauw (2001) [@hahn2001identification] formalized the modern identification theory. Imbens and Lemieux (2008), “Regression Discontinuity Designs: A Guide to Practice,” Journal of Econometrics 142, is the standard applied reference.
8 Exercises
Exercise 8.1 (\(\star\star\)). Prove that the local linear estimator has smaller boundary bias than the local constant estimator at the cutoff.
Exercise 8.2 (\(\star\star\)). Derive the MSE-optimal bandwidth for local linear regression at a boundary.
Exercise 8.3 (\(\star\)). Replicate the synthetic example with manipulation: some units just below the cutoff shift themselves to just above. Demonstrate the failure of RD.
Exercise 8.4 (\(\star\star\star\)). Using rdrobust, apply RD to a published study’s dataset (suggestions: Angrist-Lavy 1999 maimonides-rule, Lee 2008 House election RDD). Compare hand-coded local linear to CCT’s bias-corrected CI.
9 References
Calonico, S., Cattaneo, M. D., and Titiunik, R. (2014). “Robust Nonparametric Confidence Intervals for Regression-Discontinuity Designs.” Econometrica 82(6), 2295–2326.
Fan, J., and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman and Hall.
Hahn, J., Todd, P., and van der Klaauw, W. (2001). “Identification and Estimation of Treatment Effects with a Regression-Discontinuity Design.” Econometrica 69(1), 201–209.
Imbens, G. W., and Kalyanaraman, K. (2012). “Optimal Bandwidth Choice for the Regression Discontinuity Estimator.” Review of Economic Studies 79(3), 933–959.
Imbens, G. W., and Lemieux, T. (2008). “Regression Discontinuity Designs: A Guide to Practice.” Journal of Econometrics 142(2), 615–635.
McCrary, J. (2008). “Manipulation of the Running Variable in the Regression Discontinuity Design: A Density Test.” Journal of Econometrics 142(2), 698–714.
Thistlethwaite, D. L., and Campbell, D. T. (1960). “Regression-Discontinuity Analysis: An Alternative to the Ex Post Facto Experiment.” Journal of Educational Psychology 51(6), 309–317.