Regression Discontinuity Designs

Summary

RD designs exploit precise knowledge of rules determining treatment. When treatment is assigned based on a threshold in a running variable, comparing outcomes just above and below the cutoff provides credible causal estimates. Sharp RD is a selection-on-observables story; fuzzy RD is an IV strategy.

Sharp RD

Treatment is a deterministic, discontinuous function of a covariate:

$$D_i=\{\begin{array}{ll}1& \text{if }{x}_i\ge x_0\\ 0& \text{if }{x}_i<x_0\end{array}$$

The regression model:

$$Y_i=f(x_i)+\rho D_i+{\eta }_i$$

where $f(x_i)$ is a smooth function (often modeled with polynomials).

Parametric approach

$$Y_i=\alpha +{\beta }_1{x}_i+{\beta }_2{x}_i^2+...+{\beta }_p{x}_i^p+\rho D_i+{\eta }_i$$

Nonparametric approach

Compare means in a small neighborhood $[x_0-\delta ,x_0+\delta ]$:

$$\underset{\delta \to 0}{\lim }E[Y_i|x_0<x_i<x_0+\delta ]-E[Y_i|x_0-\delta <x_i<x_0]=E[Y_{1i}-Y_{0i}|x_i=x_0]$$

Key Example: Incumbency advantage (Lee, 2008)

• Running variable: vote share margin of victory
• Treatment: winning the current election
• Result: ~40 percentage point incumbency advantage in re-election probability

Fuzzy RD

Treatment probability jumps at the threshold but doesn’t go from 0 to 1:

$$P[D_i=1|x_i]=\{\begin{array}{ll}{g}_0(x_i)& \text{if }{x}_i\ge x_0\\ g_1(x_i)& \text{if }{x}_i<x_0\end{array}$$

Fuzzy RD = IV with $T_i=1(x_i\ge x_0)$ as the instrument for $D_i$.

Key Example: Class size in Israel (Angrist & Lavy, 1999)

• Maimonides’ Rule: class size capped at 40; cohort of 41 splits into two classes
• Instrument: predicted class size from the rule ($m_{sc}$)
• Result: 7-student reduction raises math scores by ~1.75 points (0.18σ)

Validity Checks

1. Pre-treatment covariates should show no jump at $x_0$
2. Density of $x_i$ should be smooth at $x_0$ (no bunching/manipulation)
3. Discontinuity sample robustness: results should be stable as the window narrows

Nonlinearity vs. Discontinuity

A sharp turn in $E[Y_{0i}|x_i]$ can be mistaken for a jump due to treatment. Use flexible functional forms and restrict to observations near the cutoff.

See Also

• Instrumental Variables — fuzzy RD is IV
• Local Average Treatment Effects — fuzzy RD estimates LATE on compliers at the cutoff
• Mostly Harmless Econometrics - Overview
• Model Checking — posterior predictive checks for formalizing RD validity tests
• The Experimental Ideal — RD approximates a local experiment at the cutoff; the ideal benchmark it approximates