External Validity in Fuzzy Regression Discontinuity Designs

Summary

Fuzzy regression discontinuity (FRD) identifies the local average treatment effect (LATE) only for compliers at the threshold — doubly local by compliance type and forcing-variable value. Bertanha and Imbens derive the testable implications of an external validity assumption (compliance type independent of potential outcomes given the forcing variable), show these implications are equivalent to continuity of $E[Y \mid X, W=w]$ at the threshold for each treatment arm separately, and recommend plotting these two treatment-arm-specific conditional expectations as a routine diagnostic alongside the standard RD graph. Two summer-school applications show partial vs. full violations of external validity.

Key Claims

• The standard Hausman test in FRD ($\tau^{\text{exo}} = \tau^{\text{frd}}$) is an obscure restriction that allows differences between compliance types to cancel out, making it hard to interpret.
• Angrist's (2004) adaptation is more natural but still tests a single scalar restriction that is not invariant to monotone transformations of the outcome.
• The appropriate pair of restrictions is: $E[Y(1)\mid \text{complier}, X=T^*] = E[Y(1)\mid \text{always-taker}, X=T^*]$ AND $E[Y(0)\mid \text{complier}, X=T^*] = E[Y(0)\mid \text{never-taker}, X=T^*]$.
• This pair is equivalent to continuity of $E[Y^{\text{obs}} \mid W=w, X]$ in $X$ at the threshold for $w \in \{0,1\}$ (Lemma 6) — a graphically inspectable nonparametric condition.
• Under the external validity assumption ($G_i \perp\!\!\!\perp (Y_i(0), Y_i(1)) \mid X_i$), the average treatment effect $\tau^{\text{ATE}}$ is identified across the full support of $X$.
• Jacob-Lefgren (Chicago): always-takers perform substantially worse than treated compliers (gap 0.36, s.e. 0.13, p=0.005); never-takers similar to non-treated compliers (gap −0.07, s.e. 0.06, p=0.21). Joint F-test: p=0.009.
• Matsudaira (NE district): both arms reject — always-takers worse than treated compliers (0.15, p<0.001) and never-takers better than non-treated compliers (−0.09, p=0.003). Joint F-test: p<0.001.

Concepts Introduced or Extended

• Fuzzy Regression Discontinuity
• Local Average Treatment Effect
• External Validity

Entities Mentioned

• Guido Imbens

Quotes

"We recommend that, in addition to those graphs, researchers present graphs containing estimates of the conditional expectations of the outcome given the forcing variable separately by treatment status. A discontinuity at the threshold in these conditional expectations provides evidence against exogeneity or unconfoundedness assumptions, and, thereby, evidence against external validity of the estimates."

My Take

A clean and practically useful paper. The main contribution is reframing external validity as a standard nonparametric continuity test — cheap to run alongside any FRD analysis and far more interpretable than a Hausman test. The paper sits somewhat outside the wiki's core Bayesian time-series focus, but the LATE / instrumental-variables (IV) identification framework is a fundamental reference in applied econometrics. One limitation: the test requires fuzzy (not sharp) RD, since it relies on observing both always-takers and never-takers near the threshold.