Robust Inference in Fuzzy Regression Discontinuity Designs

Fuzzy regression discontinuity (RD) design and instrumental variable(s) (IV) regression share similar identification strategies and numerically yield the same results under certain conditions. While the weak identification problem is widely recognized in IV regressions, it has drawn much less attention in fuzzy RD designs, where the standard t-test can also suffer from asymptotic size distortions and the confidence interval obtained by inverting such a test becomes invalid. We explicitly model fuzzy RD designs in parallel with IV regressions, and based on the extensive literature of the latter, develop tests which are robust to weak identification in fuzzy RD designs, including the Anderson-Rubin (AR) test, the Lagrange multiplier (LM) test, and the conditional likelihood ratio (CLR) test. We show that these tests have correct size regardless of the strength of identification and that their power properties are similar to those in IV regressions. Due to the similarities between a fuzzy RD design and an IV regression, one can choose either method for estimation and inference. However, we demonstrate that adopting a fuzzy RD design with our newly proposed tests has the potential to achieve more power without introducing size distortions in hypothesis testing and it is thus recommended. An extension to testing for quantile treatment effects in fuzzy RD designs is also discussed.