Identification and Estimation in Fuzzy Regression Discontinuity Designs with Covariates

We study fuzzy regression discontinuity designs with covariates and characterize the weighted averages of conditional local average treatment effects (WLATEs) that are point identified. Any identified WLATE equals a Wald ratio of conditional reduced-form and first-stage discontinuities. We highlight the Compliance-Weighted LATE (CWLATE), which weights cells by squared first-stage discontinuities and maximizes first-stage strength. For discrete covariates, we provide simple estimators and robust bias-corrected inference. In simulations calibrated to common designs, CWLATE improves stability and reduces mean squared error relative to standard fuzzy RDD estimators when compliance varies. An application to Uruguayan cash transfers during pregnancy yields precise RDD-based effects on low birthweight.