Extrapolating Treatment Effects in Multiple-Score Regression Discontinuity Designs

While regression discontinuity designs (RDDs) offer credible identification of treatment effects near cutoff points, their external validity remains inherently limited. To address this, a conditional independence assumption may be required. However, in multiple-score RDDs, this requires joint independence between all running variables and both potential outcomes and treatment assignments-a restriction often too strong to hold in practice. We relax this by assuming only mean independence between potential outcomes and the interaction of running variables, conditional on covariates. This weaker assumption allows us to link treatment effects at boundary and non-boundary points. We propose a non-parametric testing procedure and illustrate our method using simulation experiments and an empirical application in Colombian education policy, respectively.