Regression Discontinuity Designs with a Continuous Treatment

Many empirical applications of regression discontinuity (RD) designs involve a continuous treatment. This paper establishes identification and bias-corrected robust inference for such RD designs. Causal identification is achieved by utilizing changes in the distribution of the continuous treatment at the RD threshold (including the usual mean change as a special case). Applying the proposed approach, we estimate the impacts of capital holdings on bank failure in the pre-Great Depression era. Our RD design takes advantage of the minimum capital requirements which change discontinuously with town size. We find that increased capital has no impacts on the long-run failure rates of banks.