Complier Effect for Difference-in-Differences Ratio in Fuzzy Regression Discontinuity with Double Running Variables

We consider regression discontinuity (RD) design with two, not one, running/forcing variables for a single treatment. The main identified entity is a 'double-difference-based ratio', where the numerator is a 'local difference in differences (DD)' for the response, and the denominator is a local DD for the treatment. This generalizes the well-known ratio-form identified entity for RD with a single running variable. We then establish that the ratio is the effect on compliers at the cutoff, which also generalizes the complier interpretation in RD with a single running variable. We also briefly touch upon instrumental variable estimation for RD with two running variables