Multidimensional Regression Discontinuity and Regression Kink Designs with Difference-in-Differences

This paper proposes some extensions to Calonico, Cattaneo, and Titiunik’s (2014, Stata Journal 14: 909-946) commands for regression discontinuity (RD) and regression kink (RK) designs. First, the commands allow users to take a second difference in RD and RK (and higher order derivatives). That is, they combine RD/RK estimators with difference-in-differences (DID). The DID approach may be appropriate, for instance, when the density function of the running variable is discontinuous around the cutoff but steady over time. Second, they let users to specify multidimensional running variables, which is applicable, for example, to the estimation of boundary discontinuity. Finally, users can work with weighted data by specifying analytic weights and include control variables. For any of those extensions, the command ddbwsel calculates the MSE-optimal bandwidth, as proposed by Imbens and Kalyanaraman (2012, Review of Economic Studies 79: 933–959) and Calonico, Cattaneo, and Titiunik (2014, Econometrica 82: 2295-2326). The command ddrd implements the robust bias-corrected confidence intervals for any of the above extensions. The command dddens implements McCrary’s (2008, Journal of Econometrics 142: 698-714) density test for cross-sectional RD and RK, and RD and RK with DID. The paper presents applications of two-dimensional RD with DID and one-dimensional RK with DID.