  Regression discontinuity designs are extensively used for causal inference in
observational studies. However, they are usually confined to settings with
simple treatment rules, determined by a single running variable, with a single
cutoff. Motivated by the problem of estimating the impact of grade retention on
educational and juvenile crime outcomes in Chile, we propose a framework and
methods for complex discontinuity designs that encompasses multiple treatment
rules. In this framework, the observed covariates play a central role for
identification, estimation, and generalization of causal effects.
Identification is non-parametric and relies on a local strong ignorability
assumption. Estimation proceeds as in any observational study under strong
ignorability, yet in a neighborhood of the cutoffs of the running variables. We
discuss estimation approaches based on matching and weighting, including
complementary regression modeling adjustments. We present assumptions for
generalization; that is, for identification and estimation of average treatment
effects for target populations. We also describe two approaches to select the
neighborhood for analysis. We find that grade retention in Chile has a negative
impact on future grade retention, but is not associated with dropping out of
school or committing a juvenile crime.
