  Regression discontinuity designs (RDDs) are a common quasi-experiment in
economics and statistics. The most popular methodologies for analyzing RDDs
utilize continuity-based assumptions and local polynomial regression, but
recent works have developed alternative assumptions based on local
randomization. The local randomization framework avoids modeling assumptions by
instead placing assumptions on the assignment mechanism near the cutoff.
However, most works have focused on completely randomized assignment
mechanisms, which posit that propensity scores are equal for all units near the
cutoff. In our review of the local randomization framework, we extend the
framework to allow for any assignment mechanism, such that propensity scores
may differ. We outline randomization tests that can be used to select a window
around the cutoff where a particular assignment mechanism is most plausible, as
well as methodologies for estimating causal effects after a window and
assignment mechanism are chosen. We apply our methodology to a fuzzy RDD
assessing the effects of financial aid on college dropout rates in Italy. We
find that positing different assignment mechanisms within a single RDD can
provide more nuanced sensitivity analyses as well as more precise inferences
for causal effects.
