Treatment Effect Identification with Democratically Determined Treatment using DiD and RDD

Recent literature has studied how selection into treatment can be com- patible with difference in differences as an identification strategy. I build on this literature by focusing on democratic selection into treatment, and derive necessary and sufficient conditions for parallel trends to hold. I show that implementing difference in differences comparing groups that barely pass and don't pass the policy will consistently recover a local average treatment effect. I conclude by providing sufficient conditions for regression discontinuity design to recover the locally weighted average treatment effect described in Lee (2008) and micro-found this result in a model where treatment is democratically determined.