Late again, whithout Monotonicity

Monotonicity is not necessary for the Wald ratio to identify a Local Average Treatment Effect. Under random assignment and exclusion restriction, if for every value of potential outcomes there are more compliers than defiers, the Wald ratio identifies the average treatment effect within a subpopulation of compliers. I use a simple Roy selection model to show that this "less defiers than compliers" condition is substantially weaker than monotonicity. It has two implications which are testable from the data, and it is closely related to those testable implications. Similarly, the local monotonicity condition in Huber & Mellace (2012) is not necessary for their identification results to hold and can also be replaced by a substantially weaker condition