Evaluating local average and quantile treatment effects under endogeneity based on instruments: a review

This paper provides a review of methodological advancements in the evaluation of heterogeneous treatment effect models based on instrumental variable (IV) methods. We focus on models that achieve identification through a monotonicity assumption on the selection equation and analyze local average and quantile treatment effects for the subpopulation of compliers. We start with a comprehensive discussion of the binary treatment and binary instrument case which is relevant for instance in randomized experiments with imperfect compliance. We then review extensions to identification and estimation with covariates, multi-valued and multiple treatments and instruments, outcome attrition and measurement error, and the identification of direct and indirect treatment effects, among others. We also discuss testable implications and possible relaxations of the IV assumptions, approaches to extrapolate from local to global treatment effects, and the relationship to other IV approaches