Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 221 / 4 / ii / Solution

Instrumental-variable monotonicity isIt excludes defiers who would quit without the incentive but continue smoking when offered it. The possible principal strata are then never-takers $(0,0)$, compliers $(0,1)$, and always-takers $(1,1)$.

By random assignment, consistency, and exclusion,The second identity follows by checking the two possible binary exposure values. Under monotonicity, $A(1)-A(0)$ is the indicator of being a complier. Therefore the numerator iswhileTheir ratio is the complier average treatment effect, also called the local average treatment effect.