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

The instrumental variable graph iswith no arrow $Z\to Y$ and no common cause of $Z$ with $A$ or $Y$.

In potential outcome notation, a valid instrument requires:

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.

LetandBecause $U$ contains all exposure-outcome confounding and $Z$ is independent of $U$, the exclusion restriction and the law of total expectation giveLikewise,The no confounder-instrument interaction assumption says $\delta (U)=\delta$ is constant. Relevance gives $\delta \ne 0$, so the Wald ratio isthe overall average treatment effect.