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An instrumental variable $G$ supplies variation in an exposure $B$ that can be used for causal inference about its effect on $Y$. Its three core requirements are:

Relevance, independence, and exclusion are the core instrumental-variable assumptions. They do not by themselves identify an arbitrary population-average effect. For a binary instrument and treatment, instrumental-variable monotonicity and the usual consistency in causal inference assumptions can identify a local average treatment effect; stronger effect-homogeneity or structural-model assumptions are needed for other effect interpretations.

Let $Z$ indicate a recent birthday event, $A$ indicate a social gathering, and $Y$ denote subsequent household COVID-19 infection. The standard instrumental variable conditions are:

The usual consistency in causal inference, well-defined exposure, and absence of relevant interference in causal inference are also needed to interpret the result causally.

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.