Instrumental variable ($Z$)

An instrumental variable changes an exposure or treatment but affects the outcome only through that exposure. Together, instrument relevance, instrumental-variable independence, and the exclusion restriction identify causal effects under suitable structural assumptions.

Instrument relevance requires the instrumental variable to change the conditional distribution of the exposure. A nearly irrelevant instrument is a weak instrument and can produce unstable estimates.

A weak instrument has only a small association with the endogenous exposure, making instrumental-variable estimators imprecise and potentially badly biased in finite samples.

Instrumental-variable independence requires the instrument to be independent of unmeasured causes of the outcome, usually conditionally on specified observed covariates.

The exclusion restriction requires an instrumental variable to affect the outcome only through the exposure whose causal effect is being studied.

Mendelian randomization uses inherited genetic variants associated with an exposure as instrumental variables. Its causal interpretation requires instrument relevance, instrumental-variable independence, and the exclusion restriction.

Horizontal pleiotropy occurs when a genetic instrument affects an outcome through a pathway other than the exposure under study. It violates the exclusion restriction in Mendelian randomization.

Residual exposure stratification subtracts the genetically predicted component of an exposure before forming exposure strata. Under a suitable additive first-stage model, this avoids directly conditioning on the component caused by the instrument and can reduce the resulting collider bias.

A confounder-instrument interaction occurs when an instrument's effect on treatment varies with a variable that also affects the outcome. If the conditional first-stage effect is $\delta (U)=\mathbb{E}[A\mid Z=1,U]-\mathbb{E}[A\mid Z=0,U]$, absence of this interaction means that $\delta (U)$ is constant.

Instrumental-variable monotonicity requires changing the instrument in its encouraging direction never to move any unit's exposure in the opposite direction.

The local average treatment effect is the average causal effect among compliers, the units whose treatment status changes in the direction encouraged by the instrument.