Instrumental variable

An instrumental variable is used to estimate a causal effect when it changes exposure but affects the outcome only through that exposure. Its validity also requires that it be independent of unmeasured causes of the outcome.

Two-stage least squares: An instrumental-variable estimation method that predicts exposure from instruments, then relates the prediction to the outcome. It implements instrumental-variable estimation in linear regression.
Causal inference: The study of methods for estimating the effects of interventions or exposures on outcomes. Instrumental variables are one strategy for identifying effects when direct adjustment is inadequate.
Natural experiment: An observational setting in which external events or rules create variation resembling random assignment. Many instruments are drawn from naturally occurring shifts in exposure.
Randomized controlled trial: An experiment that randomly assigns participants to treatment or control groups. Random assignment directly balances treatment groups, while an instrument usually shifts exposure indirectly.
Local average treatment effect: The average treatment effect among units whose treatment status changes in response to an instrument. Under common assumptions, this is the effect identified by a binary instrument.
Wald estimator: A ratio estimator of a causal effect, dividing an instrument’s effect on the outcome by its effect on exposure. It gives the instrumental-variable estimate for a binary instrument and exposure under standard assumptions.
Causal effect: The change in an outcome caused by changing an exposure or intervention. An instrument is useful only insofar as it identifies a causal effect of exposure.
Randomized encouragement design: A study design that randomizes encouragement to receive a treatment while allowing actual uptake to vary. Random encouragement can serve as an instrument for treatment uptake.
Propensity score: The probability of receiving a treatment given measured covariates. Propensity methods address measured confounding, whereas instruments can address some unmeasured confounding.
Monotonicity assumption: An assumption that an instrument does not move some units toward treatment while moving others away. It helps define the subgroup represented by a local average treatment effect.

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