Identification and Inference with Invalid Instruments

Instrumental variables (IVs) are widely used to study the causal effect of an exposure on an outcome in the presence of unmeasured confounding. IVs require an instrument, a variable that ( $a$ ) is associated with the exposure, ( $b$ ) has no direct effect on the outcome except through the exposure, and ( $c$ ) is not related to unmeasured confounders. Unfortunately, finding variables that satisfy conditions $b$ or $c$ can be challenging in practice. This article reviews works where instruments may not satisfy conditions $b$ or $c$ , which we refer to as invalid instruments. We review identification and inference under different violations of $b$ or $c$ , specifically under linear models, nonlinear models, and heteroskedastic models. We conclude with an empirical comparison of various methods by reanalyzing the effect of body mass index on systolic blood pressure from the UK Biobank.