  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 is (A1) associated with the exposure, (A2) has
no direct effect on the outcome except through the exposure, and (A3) is not
related to unmeasured confounders. Unfortunately, finding variables that
satisfy conditions (A2) or (A3) can be challenging in practice. This paper
reviews works where instruments may not satisfy conditions (A2) or (A3), which
we refer to as invalid instruments. We review identification and inference
under different violations of (A2) or (A3), specifically under linear models,
non-linear models, and heteroskedatic models. We conclude with an empirical
comparison of various methods by re-analyzing the effect of body mass index on
systolic blood pressure from the UK Biobank.
