Instrumental Variable

When you cannot randomise and cannot measure the confounder, sometimes nature has randomised something for you. An instrument is a variable that moves the treatment, has no other route to the outcome, and is not itself confounded — a lottery, a distance, an arbitrary cutoff. Its randomness is borrowed to identify an effect you could not otherwise reach.

The strength of the design is that it needs no measurement of the confounder at all. Its weakness is that the exclusion restriction — the instrument affects the outcome only through the treatment — is untestable, and a weak instrument amplifies any violation of it enormously. An instrument that moves the treatment only slightly gives an estimate whose bias can exceed the one it was meant to remove.

With instrument Z, treatment X and outcome Y, the Wald estimator is Cov(Y,Z)/Cov(X,Z). The denominator is instrument strength, so a weak instrument divides by something near zero and inflates both variance and any bias from a violated exclusion restriction. What it identifies is a local effect — the effect among those whose treatment the instrument actually moved, not the population average.