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ResearchEconometrics / Causal inference / IV / Method notes

Instrumental variables: when the regressor is part of the problem

Econometrics03 / NOTE

If an explanatory variable is correlated with the error, OLS is inconsistent. An instrument that moves the regressor without touching the error restores identification — under one assumption you can never test.

OLS is consistent only if the regressors are uncorrelated with the error. When they aren’t — through omitted variables, simultaneity, measurement error, or the sorting of people across choices — the estimate is biased, and no amount of extra data fixes it. Instrumental variables (IV) restore consistency by finding a variable z that moves the troublesome regressor x but is otherwise unrelated to the outcome.

An instrument must satisfy two conditions:

  1. Relevance — Cov(z, x) ≠ 0. The instrument actually shifts x. This is testable (the first-stage F-statistic).
  2. Exclusion / exogeneity — Cov(z, ε) = 0. The instrument affects y only through x, never directly or through omitted factors. This is not testable; it has to be argued from theory or design.

The workhorse estimator, two-stage least squares (2SLS):

Stage 1: regress x on z (and controls) → fitted x̂
Stage 2: regress y on x̂

Intuitively, you discard the endogenous part of x and keep only the variation the instrument induces, which is — by assumption — clean. With more instruments than endogenous regressors the model is over-identified, and a Hansen / Sargan J-test checks the over-identifying restrictions.

The traps are real and specific:

  • Weak instruments. If relevance is weak (a low first-stage F — a rule of thumb is F > 10), the finite-sample bias explodes and the standard errors mislead. Use weak-instrument-robust inference when in doubt.
  • The exclusion restriction can’t be verified. It is an argument, not a test result — and the whole causal claim rests on it. A plausible-sounding instrument with a back door to the outcome is worse than none.
  • Heterogeneous effects. When effects vary across units, IV identifies a local average treatment effect — the effect for the “compliers” the instrument actually moves — not the average over everyone.

IV is the natural upgrade path when a descriptive regression can’t support a causal claim: a source of quasi-random variation in the regressor — a threshold, a boundary, a lottery, a policy switch — can serve as an instrument or a discontinuity design. It fixes the endogeneity that weighting and fixed effects cannot — but only as far as the exclusion restriction genuinely holds.