How does two-stage least squares solve endogeneity?

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AcadiFi · Accepted Answer

Endogeneity arises when a regressor X is correlated with the error term, making OLS biased. Sources: reverse causality, omitted variables, measurement error, self-selection. Instrumental variables (IV) solves this using an instrument Z that satisfies two conditions: (1) relevance: Cov(Z, X) ≠ 0, (2) exogeneity: Cov(Z, e) = 0 (Z affects Y only through X).

Two-stage least squares (2SLS): Stage 1: regress X on Z (and exogenous controls): X_hat = alpha + gamma*Z + delta*W. Stage 2: regress Y on X_hat: Y = beta_0 + beta_1 * X_hat + beta_2 * W + u. The 2SLS estimator of beta_1 is consistent under the IV assumptions.

For Sterling's training evaluation, workers near regional training offices are more likely to attend (relevance), and office location is unrelated to unobserved worker productivity (exogeneity). Using distance-to-office as IV: Stage 1 predicts training probability from distance and controls; Stage 2 estimates productivity effect using predicted training.

Diagnostics: (1) first-stage F-statistic > 10 (Staiger-Stock rule) rules out weak instruments that inflate 2SLS SE and bias toward OLS; (2) Sargan-Hansen J-test when overidentified (more instruments than endogenous vars) tests exogeneity; (3) Hausman test compares OLS vs 2SLS—if equal, OLS is consistent and more efficient.

Famous applications: Angrist-Krueger use quarter of birth to instrument for schooling in estimating returns to education; Acemoglu uses colonial settler mortality to instrument for institutions. Weak instruments are the most common failure mode—always report first-stage F.

How does two-stage least squares solve endogeneity?

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