Wu-Hausman Test

The Wu-Hausman test asks whether a regressor is actually endogenous. The intuition: under the null of exogeneity, OLS and 2SLS estimate the same parameter (OLS is even more efficient); under the alternative of endogeneity, the two estimators converge to different things. The test statistic compares $\hat\beta_{\text{OLS}}$ and $\hat\beta_{\text{IV}}$ and rejects when their difference is too large to be sampling noise.

When to use

Run after any IV estimation: if the test rejects, the regressor is endogenous and IV was necessary (OLS would have been biased). If it fails to reject, OLS and IV agree and you can prefer OLS for its efficiency. PS_2 reports Wu-Hausman p = 9.4×10⁻¹⁰ for the fertility-education regression, decisively confirming endogeneity. Available in R as summary(ivreg_model, diagnostics = TRUE) — the "Wu-Hausman" row.