DurbinWuHausman

The Durbin-Wu-Hausman test is a statistical procedure in econometrics used to assess whether a regressor in a regression model is endogenous, meaning correlated with the error term. The test is named after James Durbin, Shen Wu, and J. Hausman, and is widely applied to decide between estimators that are efficient under exogeneity (such as ordinary least squares, OLS) and estimators that remain consistent under endogeneity (such as instrumental variables, IV, or two-stage least squares, 2SLS).

In essence, the test contrasts two estimation approaches. Under the null hypothesis that the regressor is exogenous, OLS and IV/2SLS provide consistent estimates that are asymptotically similar. If the regressor is endogenous, the IV/2SLS estimator is consistent while OLS remains biased and inconsistent, leading to a detectable difference between the estimators.

Common implementation methods include:

- Residual-based approach: regress the endogenous variable on instruments and exogenous variables to obtain residuals, then include these residuals as an extra regressor in the main equation estimated by OLS. A statistically significant coefficient on the residuals indicates endogeneity.

- Difference-in-coefficients approach: compute the difference between OLS and IV/2SLS estimates and test whether this difference is statistically different from zero, using a chi-square or F statistic with an appropriate variance-covariance estimate.

Interpretation hinges on model validity: rejection of exogeneity suggests the presence of endogeneity and supports using IV/2SLS or other robust methods, while failure to reject supports continued use of OLS assuming exogeneity. Cautions include instrument relevance and strength, as weak instruments can distort the test.