The Hausman Test, and Some Alternatives, with Heteroskedastic Data

The Hausman test is used in applied economic work as a test of misspecification. It is most commonly thought of as a test of whether one or more explanatory variables in a regression model are endogenous. The usual Hausman contrast test requires one estimator to be efficient under the null hypothesis. If data are heteroskedastic, the least squares estimator is no longer efficient. The first option is to estimate the covariance matrix of the difference of the contrasted estimators, as suggested by Hahn, Ham, and Moon (2011). Other options for carrying out a Hausman-like test in this case include estimating an artificial regression and using robust standard errors. Alternatively, we might seek additional power by estimating the artificial regression using feasible generalized least squares. Finally, we might stack moment conditions leading to the two estimators and estimate the resulting system by GMM. We examine these options in a Monte Carlo experiment. We conclude that the test based on the procedure by Hahn, Ham, and Moon has good properties. The generalized least squares-based tests have higher size-corrected power when heteroskedasticity is detected in the DWH regression, and the heteroskedasticity is associated with a strong external IV. We do not consider the properties of the implied pretest estimator.