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 (wrongly some would say) as a test of whether one or more explanatory variables in a regression model is endogenous. There are several versions of the test available with modern software, some of them suggesting opposite conclusions about the null hypothesis. We explore the size and power of the alternative tests to find the best option. Secondly, 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. Options for carrying out a Hausman-like test in this case include estimating an artificial regression and using robust standard errors, or bootstrapping the covariance matrix of the two estimators used in the contrast, or stacking moment conditions leading to two estimators and estimating them as a system. We examine these options in a Monte Carlo experiment. We conclude that in both these cases the preferred test is based on an artificial regression, perhaps using a robust covariance matrix estimator if heteroskedasticity is suspected. If instruments are weak (not highly correlated with the endogenous regressors), however, no test procedure is reliable. If the test is designed to choose between the least squares estimator and a consistent alternative, the least desirable test has some positive aspects. We also investigate the impact of various types of bootstrapping. Our results suggest that in large samples, wild (correcting for heteroskedasticity) bootstrapping is a slight improvement over asymptotics in models with weak instruments. Lastly, we consider another model where heteroskedasticity is present - the count data model. Our Monte Carlo experiment shows that the test using stacked moment conditions and the second round estimator has the best performance, but which could still be improved upon by bootstrapping