Hausman test

The Durbin–Wu–Hausman test (also called Hausman specification test) is a statistical hypothesis test in econometrics named after James Durbin, De-Min Wu, and Jerry A. Hausman. The test evaluates the consistency of an estimator when compared to an alternative, less efficient estimator which is already known to be consistent. It helps one evaluate if a statistical model corresponds to the data.Consider the linear model y = bX + e, where y is the dependent variable and X is vector of regressors, b is a vector of coefficients and e is the error term. We have two estimators for b: b0 and b1. Under the null hypothesis, both of these estimators are consistent, but b1 is efficient (has the smallest asymptotic variance), at least in the class of estimators containing b0. Under the alternative hypothesis, b0 is consistent, whereas b1 isn’t.Assuming joint normality of the estimators.The Hausman test can be also used to differentiate between fixed effects model and random effects model in panel data. In this case, Random effects (RE) is preferred under the null hypothesis due to higher efficiency, while under the alternative Fixed effects (FE) is at least as consistent and thus preferred.