Hausman test

The Wu-Hausman test (also called the Hausman test, Hausman specification test, and Durbin-Wu-Hausman test) is a statistical hypothesis test in econometrics named after De-Min Wu and Jerry A. Hausman.^{[citation needed]} The test evaluates the significance of an estimator versus an alternative estimator. It helps one evaluate if a statistical model corresponds to the data.

Details

Consider the linear model y = bX + e, where y is univariate 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.

Then the Wu-Hausman statistic is:

$H=(b_{1}-b_{0})'\big(\operatorname{Var}(b_{0})-\operatorname{Var}(b_{1})\big)^\dagger(b_{1}-b_{0}),$

where ^† denotes the Moore–Penrose pseudoinverse. This statistic has asymptotically the chi-squared distribution with the number of degrees of freedom equal to the rank of matrix Var(b₀) − Var(b₁).

If we reject the null hypothesis, one or both of the estimators is inconsistent. This test can be used to check for the endogeneity of a variable (by comparing instrumental variable (IV) estimates to ordinary least squares (OLS) estimates). It can also be used to check the validity of extra instruments by comparing IV estimates using a full set of instruments Z to IV estimates that use a proper subset of Z. Note that in order for the test to work in the latter case, we must be certain of the validity of the subset of Z and that subset must have enough instruments to identify the parameters of the equation.

Hausman also showed that the covariance between an efficient estimator and the difference of an efficient and inefficient estimator is zero.

See also

• Regression model validation