Hausman test

The Hausman test is a test in econometrics named after Jerry Hausman. The test evaluates the significance of an estimators versus an alternative estimator.

If 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, $$b_{0}$$ and $$b_{1}$$. Under the null hypothesis, both of these estimators are consistent, but $$b_{1}$$ is more efficient (has smaller asymptotic variance) than $$b_{0}$$. Under the alternative hypothesis, one or both of these estimators is inconsistent.

We can derive the statistic


 * $$H=T(b_{0}-b_{1})'Var(b_{0}-b_{1})^{-1}(b_{0}-b_{1}),$$

where T is the number of observations. This statistic has chi-square distribution with k (length of b) degrees of freedom. You can find more about the $$\chi^2$$ (chi-square distribution) distribution at the SOCR Resource Distributions.

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 IV estimates to 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 using 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.

Literature
Hausman, J. A. (1978). Specification Tests in Econometrics, Econometrica, Vol. 46, No. 6. (Nov., 1978), pp. 1251-1271.

Hausman-Test