[5] It helps one evaluate if a statistical model corresponds to the data.
Under the null hypothesis, 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).
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.
By the delta method Using the commonly used result, showed by Hausman, that the covariance of an efficient estimator with its difference from an inefficient estimator is zero yields The chi-squared test is based on the Wald criterion where † denotes the Moore–Penrose pseudoinverse and K denotes the dimension of vector b.
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.