# Hausman test for 2SLS vs 3SLS

Can we do a Hausman test for 2SLS vs 3SLS? I know that we can do a BP test for the cross-equation correlation of errors, but what should the null and alternative hypotheses of a Hausman test be?

I do not really see how this is to work. The Hausman test is designed for comparing the difference between two estimates, where one of the estimators is consistent under $H_1$ and the other one is not. The classical version assumes that the estimator which is inconsistent under $H_1$ is efficient under $H_0$, whereas the one that is also consistent under $H_1$ is not efficient under $H_0$.
In so doing, it provides a test of the underlying moment conditions used to identify a parameter of interest, based on the idea that if we observe a large difference, then we are probably faced with a situation in which the underlying $H_0$ is violated, as, if $H_0$ is true, the estimates should be "close" as measured by the $\chi^2$-distribution.
EDIT: the systemfit package provides an implementation of the test I am referring to in the last paragraph.