A reinterpretation of the tests of overidentifying restrictions

The Anderson-Rubin-Sargan-Basmann tests are usually considered as testing the validity of overidentifying structural restrictions. In this paper we show that a more eligible interpretation is that they test the validity of overidentifying orthogonality conditions. The tests are second-order unbiased against the latter alternative, while under the first alternative there is a subset of the parameter space where the tests are biased. The third-order local power under the alternative of false orthogonality conditions is derived, and is used to find the implicit null hypothesis of the tests and to propose a more informative interpretation. Several methods of size correction are compared by Monte Carlo experiments.