A Comment on the Test of Overidentifying Restrictions

THE TEST OF OVERIDENTIFYING restrictions of one equation in a simultaneous system proposed by Anderson and Rubin [1] and amplified by Koopmans and Hood [7] has been a source of some confusion in the literature. For instance, Liu and Breen [8] claimed that "It is ... clear that the test does not really test the null hypothesis (of zero restrictions on the endogenous and exogenous variables)" because, they thought the restrictions on the endogenous variables were included in the computation of the likelihood under the alternative hypothesis. After Fisher and Kadane [3] gave a verbal argument showing that the test is consistent over a wide class of alternatives, Liu and Breen [9] withdrew their earlier view. Nonetheless there is a problem in that generally the null hypothesis is expressed in terms of the structural form, while generally consistency is a matter of the reduced form. Our purpose is to reexamine this problem, and prove two theorems showing the equivalence of various conditions in the literature. We suggest that the null hypothesis be extended.