title: A Theorem on the Rank of a Product of Matrices with Illustration of Its Use in Goodness of Fit Testing.
abstract: This paper develops a theorem that facilitates computing the degrees of freedom of Wald-type chi-square tests for moment restrictions when there is rank deficiency of key matrices involved in the definition of the test. An if and only if (iff) condition is developed for a simple rule of difference of ranks to be used when computing the desired degrees of freedom of the test. The theorem is developed exploiting basics tools of matrix algebra. The theorem is shown to play a key role in proving the asymptotic chi-squaredness of a goodness of fit test in moment structure analysis, and in finding the degrees of freedom of this chi-square statistic.
authors.affiliation: Department of Economics and Business, Universitat Pompeu Fabra, Ramon Trias Fargas, 25-27, 08005 , Barcelona, Spain. albert.satorra@upf.edu.
authors.forename: Albert
authors.initials: A
authors.lastname: Satorra
authors.affiliation: University of Amsterdam, Amsterdam, The Netherlands.
authors.forename: Heinz
authors.initials: H
authors.lastname: Neudecker
mesh_terms: D000465:Algorithms; D016009:Chi-Square Distribution; D011594:Psychometrics / Q000706:statistics & numerical data
keywords: augmented moment structures; chi-square goodness of fit test; matrix algebra; rank deficiency; structural equation modeling; wald test