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A Theorem on the Rank of a Product of Matrices with Illustration of Its Use in Goodness of Fit Testing

Published online by Cambridge University Press:  01 January 2025

Albert Satorra  and
Heinz Neudecker
Albert Satorra*
Affiliation:
Universitat Pompeu Fabra
Heinz Neudecker
Affiliation:
University of Amsterdam
*
Correspondence should be made to Albert Satorra, Department of Economics and Business, Universitat Pompeu Fabra, Ramon Trias Fargas, 25-27, 08005 Barcelona, Spain. Email: albert.satorra@upf.edu https://www.econ.upf.edu/@satorra
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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.

Keywords

matrix algebrarank deficiencywald testchi-square goodness of fit testaugmented moment structuresstructural equation modeling
Type
Original Paper
Information
Psychometrika , Volume 80 , Issue 4 , December 2015 , pp. 938 - 948
Copyright
Copyright © 2015 The Psychometric Society

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