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

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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. Copyright The Psychometric Society 2015

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  • Albert Satorra & Heinz Neudecker, 2015. "A Theorem on the Rank of a Product of Matrices with Illustration of Its Use in Goodness of Fit Testing," Psychometrika, Springer;The Psychometric Society, vol. 80(4), pages 938-948, December.
    • Handle: RePEc:spr:psycho:v:80:y:2015:i:4:p:938-948
    DOI: 10.1007/s11336-014-9438-5
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    1. Andrews, Donald W. K., 1987. "Asymptotic Results for Generalized Wald Tests," Econometric Theory, Cambridge University Press, vol. 3(3), pages 348-358, June.
    2. Puntanen, Simo & Styan, George P.H. & Werner, Hans Joachim, 2003. "02.4.2. On the Rank of a Matrix Useful in Goodness-of-Fit Testing of Structural Equation Models —Solution," Econometric Theory, Cambridge University Press, vol. 19(4), pages 704-705, August.
    3. Robert Jennrich & Albert Satorra, 2013. "Erratum to: Continuous orthogonal complement functions and distribution-free goodness of fit tests in moment structure analysis," Psychometrika, Springer;The Psychometric Society, vol. 78(3), pages 553-553, July.
    4. Robert Jennrich & Albert Satorra, 2013. "Continuous Orthogonal Complement Functions and Distribution-Free Goodness of Fit Tests in Moment Structure Analysis," Psychometrika, Springer;The Psychometric Society, vol. 78(3), pages 545-552, July.
    5. Albert Satorra, 1989. "Alternative test criteria in covariance structure analysis: A unified approach," Psychometrika, Springer;The Psychometric Society, vol. 54(1), pages 131-151, March.
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