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Bayes Factors for Testing the Equality of Covariance Matrix Eigenvalues

In: Modelling and Prediction Honoring Seymour Geisser

Author

Listed:
  • Robert E. McCulloch

    (University of Chicago, Graduate School of Business)

  • Peter E. Rossi

    (University of Chicago, Graduate School of Business)

Abstract

We use the methods of McCulloch and Rossi (1992) to construct a Bayes Factor for the hypothesis of the equality of eigenvalues of a covariance matrix. If the smallest s eigenvalues of a covariance matrix are equal, then we can think of the p dimensional sample data as arising from a reduced rank model. Tests of this sort are often used to identify the number of components used in a Principal Components analysis. The Bayes Factor approach requires specification of prior distributions for both the restricted and unrestricted covariance matrices. The problem of specifying a prior distribution on the restricted covariance matrix is solved via a projection method. We exploit the duality between sufficient statistics and parameters in exponential families to use the restricted MLE as a projection device. We illustrate this method with simulated and actual data. Our real data example addresses the question of the number of factors underlying stock return data.

Suggested Citation

  • Robert E. McCulloch & Peter E. Rossi, 1996. "Bayes Factors for Testing the Equality of Covariance Matrix Eigenvalues," Springer Books, in: Jack C. Lee & Wesley O. Johnson & Arnold Zellner (ed.), Modelling and Prediction Honoring Seymour Geisser, pages 305-314, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-2414-3_19
    DOI: 10.1007/978-1-4612-2414-3_19
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