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Pythagorean Generalization of Testing the Equality of Two Symmetric Positive Definite Matrices

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  • JIN SEO CHO

    (Yonsei University)

  • PETER C.B. PHILLIPS

    (Yale University University of Auckland, Singapore Management University & University of Southampton)

Abstract

We provide a new test for equality of two symmetric positive-definite matrices that leads to a convenient mechanism for testing specification using the information matrix equality and the sandwich asymptotic covariance matrix of the GMM estimator. The test relies on a new characterization of equality between two k dimensional symmetric positive-definite matrices A and B: the traces of AB-1 and BA-1 are equal to k if and only if A = B. Using this criterion, we introduce a class of omnibus test statistics for equality and examine their null and local alternative approximations under some mild regularity conditions. A preferred test in the class with good omni-directional power is recommended for practical work. Monte Carlo experiments are conducted to explore performance characteristics under the null and local as well as fixed alternatives. The test is applicable in many settings, including GMM estimation, SVAR models and high dimensional variance matrix settings.

Suggested Citation

  • Jin Seo Cho & Peter C.B. Phillips, 2016. "Pythagorean Generalization of Testing the Equality of Two Symmetric Positive Definite Matrices," Working papers 2016rwp-89, Yonsei University, Yonsei Economics Research Institute.
    • Handle: RePEc:yon:wpaper:2016rwp-89
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    Cited by:

    1. MacKinnon, James G. & Nielsen, Morten Ørregaard & Webb, Matthew D., 2023. "Testing for the appropriate level of clustering in linear regression models," Journal of Econometrics, Elsevier, vol. 235(2), pages 2027-2056.
    2. Lijuan Huo & Jin Seo Cho, 2019. "Testing for the Sandwich-Form Covariance Matrix Applied to Quasi-Maximum Likelihood Estimation Using Economic and Energy Price Growth Rates," Working papers 2019rwp-152, Yonsei University, Yonsei Economics Research Institute.
    3. Jin Seo Cho & Peter C.B. Phillips, 2016. "Online Supplement to "Pythagorean Generalization of Testing the Equality of Two Symmetric Positive Definite Matrices"," Working papers 2016rwp-89a, Yonsei University, Yonsei Economics Research Institute.
    4. Sin, C.Y. (Chor-yiu) & Lee, Cheng-Few, 2021. "Using heteroscedasticity-non-consistent or heteroscedasticity-consistent variances in linear regression," Econometrics and Statistics, Elsevier, vol. 18(C), pages 117-142.
    5. Linton, O. & Tang, H., 2020. "Estimation of the Kronecker Covariance Model by Quadratic Form," Cambridge Working Papers in Economics 2050, Faculty of Economics, University of Cambridge.
    6. Richard M. Golden & Steven S. Henley & Halbert White & T. Michael Kashner, 2019. "Consequences of Model Misspecification for Maximum Likelihood Estimation with Missing Data," Econometrics, MDPI, vol. 7(3), pages 1-27, September.

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    More about this item

    Keywords

    Matrix equality; Trace; Determinant; Arithmetic mean; Geometric mean; Harmonic mean; Sandwich covariance matrix; Eigenvalues.;
    All these keywords.

    JEL classification:

    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

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