IDEAS home Printed from https://ideas.repec.org/a/eee/econom/v202y2018i1p45-56.html
   My bibliography  Save this article

Pythagorean generalization of testing the equality of two symmetric positive definite matrices

Author

Listed:
  • Cho, Jin Seo
  • Phillips, Peter C.B.

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 or 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 simple 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

  • Cho, Jin Seo & Phillips, Peter C.B., 2018. "Pythagorean generalization of testing the equality of two symmetric positive definite matrices," Journal of Econometrics, Elsevier, vol. 202(1), pages 45-56.
  • Handle: RePEc:eee:econom:v:202:y:2018:i:1:p:45-56
    DOI: 10.1016/j.jeconom.2017.05.020
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304407617301902
    Download Restriction: Full text for ScienceDirect subscribers only

    As the access to this document is restricted, you may want to look for a different version below or search for a different version of it.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Nagarsenker, B. N. & Pillai, K. C. S., 1973. "The distribution of the sphericity test criterion," Journal of Multivariate Analysis, Elsevier, vol. 3(2), pages 226-235, June.
    2. repec:oup:jfinec:v:15:y:2017:i:2:p:173-222. is not listed on IDEAS
    3. Baek, Yae In & Cho, Jin Seo & Phillips, Peter C.B., 2015. "Testing linearity using power transforms of regressors," Journal of Econometrics, Elsevier, vol. 187(1), pages 376-384.
    4. Jin Seo Cho & Halbert White, 2014. "Notations in "Testing the Equality of Two Positive-Definite Matrices with Application to Information Matrix Testing" by Cho and White (2014)," Working papers 2014rwp-67a, Yonsei University, Yonsei Economics Research Institute.
    5. Magnus, Jan R., 1985. "On Differentiating Eigenvalues and Eigenvectors," Econometric Theory, Cambridge University Press, vol. 1(02), pages 179-191, August.
    6. Dhaene, Geert & Hoorelbeke, Dirk, 2004. "The information matrix test with bootstrap-based covariance matrix estimation," Economics Letters, Elsevier, vol. 82(3), pages 341-347, March.
    7. Horowitz, Joel L., 1994. "Bootstrap-based critical values for the information matrix test," Journal of Econometrics, Elsevier, vol. 61(2), pages 395-411, April.
    8. Magnus, J.R., 1985. "On differentiating eigenvalues and eigenvectors," Other publications TiSEM f410e3a5-ba9b-4787-b8cc-4, Tilburg University, School of Economics and Management.
    9. Cho, Jin Seo & White, Halbert, 2011. "Generalized runs tests for the IID hypothesis," Journal of Econometrics, Elsevier, vol. 162(2), pages 326-344, June.
    10. Orme, Christopher, 1988. "The Calculation of the Information Matrix Test for Binary Data Models," The Manchester School of Economic & Social Studies, University of Manchester, vol. 56(4), pages 370-376, December.
    11. Alastair Hall, 1987. "The Information Matrix Test for the Linear Model," Review of Economic Studies, Oxford University Press, vol. 54(2), pages 257-263.
    12. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
    13. Chesher, Andrew & Spady, Richard, 1991. "Asymptotic Expansions of the Information Matrix Test Statistic," Econometrica, Econometric Society, vol. 59(3), pages 787-815, May.
    14. Jin Seo Cho & Halbert White, 2014. "Testing the Equality of Two Positive-Definite Matrices with Application to Information Matrix Testing," Working papers 2014rwp-67, Yonsei University, Yonsei Economics Research Institute.
    Full references (including those not matched with items on IDEAS)

    More about this item

    Keywords

    Matrix equality; Trace; Determinant; Arithmetic mean; Geometric mean; Harmonic mean; Sandwich covariance matrix; Eigenvalues;

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:econom:v:202:y:2018:i:1:p:45-56. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Dana Niculescu). General contact details of provider: http://www.elsevier.com/locate/jeconom .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.