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Testing the Equality of Two Positive-Definite Matrices with Application to Information Matrix Testing



    (Yonsei University)


    (University of California, San Diego)


We provide a new characterization of the equality of two positive-definite matrices A and B, and we use this to propose several new computationally convenient statistical tests for the equality of two unknown positive-definite matrices. Our primary focus is on testing the information matrix equality (e.g., White, 1982, 1994). We characterize the asymptotic behavior of our new trace-determinant information matrix test statistics under the null and the alternative and investigate their finite-sample performance for a variety of models: linear regression, exponential duration, probit, and Tobit. The parametric bootstrap suggested by Horowitz (1994) delivers critical values that provide admirable level behavior, even in samples as small as n ¨¡ 50. Our new tests often have better power than the parametric-bootstrap version of the traditional IMT; when they do not, they nevertheless perform respectably.

Suggested Citation

  • 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.
  • Handle: RePEc:yon:wpaper:2014rwp-67

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    References listed on IDEAS

    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. Taylor, Larry W., 1987. "The size bias of White's information matrix test," Economics Letters, Elsevier, vol. 24(1), pages 63-67.
    3. Cho, Jin Seo & White, Halbert, 2010. "Testing for unobserved heterogeneity in exponential and Weibull duration models," Journal of Econometrics, Elsevier, vol. 157(2), pages 458-480, August.
    4. Horowitz, Joel L., 1994. "Bootstrap-based critical values for the information matrix test," Journal of Econometrics, Elsevier, vol. 61(2), pages 395-411, April.
    5. White,Halbert, 1996. "Estimation, Inference and Specification Analysis," Cambridge Books, Cambridge University Press, number 9780521574464, October.
    6. Abadir,Karim M. & Magnus,Jan R., 2005. "Matrix Algebra," Cambridge Books, Cambridge University Press, number 9780521537469, August.
    7. Gerard J. van den Berg & Geert Ridder, 1998. "An Empirical Equilibrium Search Model of the Labor Market," Econometrica, Econometric Society, vol. 66(5), pages 1183-1222, September.
    8. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504, August.
    9. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
    10. Jin Seo Cho & Halbert White, 2007. "Testing for Regime Switching," Econometrica, Econometric Society, vol. 75(6), pages 1671-1720, November.
    11. Strawderman, Robert L., 1994. "A note on necessary and sufficient conditions for proving that a random symmetric matrix converges to a given limit," Statistics & Probability Letters, Elsevier, vol. 21(5), pages 367-370, December.
    12. Orme, Chris, 1990. "The small-sample performance of the information-matrix test," Journal of Econometrics, Elsevier, vol. 46(3), pages 309-331, December.
    13. Pillai, K. C. S. & Nagarsenker, B. N., 1972. "On the distributions of a class of statistics in multivariate analysis," Journal of Multivariate Analysis, Elsevier, vol. 2(1), pages 96-114, March.
    14. Maxwell L. King & Xibin Zhang & Muhammad Akram, 2011. "A New Procedure For Multiple Testing Of Econometric Models," Monash Econometrics and Business Statistics Working Papers 7/11, Monash University, Department of Econometrics and Business Statistics.
    15. repec:cup:cbooks:9780521822893 is not listed on IDEAS
    16. Lancaster, Tony, 1979. "Econometric Methods for the Duration of Unemployment," Econometrica, Econometric Society, vol. 47(4), pages 939-956, July.
    17. Alastair Hall, 1987. "The Information Matrix Test for the Linear Model," Review of Economic Studies, Oxford University Press, vol. 54(2), pages 257-263.
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    Cited by:

    1. repec:eee:econom:v:202:y:2018:i:1:p:45-56 is not listed on IDEAS
    2. 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.
    3. Seok Young Hong & Oliver Linton & Hui Jun Zhang, 2015. "An investigation into Multivariate Variance Ratio Statistics and their application to Stock Market Predictability," Cambridge Working Papers in Economics 1552, Faculty of Economics, University of Cambridge.

    More about this item


    Matrix equality; Information matrix test; Eigenvalues; Trace; Determinant; Eigenspectrum test; Parametric Bootstrap.;

    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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