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Multivariate Hermite polynomials and information matrix tests

Author

Listed:
  • Dante Amengual

    (CEMFI, Casado del Alisal 5, E-28014 Madrid, Spain)

  • Gabriele Fiorentini

    (Università di Firenze and RCEA, Viale Morgagni 59, I-50134 Firenze, Italy)

  • Enrique Sentana

    (CEMFI, Casado del Alisal 5, E-28014 Madrid, Spain)

Abstract

We show that the information matrix test for a multivariate normal random vector coincides with the sum of the two moment tests that look at the means of all the different third- and fourth-order multivariate Hermite polynomials, respectively. We also explain how to simulate its exact, parameter-free, finite sample distribution to any desired degree of accuracy for any dimension of the random vector and sample size. Specifically, we exploit the numerical invariance of the test statistic to affine transformations of the observed variables to simulate draws extremely quickly.

Suggested Citation

  • Dante Amengual & Gabriele Fiorentini & Enrique Sentana, 2021. "Multivariate Hermite polynomials and information matrix tests," Econometrics Working Papers Archive 2021_07, Universita' degli Studi di Firenze, Dipartimento di Statistica, Informatica, Applicazioni "G. Parenti".
    • Handle: RePEc:fir:econom:wp2021_07
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    File URL: https://labdisia.disia.unifi.it/wp_disia/2021/wp_disia_2021_07.pdf
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    References listed on IDEAS

    as
    1. Mencía, Javier & Sentana, Enrique, 2009. "Multivariate location-scale mixtures of normals and mean-variance-skewness portfolio allocation," Journal of Econometrics, Elsevier, vol. 153(2), pages 105-121, December.
    2. Engle, Robert F & Kozicki, Sharon, 1993. "Testing for Common Features," Journal of Business & Economic Statistics, American Statistical Association, vol. 11(4), pages 369-380, October.
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    4. Norbert Henze, 2002. "Invariant tests for multivariate normality: a critical review," Statistical Papers, Springer, vol. 43(4), pages 467-506, October.
    5. Best, D. J. & Rayner, J. C. W., 1988. "A test for bivariate normality," Statistics & Probability Letters, Elsevier, vol. 6(6), pages 407-412, May.
    6. J. Koziol, 1987. "An alternative formulation of Neyman’s smooth goodness of fit tests under composite alternatives," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 34(1), pages 17-24, December.
    7. Chesher, Andrew D, 1984. "Testing for Neglected Heterogeneity," Econometrica, Econometric Society, vol. 52(4), pages 865-872, July.
    8. Kiefer, Nicholas M. & Salmon, Mark, 1983. "Testing normality in econometric models," Economics Letters, Elsevier, vol. 11(1-2), pages 123-127.
    9. Pietro BALESTRA & Alberto HOLLY, 1990. "A General Kronecker Formula for the Moments of the Multivariate Normal Distribution," Cahiers de Recherches Economiques du Département d'économie 9002, Université de Lausanne, Faculté des HEC, Département d’économie.
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    Cited by:

    1. Dante Amengual & Gabriele Fiorentini & Enrique Sentana, 2022. "Tests for Random Coefficient Variation in Vector Autoregressive Models," Advances in Econometrics, in: Essays in Honour of Fabio Canova, volume 44, pages 1-35, Emerald Group Publishing Limited.

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    More about this item

    Keywords

    Exact test; Hessian matrix; Multivariate normality; Outer product of the score;
    All these keywords.

    JEL classification:

    • C30 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - General
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

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