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Invertibility Condition of the Fisher Information Matrix of a VARMAX Process and the Tensor Sylvester Matrix

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  • André Klein
  • Guy Melard

Abstract

In this paper the invertibility condition of the asymptotic Fisher information matrix of a controlled vector autoregressive moving average stationary process, VARMAX, is displayed in a theorem. It is shown that the Fisher information matrix of a VARMAX process becomes invertible if the VARMAX matrix polynomials have no common eigenvalue. Contrarily to what was mentioned previously in a VARMA framework, the reciprocal property is untrue. We make use of tensor Sylvester matrices since checking equality of the eigenvalues of matrix polynomials is most easily done in that way. A tensor Sylvester matrix is a block Sylvester matrix with blocks obtained by Kronecker products of the polynomial coefficients by an identity matrix, on the left for one polynomial and on the right for the other one. The results are illustrated by numerical computations.

Suggested Citation

  • André Klein & Guy Melard, 2020. "Invertibility Condition of the Fisher Information Matrix of a VARMAX Process and the Tensor Sylvester Matrix," Working Papers ECARES 2020-11, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/304274
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    References listed on IDEAS

    as
    1. Leon Wegge, 2012. "ARMAX(p,r,q) Parameter Identifiability Without Coprimeness," Working Papers 1217, University of California, Davis, Department of Economics.
    2. Athanasopoulos, George & Vahid, Farshid, 2008. "VARMA versus VAR for Macroeconomic Forecasting," Journal of Business & Economic Statistics, American Statistical Association, vol. 26, pages 237-252, April.
    3. Peter Brockwell & Alexander Lindner & Bernd Vollenbröker, 2012. "Strictly stationary solutions of multivariate ARMA equations with i.i.d. noise," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(6), pages 1089-1119, December.
    4. Hannan, E J, 1971. "The Identification Problem for Multiple Equation Systems with Moving Average Errors," Econometrica, Econometric Society, vol. 39(5), pages 751-765, September.
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    Keywords

    Tensor Sylvester matrix; Matrix polynomial; Common eigenvalues; Fisher in- formation matrix; Stationary VARMAX process;
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