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Bartlett's formula for a general class of nonlinear processes

Author

Listed:
  • Christian Francq

    (CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - GENES - Groupe des Écoles Nationales d'Économie et Statistique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - GENES - Groupe des Écoles Nationales d'Économie et Statistique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique, IP Paris - Institut Polytechnique de Paris)

  • Jean‐michel Zakoïan

    (CREST - Centre de Recherche en Economie et Statistique [Bruz] - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - GENES - Groupe des Écoles Nationales d'Économie et Statistique, IP Paris - Institut Polytechnique de Paris)

Abstract

. A Bartlett‐type formula is proposed for the asymptotic distribution of the sample autocorrelations of nonlinear processes. The asymptotic covariances between sample autocorrelations are expressed as the sum of two terms. The first term corresponds to the standard Bartlett's formula for linear processes, involving only the autocorrelation function of the observed process. The second term, which is specific to nonlinear processes, involves the autocorrelation function of the observed process, the kurtosis of the linear innovation process and the autocorrelation function of its square. This formula is obtained under a symmetry assumption on the linear innovation process. It is illustrated on ARMA–GARCH models and compared to the standard formula. An empirical application on financial time series is proposed.

Suggested Citation

  • Christian Francq & Jean‐michel Zakoïan, 2009. "Bartlett's formula for a general class of nonlinear processes," Post-Print hal-05417890, HAL.
    • Handle: RePEc:hal:journl:hal-05417890
    DOI: 10.1111/j.1467-9892.2009.00623.x
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