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On Bartlett’s Formula for Non‐linear Processes

Author

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  • Alain Berlinet
  • Christian Francq

Abstract

Bartlett’s formula is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocovariances. However, it is derived under precise assumptions (namely linearity of the underlying process and vanishing of its fourth‐order cumulants) and effectiv e computations show that the value given by this formula can deviate markedly from the true asymptotic covariance when the requirements on the underlying process are not satisfied. This is the case for a large class of models, for instance bilinear and autoregressive conditionally heteroscedastic processes. For these reasons we investigate the behaviour of smoothed empirical estimates of the covariance between two sample autocovariance s. We prove L2 and strong consistency for strongly mixing stationary processes and define for the covariance matrix of a vector of sample autocovariances a consistent estimate which is a non‐negative definite matrix. The choice of the parameters is discussed, applications are given and comparisons are made through a simulation study

Suggested Citation

  • Alain Berlinet & Christian Francq, 1997. "On Bartlett’s Formula for Non‐linear Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 18(6), pages 535-552, November.
  • Handle: RePEc:bla:jtsera:v:18:y:1997:i:6:p:535-552
    DOI: 10.1111/1467-9892.00067
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    Cited by:

    1. Christian Francq & Jean-Michel Zakoïan, 2008. "Barlett’s Formula for Non Linear Processes," Working Papers 2008-05, Center for Research in Economics and Statistics.
    2. Hella, Heikki, 2003. "On robust ESACF identification of mixed ARIMA models," Bank of Finland Scientific Monographs, Bank of Finland, volume 0, number sm2003_027, March.

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