On Bartlett’s Formula for Non‐linear Processes

Alain Berlinet and Christian Francq ()
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Alain Berlinet: I3M - Institut de Mathématiques et de Modélisation de Montpellier - UM2 - Université Montpellier 2 - Sciences et Techniques - UM - Université de Montpellier - CNRS - Centre National de la Recherche Scientifique, BME - Budapest University of Technology and Economics [Budapest]
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

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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 L 2 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

Date: 2001-12-26
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Published in Journal of Time Series Analysis, 2001, 18 (6), pp.535-552. ⟨10.1111/1467-9892.00067⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-05431305

DOI: 10.1111/1467-9892.00067

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