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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] - Groupe ENSAE-ENSAI - 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 - Groupe ENSAE-ENSAI - 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] - Groupe ENSAE-ENSAI - 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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    Cited by:

    1. is not listed on IDEAS
    2. Christian Francq & Roch Roy & Abdessamad Saidi, 2011. "Asymptotic Properties of Weighted Least Squares Estimation in Weak PARMA Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 32(6), pages 699-723, November.
    3. Deniz Erdemlioglu & Sébastien Laurent & Christopher J. Neely, 2013. "Econometric modeling of exchange rate volatility and jumps," Chapters, in: Adrian R. Bell & Chris Brooks & Marcel Prokopczuk (ed.), Handbook of Research Methods and Applications in Empirical Finance, chapter 16, pages 373-427, Edward Elgar Publishing.
    4. Pierluigi Vallarino, 2024. "Dynamic kernel models," Tinbergen Institute Discussion Papers 24-082/III, Tinbergen Institute.
    5. Susana Martins & Cristina Amado, 2018. "Financial Market Contagion and the Sovereign Debt Crisis: A Smooth Transition Approach," NIPE Working Papers 08/2018, NIPE - Universidade do Minho.
    6. Campos-Martins, Susana & Amado, Cristina, 2025. "Modelling dynamic interdependence in nonstationary variances with an application to carbon markets," Journal of Economic Dynamics and Control, Elsevier, vol. 173(C).
    7. Dalla, Violetta & Giraitis, Liudas & Phillips, Peter C. B., 2022. "Robust Tests For White Noise And Cross-Correlation," Econometric Theory, Cambridge University Press, vol. 38(5), pages 913-941, October.
    8. Carlos Trucíos & James W. Taylor, 2023. "A comparison of methods for forecasting value at risk and expected shortfall of cryptocurrencies," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 42(4), pages 989-1007, July.
    9. Olusanya E. Olubusoye & OlaOluwa S. Yaya, 2016. "Time series analysis of volatility in the petroleum pricing markets: the persistence, asymmetry and jumps in the returns series," OPEC Energy Review, Organization of the Petroleum Exporting Countries, vol. 40(3), pages 235-262, September.
    10. Bauwens, L. & Hafner C. & Laurent, S., 2011. "Volatility Models," LIDAM Discussion Papers ISBA 2011044, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
      • BAUWENS, Luc & HAFNER, Christian & LAURENT, Sébastien, 2011. "Volatility models," LIDAM Discussion Papers CORE 2011058, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
      • Bauwens, L. & Hafner, C. & Laurent, S., 2012. "Volatility Models," LIDAM Reprints ISBA 2012028, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    11. Su, Nan & Lund, Robert, 2012. "Multivariate versions of Bartlett’s formula," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 18-31.
    12. ALAMI CHENTOUFI, Reda, 2024. "Penalized Convex Estimation in Dynamic Location-Scale models," MPRA Paper 123283, University Library of Munich, Germany.
    13. Trucíos, Carlos, 2019. "Forecasting Bitcoin risk measures: A robust approach," International Journal of Forecasting, Elsevier, vol. 35(3), pages 836-847.
    14. Uwe Hassler & Marc-Oliver Pohle & Tanja Zahn, 2025. "Simultaneous Inference Bands for Autocorrelations," Papers 2503.18560, arXiv.org, revised Aug 2025.
    15. Daniel Cirkovic & Thomas J. Fisher, 2021. "On testing for the equality of autocovariance in time series," Environmetrics, John Wiley & Sons, Ltd., vol. 32(7), November.
    16. Boubacar Mainassara, Y. & Carbon, M. & Francq, C., 2012. "Computing and estimating information matrices of weak ARMA models," Computational Statistics & Data Analysis, Elsevier, vol. 56(2), pages 345-361.
    17. Regnard, Nazim & Zakoïan, Jean-Michel, 2011. "A conditionally heteroskedastic model with time-varying coefficients for daily gas spot prices," Energy Economics, Elsevier, vol. 33(6), pages 1240-1251.
    18. repec:dau:papers:123456789/2603 is not listed on IDEAS
    19. Proietti, Tommaso & Luati, Alessandra, 2015. "The generalised autocovariance function," Journal of Econometrics, Elsevier, vol. 186(1), pages 245-257.
    20. Campos-Martins, Susana & Amado, Cristina, 2022. "Financial market linkages and the sovereign debt crisis," Journal of International Money and Finance, Elsevier, vol. 123(C).

    More about this item

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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