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Bartlett's formula for a general class of non linear processes

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  • Francq, Christian
  • Zakoian, Jean-Michel

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. An application to GARCH models is proposed.

Suggested Citation

  • Francq, Christian & Zakoian, Jean-Michel, 2009. "Bartlett's formula for a general class of non linear processes," MPRA Paper 13224, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:13224
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    File URL: https://mpra.ub.uni-muenchen.de/13224/1/MPRA_paper_13224.pdf
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    References listed on IDEAS

    as
    1. Ling, Shiqing & McAleer, Michael, 2002. "NECESSARY AND SUFFICIENT MOMENT CONDITIONS FOR THE GARCH(r,s) AND ASYMMETRIC POWER GARCH(r,s) MODELS," Econometric Theory, Cambridge University Press, vol. 18(03), pages 722-729, June.
    2. Kokoszka, Piotr S. & Politis, D N, 2008. "The Variance of Sample Autocorrelations: Does Barlett's Formula Work With ARCH Data?," University of California at San Diego, Economics Working Paper Series qt68c247dp, Department of Economics, UC San Diego.
    3. Christian Francq & Jean-Michel Zakoïan, 2008. "Barlett’s Formula for Non Linear Processes," Working Papers 2008-05, Center for Research in Economics and Statistics.
    4. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    5. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. 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.
    2. Deniz Erdemlioglu & Sébastien Laurent & Christopher J. Neely, 2013. "Econometric modeling of exchange rate volatility and jumps," Chapters,in: Handbook of Research Methods and Applications in Empirical Finance, chapter 16, pages 373-427 Edward Elgar Publishing.
    3. 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.
    4. Proietti, Tommaso & Luati, Alessandra, 2015. "The generalised autocovariance function," Journal of Econometrics, Elsevier, vol. 186(1), pages 245-257.
    5. 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.
    6. BAUWENS, Luc & HAFNER, Christian & LAURENT, Sébastien, 2011. "Volatility models," CORE Discussion Papers 2011058, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. Su, Nan & Lund, Robert, 2012. "Multivariate versions of Bartlett’s formula," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 18-31.
    8. 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.
    9. repec:dau:papers:123456789/2603 is not listed on IDEAS

    More about this item

    Keywords

    Bartlett's formula; nonlinear time series model; sample autocorrelation; GARCH model; weak white noise;

    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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