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Asymptotic properties of weighted least squares estimation in weak parma models

Author

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  • Francq, Christian
  • Roy, Roch
  • Saidi, Abdessamad

Abstract

The aim of this work is to investigate the asymptotic properties of weighted least squares (WLS) estimation for causal and invertible periodic autoregressive moving average (PARMA) models with uncorrelated but dependent errors. Under mild assumptions, it is shown that the WLS estimators of PARMA models are strongly consistent and asymptotically normal. It extends Theorem 3.1 of Basawa and Lund (2001) on least squares estimation of PARMA models with independent errors. It is seen that the asymptotic covariance matrix of the WLS estimators obtained under dependent errors is generally different from that obtained with independent errors. The impact can be dramatic on the standard inference methods based on independent errors when the latter are dependent. Examples and simulation results illustrate the practical relevance of our findings. An application to financial data is also presented.

Suggested Citation

  • Francq, Christian & Roy, Roch & Saidi, Abdessamad, 2011. "Asymptotic properties of weighted least squares estimation in weak parma models," MPRA Paper 28721, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:28721
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    References listed on IDEAS

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    1. Christian Francq & Jean‐Michel Zakoïan, 2009. "Bartlett's formula for a general class of nonlinear processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(4), pages 449-465, July.
    2. Peiro, Amado, 1994. "Daily seasonality in stock returns : Further international evidence," Economics Letters, Elsevier, vol. 45(2), pages 227-232, June.
    3. 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(3), pages 722-729, June.
    4. I. V. Basawa & Robert Lund, 2001. "Large Sample Properties of Parameter Estimates for Periodic ARMA Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 22(6), pages 651-663, November.
    5. Francq, Christian & Zakoïan, Jean-Michel, 2000. "Estimating Weak Garch Representations," Econometric Theory, Cambridge University Press, vol. 16(5), pages 692-728, October.
    6. Christian Francq & Jean-Michel Zakoïan, 2008. "Barlett’s Formula for Non Linear Processes," Working Papers 2008-05, Center for Research in Economics and Statistics.
    7. Robert Lund & I. V. Basawa, 2000. "Recursive Prediction and Likelihood Evaluation for Periodic ARMA Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 21(1), pages 75-93, January.
    8. Francq, Christian & Roy, Roch & Zakoian, Jean-Michel, 2005. "Diagnostic Checking in ARMA Models With Uncorrelated Errors," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 532-544, June.
    9. Philip Hans Franses & Richard Paap, 2000. "Modelling day-of-the-week seasonality in the S&P 500 index," Applied Financial Economics, Taylor & Francis Journals, vol. 10(5), pages 483-488.
    10. Roy, Roch & Saidi, Abdessamad, 2008. "Aggregation and systematic sampling of periodic ARMA processes," Computational Statistics & Data Analysis, Elsevier, vol. 52(9), pages 4287-4304, May.
    11. A. I. McLeod, 1994. "Diagnostic Checking Of Periodic Autoregression Models With Application," Journal of Time Series Analysis, Wiley Blackwell, vol. 15(2), pages 221-233, March.
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    14. Abdelhakim Aknouche & Abdelouahab Bibi, 2009. "Quasi‐maximum likelihood estimation of periodic GARCH and periodic ARMA‐GARCH processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(1), pages 19-46, January.
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    Cited by:

    1. Abdelhakim Aknouche & Bader Almohaimeed & Stefanos Dimitrakopoulos, 2022. "Periodic autoregressive conditional duration," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(1), pages 5-29, January.
    2. Daniel Dzikowski & Carsten Jentsch, 2024. "Structural Periodic Vector Autoregressions," Papers 2401.14545, arXiv.org.
    3. Abdelouahab Bibi & Ahmed Ghezal, 2016. "On periodic time-varying bilinear processes: structure and asymptotic inference," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 25(3), pages 395-420, August.
    4. Aknouche, Abdelhakim & Almohaimeed, Bader & Dimitrakopoulos, Stefanos, 2020. "Periodic autoregressive conditional duration," MPRA Paper 101696, University Library of Munich, Germany, revised 08 Jul 2020.

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    More about this item

    Keywords

    Weak periodic autoregressive moving average models; Seasonality; Weighted least squares; Asymptotic normality; Strong consistency; Weak periodic white noise; Strong mixing.;
    All these keywords.

    JEL classification:

    • 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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