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Asymptotic Properties of Weighted Least Squares Estimation in Weak PARMA Models

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)

  • Roch Roy
  • Abdessamad Saidi

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.
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Suggested Citation

  • Christian Francq & Roch Roy & Abdessamad Saidi, 2011. "Asymptotic Properties of Weighted Least Squares Estimation in Weak PARMA Models," Post-Print hal-05417839, HAL.
    • Handle: RePEc:hal:journl:hal-05417839
    DOI: 10.1111/j.1467-9892.2011.00728.x
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    Cited by:

    1. Y. Boubacar Maïnassara & A. Ilmi Amir, 2024. "Portmanteau tests for periodic ARMA models with dependent errors," Journal of Time Series Analysis, Wiley Blackwell, vol. 45(2), pages 164-188, March.
    2. 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.
    3. Daniel Dzikowski & Carsten Jentsch, 2024. "Structural Periodic Vector Autoregressions," Papers 2401.14545, arXiv.org, revised Aug 2025.
    4. Aknouche, Abdelhakim & Almohaimeed, Bader & Dimitrakopoulos, Stefanos, 2020. "Periodic autoregressive conditional duration," MPRA Paper 101696, University Library of Munich, Germany, revised 08 Jul 2020.
    5. Dzikowski, Daniel & Jentsch, Carsten, 2025. "Structural periodic vector autoregressions," Journal of Econometrics, Elsevier, vol. 252(PA).
    6. 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.
    7. Boubacar Maïnassara, Yacouba & Ursu, Eugen, 2025. "Diagnostic checking of periodic vector autoregressive time series models with dependent errors," Journal of Multivariate Analysis, Elsevier, vol. 205(C).
    8. Yacouba Boubacar Maïnassara & Eugen Ursu, 2023. "Estimating weak periodic vector autoregressive time series," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(3), pages 958-997, September.

    More about this item

    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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