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A k‐Factor GARMA Long‐memory Model

Author

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  • Wayne A. Woodward
  • Q. C. Cheng
  • H. L. Gray

Abstract

Long‐memory models have been used by several authors to model data with persistent autocorrelations. The fractional and fractional autoregressive moving‐average (FARMA) models describe long‐memory behavior associated with an infinite peak in the spectrum at f = 0. The Gegenbauer and Gegenbauer ARMA (GARMA) processes of Gray, Zhang and Woodward (On generalized fractional processes. J. Time Ser. Anal. 10 (1989), 233–57) can model long‐term periodic behavior for any frequency 0 ≤f≤ 0.5. In this paper we introduce a k‐factor extension of the Gegenbauer and GARMA models that allows for long‐memory behavior to be associated with each of k frequencies in [0, 0.5]. We prove stationarity conditions for the k‐factor model and discuss issues such as parameter estimation, model iden‐ tification, realization generation and forecasting. A two‐factor GARMA model is then applied to the Mauna Loa atmospheric CO2 data. It is shown that this model provides a reasonable fit to the CO2 data and produces excellent forecasts.

Suggested Citation

  • Wayne A. Woodward & Q. C. Cheng & H. L. Gray, 1998. "A k‐Factor GARMA Long‐memory Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 19(4), pages 485-504, July.
    • Handle: RePEc:bla:jtsera:v:19:y:1998:i:4:p:485-504
    DOI: 10.1111/j.1467-9892.1998.00105.x
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    Cited by:

    1. Laurent Ferrara & Dominique Guegan & Zhiping Lu, 2008. "Testing fractional order of long memory processes: a Monte Carlo study," Documents de travail du Centre d'Economie de la Sorbonne b08012, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    2. Dominique Guegan & Laurent Ferrara, 2008. "Fractional and seasonal filtering," Post-Print halshs-00646178, HAL.
    3. Laurent Ferrara & Dominique Guégan, 2008. "Business surveys modelling with Seasonal-Cyclical Long Memory models," Economics Bulletin, AccessEcon, vol. 3(29), pages 1-10.
    4. Asai, Manabu & McAleer, Michael & Peiris, Shelton, 2020. "Realized stochastic volatility models with generalized Gegenbauer long memory," Econometrics and Statistics, Elsevier, vol. 16(C), pages 42-54.
    5. Voges, Michelle & Sibbertsen, Philipp, 2021. "Cyclical fractional cointegration," Econometrics and Statistics, Elsevier, vol. 19(C), pages 114-129.
    6. Asai Manabu & Peiris Shelton & McAleer Michael & Allen David E., 2020. "Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates," Journal of Time Series Econometrics, De Gruyter, vol. 12(1), pages 1-18, January.
    7. Beaumont, Paul & Smallwood, Aaron, 2019. "Inference for likelihood-based estimators of generalized long-memory processes," MPRA Paper 96313, University Library of Munich, Germany.
    8. Massimiliano Caporin & Francesco Lisi, 2010. "Misspecification tests for periodic long memory GARCH models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 19(1), pages 47-62, March.
    9. Dmitriy Ivanov & Aleksandr Zhdanov, 2021. "Symmetrical Augmented System of Equations for the Parameter Identification of Discrete Fractional Systems by Generalized Total Least Squares," Mathematics, MDPI, vol. 9(24), pages 1-13, December.
    10. Proietti, Tommaso & Maddanu, Federico, 2024. "Modelling cycles in climate series: The fractional sinusoidal waveform process," Journal of Econometrics, Elsevier, vol. 239(1).
    11. Oumaima Essefiani & Rachid El Halimi & Said Hamdoune, 2024. "Some Estimation Methods for a Random Coefficient in the Gegenbauer Autoregressive Moving-Average Model," Mathematics, MDPI, vol. 12(11), pages 1-16, May.
    12. Souhir, Ben Amor & Heni, Boubaker & Lotfi, Belkacem, 2019. "Price risk and hedging strategies in Nord Pool electricity market evidence with sector indexes," Energy Economics, Elsevier, vol. 80(C), pages 635-655.
    13. Beaumont, Paul & Smallwood, Aaron, 2019. "Conditional Sum of Squares Estimation of Multiple Frequency Long Memory Models," MPRA Paper 96314, University Library of Munich, Germany.
    14. Nicholas Apergis & Andrea Mervar & James E. Payne, 2017. "Forecasting disaggregated tourist arrivals in Croatia," Tourism Economics, , vol. 23(1), pages 78-98, February.
    15. Stéphane Goutte & David Guerreiro & Bilel Sanhaji & Sophie Saglio & Julien Chevallier, 2019. "International Financial Markets," Post-Print halshs-02183053, HAL.
    16. Richard Hunt & Shelton Peiris & Neville Weber, 2022. "Estimation methods for stationary Gegenbauer processes," Statistical Papers, Springer, vol. 63(6), pages 1707-1741, December.

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