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A Generalized Fractionally Integrated Autoregressive Moving‐Average Process

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  • Ching‐Fan Chung

Abstract

. This paper considers the long memory Gegenbauer autoregressive movingaverage (GARMA) process that generalizes the fractionally integrated ARMA (ARFIMA) process to allow for hyperbolic and sinusoidal decay in autocorrelations. We propose the conditional sum of squares method for estimation (which is asymptotically equivalent to the maximum likelihood estimation) and develop the asymptotic theory. Many results are in sharp contrast to those of the ARFIMA model. Simulations are conducted to assess the performance of the proposed estimators in small sample applications. Two applications to the sunspot data and the US inflation rates based on the wholesale price index are provided.

Suggested Citation

  • Ching‐Fan Chung, 1996. "A Generalized Fractionally Integrated Autoregressive Moving‐Average Process," Journal of Time Series Analysis, Wiley Blackwell, vol. 17(2), pages 111-140, March.
    • Handle: RePEc:bla:jtsera:v:17:y:1996:i:2:p:111-140
    DOI: 10.1111/j.1467-9892.1996.tb00268.x
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    Cited by:

    1. Federico Maddanu, 2022. "A harmonically weighted filter for cyclical long memory processes," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(1), pages 49-78, March.
    2. Shelton Peiris & Manabu Asai & Michael McAleer, 2017. "Estimating and Forecasting Generalized Fractional Long Memory Stochastic Volatility Models," JRFM, MDPI, vol. 10(4), pages 1-16, December.
    3. Guglielmo Maria Caporale & Luis A. Gil-Alana & Carlos Poza, 2021. "Cycles and Long-Range Behaviour in the European Stock Markets," Dynamic Modeling and Econometrics in Economics and Finance, in: Gilles Dufrénot & Takashi Matsuki (ed.), Recent Econometric Techniques for Macroeconomic and Financial Data, pages 293-302, Springer.
    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. Paul M. Beaumont & Aaron D. Smallwood, 2024. "Conditional sum of squares estimation of k-factor GARMA models," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 108(3), pages 501-543, September.
    6. Guglielmo Maria Caporale & Luis Alberiko Gil-Alana, 2023. "Long-Run Trends and Cycles in US House Prices," CESifo Working Paper Series 10751, CESifo.
    7. Guglielmo Maria Caporale & Luis Gil-Alana, 2012. "Long Memory and Volatility Dynamics in the US Dollar Exchange Rate," Multinational Finance Journal, Multinational Finance Journal, vol. 16(1-2), pages 105-136, March - J.
    8. Beaumont, Paul & Smallwood, Aaron, 2019. "Inference for likelihood-based estimators of generalized long-memory processes," MPRA Paper 96313, University Library of Munich, Germany.
    9. Juan Carlos Cuestas & Luis A. Gil-Alana, 2012. "A Non-Linear Approach with Long Range Dependence Based on Chebyshev Polynomials," Working Papers 2012013, The University of Sheffield, Department of Economics.
    10. Aaron D. Smallwood & Stefan C. Norrbin, 2006. "Generalized long memory processes, failure of cointegration tests and exchange rate dynamics," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 21(4), pages 409-417, May.
    11. Proietti, Tommaso & Maddanu, Federico, 2024. "Modelling cycles in climate series: The fractional sinusoidal waveform process," Journal of Econometrics, Elsevier, vol. 239(1).
    12. Gil-Alana, Luis A. & Trani, Tommaso, 2019. "The cyclical structure of the UK inflation rate: 1210–2016," Economics Letters, Elsevier, vol. 181(C), pages 182-185.
    13. Giorgio Canarella & Luis A. Gil-Alana & Rangan Gupta & Stephen M. Miller, 2020. "Modeling US historical time-series prices and inflation using alternative long-memory approaches," Empirical Economics, Springer, vol. 58(4), pages 1491-1511, April.
    14. Wilfredo Palma & Ngai Hang Chan, 2005. "Efficient Estimation of Seasonal Long‐Range‐Dependent Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(6), pages 863-892, November.
    15. Beaumont, Paul & Smallwood, Aaron, 2019. "Conditional Sum of Squares Estimation of Multiple Frequency Long Memory Models," MPRA Paper 96314, University Library of Munich, Germany.
    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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