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ARCH model and fractional Brownian motion

Author

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  • Bahamonde, Natalia
  • Torres, Soledad
  • Tudor, Ciprian A.

Abstract

We study an extension of the ARCH model that includes the squared fractional Brownian motion. We study the statistical properties of the model as the conditions for the existence of a stationary solution and the moments of the process. We study their asymptotic behavior of the autocorrelation function of the squared of the process and we prove that the long memory property of the model holds. We illustrate our results by numerical simulations.

Suggested Citation

  • Bahamonde, Natalia & Torres, Soledad & Tudor, Ciprian A., 2018. "ARCH model and fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 134(C), pages 70-78.
    • Handle: RePEc:eee:stapro:v:134:y:2018:i:c:p:70-78
    DOI: 10.1016/j.spl.2017.10.003
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    References listed on IDEAS

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    1. Arup Bose & Kanchan Mukherjee, 2003. "Estimating The Arch Parameters By Solving Linear Equations," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(2), pages 127-136, March.
    2. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    3. 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.
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    Cited by:

    1. Marko Voutilainen & Pauliina Ilmonen & Soledad Torres & Ciprian Tudor & Lauri Viitasaari, 2021. "On the ARCH model with stationary liquidity," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(2), pages 195-224, February.
    2. Sikora, Grzegorz, 2018. "Statistical test for fractional Brownian motion based on detrending moving average algorithm," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 54-62.

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