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Type I and Type II Fractional Brownian Motions: a Reconsideration

Author

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  • James Davidson

    (Department of Economics, University of Exeter)

  • Nigar Hashimzade

    (University of Reading)

Abstract

The so-called type I and type II fractional Brownian motions are limit distributions associated with the fractional integration model in which pre-sample shocks are either included in the lag structure, or suppressed. There can be substantial differences between the distributions of these two processes and of functionals derived from them, so that it becomes an important issue to decide which model to use as a basis for inference. Alternative methods for simulating the type I case are contrasted, and for models close to the nonstationarity boundary, truncating infinite sums is shown to result in a significant distortion of the distribution. A simple simulation method that overcomes this problem is described and implemented. The approach also has implications for the estimation of type I ARFIMA models, and a new conditional ML estimator is proposed, using the annual Nile minima series for illustration.

Suggested Citation

  • James Davidson & Nigar Hashimzade, 2008. "Type I and Type II Fractional Brownian Motions: a Reconsideration," Discussion Papers 0816, University of Exeter, Department of Economics.
    • Handle: RePEc:exe:wpaper:0816
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    7. Frank S. Nielsen, 2011. "Local Whittle estimation of multi‐variate fractionally integrated processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 32(3), pages 317-335, May.
    8. Ergemen, Yunus Emre & Haldrup, Niels & Rodríguez-Caballero, Carlos Vladimir, 2016. "Common long-range dependence in a panel of hourly Nord Pool electricity prices and loads," Energy Economics, Elsevier, vol. 60(C), pages 79-96.
    9. 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.
    10. Caporin, Massimiliano & Ranaldo, Angelo & Santucci de Magistris, Paolo, 2013. "On the predictability of stock prices: A case for high and low prices," Journal of Banking & Finance, Elsevier, vol. 37(12), pages 5132-5146.
    11. Monge, Manuel & Romero Rojo, María Fátima & Gil-Alana, Luis Alberiko, 2023. "The impact of geopolitical risk on the behavior of oil prices and freight rates," Energy, Elsevier, vol. 269(C).
    12. Peter C. B. Phillips, 2023. "Discrete Fourier Transforms of Fractional Processes with Econometric Applications," Advances in Econometrics, in: Essays in Honor of Joon Y. Park: Econometric Theory, volume 45, pages 3-71, Emerald Group Publishing Limited.

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