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Representation and Weak Convergence of Stochastic Integrals with Fractional Integrator Processes

Author

Listed:
  • James Davidson
  • Nigar Hashimzade

    (School of Economics and Management, University of Aarhus, Denmark and CREATES)

Abstract

This paper considers the asymptotic distribution of the covariance of a nonstationary frac- tionally integrated process with the stationary increments of another such process - possibly, itself. Questions of interest include the relationship between the harmonic representation of these random variables, which we have analysed in a previous paper, and the construction derived from moving average representations in the time domain. The limiting integrals are shown to be expressible in terms of functionals of Itô integrals with respect to two distinct Brownian motions. Their mean is nonetheless shown to match that of the harmonic rep- resentation, and they satisfy the required integration by parts rule. The advantages of our approach over the harmonic analysis include the facts that our formulae are valid for the full range of the long memory parameters, and extend to non-Gaussian processes.

Suggested Citation

  • James Davidson & Nigar Hashimzade, 2007. "Representation and Weak Convergence of Stochastic Integrals with Fractional Integrator Processes," CREATES Research Papers 2007-45, Department of Economics and Business Economics, Aarhus University.
    • Handle: RePEc:aah:create:2007-45
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    References listed on IDEAS

    as
    1. Bender, Christian, 2003. "An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter," Stochastic Processes and their Applications, Elsevier, vol. 104(1), pages 81-106, March.
    2. Vladas Pipiras & Murad S. Taqqu, 2002. "Deconvolution of fractional brownian motion," Journal of Time Series Analysis, Wiley Blackwell, vol. 23(4), pages 487-501, July.
    3. Davidson, James & de Jong, Robert M., 2000. "The Functional Central Limit Theorem And Weak Convergence To Stochastic Integrals Ii," Econometric Theory, Cambridge University Press, vol. 16(5), pages 643-666, October.
    4. Davidson, James & Hashimzade, Nigar, 2008. "Alternative Frequency And Time Domain Versions Of Fractional Brownian Motion," Econometric Theory, Cambridge University Press, vol. 24(1), pages 256-293, February.
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    2. Christensen, Bent Jesper & Kruse, Robinson & Sibbertsen, Philipp, 2013. "A unified framework for testing in the linear regression model under unknown order of fractional integration," Hannover Economic Papers (HEP) dp-519, Leibniz Universität Hannover, Wirtschaftswissenschaftliche Fakultät.
    3. Buchmann, Boris & Chan, Ngai Hang, 2013. "Unified asymptotic theory for nearly unstable AR(p) processes," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 952-985.

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    Keywords

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    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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