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A necessary moment condition for the fractional functional central limit theorem

Author

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  • Søren Johansen

    () (University of Copenhagen and CREATES)

  • Morten Ørregaard Nielsen

    () (Queen's University and CREATES)

Abstract

We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x_{t} = Delta^{-d} u_{t}, where d in (-1/2,1/2) is the fractional integration parameter and u_{t} is weakly dependent. The classical condition is existence of q≥2 and q>1/(d+1/2) moments of the innovation sequence. When d is close to -1/2 this moment condition is very strong. Our main result is to show that when d in (-1/2,0) and under some relatively weak conditions on u_{t}, the existence of q≥1/(d+1/2) moments is in fact necessary for the FCLT for fractionally integrated processes, and that q>1/(d+1/2) moments are necessary for more general fractional processes. Davidson and de Jong (2000) presented a fractional FCLT where only q>2 finite moments are assumed. As a corollary to our main theorem we show that their moment condition is not sufficient, and hence that their result is incorrect.

Suggested Citation

  • Søren Johansen & Morten Ørregaard Nielsen, 2010. "A necessary moment condition for the fractional functional central limit theorem," Working Papers 1244, Queen's University, Department of Economics.
  • Handle: RePEc:qed:wpaper:1244
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    File URL: http://qed.econ.queensu.ca/working_papers/papers/qed_wp_1244.pdf
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    References listed on IDEAS

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    1. de Jong, Robert M. & Davidson, James, 2000. "The Functional Central Limit Theorem And Weak Convergence To Stochastic Integrals I," Econometric Theory, Cambridge University Press, vol. 16(05), pages 621-642, October.
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    Cited by:

    1. Søren Johansen & Morten Ørregaard Nielsen, 0111. "Testing the CVAR in the fractional CVAR model," CREATES Research Papers 2017-37, Department of Economics and Business Economics, Aarhus University.
    2. Morten Ørregaard Nielsen, 2015. "Asymptotics for the Conditional-Sum-of-Squares Estimator in Multivariate Fractional Time-Series Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(2), pages 154-188, March.
    3. Man Wang & Ngai Hang Chan, 2016. "Testing for the Equality of Integration Orders of Multiple Series," Econometrics, MDPI, Open Access Journal, vol. 4(4), pages 1-10, December.

    More about this item

    Keywords

    Fractional integration; functional central limit theorem; long memory; moment condition; necessary condition;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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