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A Necessary Moment Condition For The Fractional Functional Central Limit Theorem


  • Johansen, Søren
  • Ørregaard Nielsen, Morten


We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x = Δ − u , where null is the fractional integration parameter and u is weakly dependent. The classical condition is existence of q ≥ 2 and null moments of the innovation sequence. When d is close to null this moment condition is very strong. Our main result is to show that when null and under some relatively weak conditions on u , the existence of null moments is in fact necessary for the FCLT for fractionally integrated processes and that null moments are necessary for more general fractional processes. Davidson and de Jong (2000, Econometric Theory 16, 643–666) 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

  • Johansen, Søren & Ørregaard Nielsen, Morten, 2012. "A Necessary Moment Condition For The Fractional Functional Central Limit Theorem," Econometric Theory, Cambridge University Press, vol. 28(03), pages 671-679, June.
  • Handle: RePEc:cup:etheor:v:28:y:2012:i:03:p:671-679_00

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    References listed on IDEAS

    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.

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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


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