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A necessary moment condition for the fractional functional central limit theorem

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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>max(2,(d+1/2)^{-1}) 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 under some relatively weak conditions on u_{t}, the existence of q=max(2,(d+1/2)^{-1}) is in fact necessary for the FCLT for fractionally integrated processes and that q>max(2,(d+1/2)^{-1}) moments are necessary and sufficient for more general fractional processes. Davidson and de Jong (2000) presented a fractional FCLT where only q>2 finite moments are assumed, which is remarkable because it is the only FCLT where the moment condition has been weakened relative to the earlier condition. As a corollary to our main theorem we show that their moment condition is not sufficient.

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Bibliographic Info

Paper provided by School of Economics and Management, University of Aarhus in its series CREATES Research Papers with number 2010-70.

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Length: 8
Date of creation: 10 2010
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Handle: RePEc:aah:create:2010-70

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Web page: http://www.econ.au.dk/afn/

Related research

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

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  1. 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(05), pages 643-666, October.
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