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A Necessary Moment Condition for the Fractional Functional Central Limit Theorem

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  • Søren Johansen

    (Department of Economics, University of Copenhagen)

  • Morten Ørregaard Nielsen

    (Department of Economics, Queen's University, Kingston, Ontario)

Abstract

We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x(t)=Δ^(-d)u(t), where d є (-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 University of Copenhagen. Department of Economics in its series Discussion Papers with number 10-29.

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Length: 8 pages
Date of creation: Oct 2010
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Handle: RePEc:kud:kuiedp:1029

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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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