## A Necessary Moment Condition for the Fractional Functional Central Limit Theorem

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**A Necessary Moment Condition for the Fractional Functional Central Limit Theorem.** / Johansen, Søren; Nielsen, Morten Ørregaard.

Research output: Working paper › Research

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*A Necessary Moment Condition for the Fractional Functional Central Limit Theorem*. Department of Economics, University of Copenhagen.

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TY - UNPB

T1 - A Necessary Moment Condition for the Fractional Functional Central Limit Theorem

AU - Johansen, Søren

AU - Nielsen, Morten Ørregaard

N1 - JEL classification: C22

PY - 2010

Y1 - 2010

N2 - We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x(t)=¿^(-d)u(t), where d ¿ (-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.

AB - We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x(t)=¿^(-d)u(t), where d ¿ (-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.

KW - Faculty of Social Sciences

M3 - Working paper

BT - A Necessary Moment Condition for the Fractional Functional Central Limit Theorem

PB - Department of Economics, University of Copenhagen

ER -

ID: 22773767