Abstract
We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x(t)=¿^{-d}u(t) , where -1/2<d<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 -1/2<d<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, Econometric Theory 16, 643-- 666) presented a fractional FCLT where onlyq>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.
| Original language | English |
|---|---|
| Journal | Econometric Theory |
| Issue number | 28 |
| Pages (from-to) | 671-679 |
| Number of pages | 9 |
| ISSN | 0266-4666 |
| DOIs | |
| Publication status | Published - 2012 |