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A NECESSARY MOMENT CONDITION FOR THEFRACTIONAL FUNCTIONAL CENTRAL LIMITTHEOREM

Published online by Cambridge University Press:  25 November 2011

Søren Johansen  and
Morten Ørregaard Nielsen
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Abstract

We discuss the moment condition for the fractionalfunctional central limit theorem (FCLT) for partialsums of xt =Δ−dut,where isthe fractional integration parameter andut is weaklydependent. The classical condition is existence ofq ≥ 2 and moments of the innovation sequence. Whend is close to thismoment condition is very strong. Our main result isto show that when and undersome relatively weak conditions onut, the existence ofmoments is in fact necessary for the FCLT forfractionally integrated processes and thatmoments are necessary for more general fractionalprocesses. Davidson and de Jong (2000,Econometric Theory 16, 643–666)presented a fractional FCLT where onlyq > 2 finite moments areassumed. As a corollary to our main theorem we showthat their moment condition is not sufficient andhence that their result is incorrect.

Information

Type
Brief Report
Information
Econometric Theory , Volume 28 , Issue 3 , June 2012 , pp. 671 - 679
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

We are grateful to Benedikt Pötscher, threeanonymous referees, and James Davidson forcomments and to the Social Sciences and HumanitiesResearch Council of Canada (SSHRC grant410-2009-0183) and the Center for Research inEconometric Analysis of Time Series, (CREATES,funded by the Danish National Research Foundation)for financial support.

References

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