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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

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We discuss the moment condition for the fractionalfunctional central limit theorem (FCLT) for partialsums of xt =Δ−dut,where $d\, \in \,\left({ - {1 \over 2}\,,\,{1 \over 2}} \right)$ isthe fractional integration parameter andut is weaklydependent. The classical condition is existence ofq ≥ 2 and $q\, > \,\left( {d\, + \,{1 \over 2}} \right)^{ - 1} $moments of the innovation sequence. Whend is close to $ - {1 \over 2}$ thismoment condition is very strong. Our main result isto show that when $d\, \in \,\left({ - \,{1 \over 2},\,0} \right)$ and undersome relatively weak conditions onut, the existence of$q\, \ge \,\left({d\, + \,{1 \over 2}} \right)^{ - 1} $moments is in fact necessary for the FCLT forfractionally integrated processes and that$q\, > \,\left( {d\, + \,{1 \over 2}} \right)^{ - 1} $moments 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

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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