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A NECESSARY MOMENT CONDITION FOR THE FRACTIONAL FUNCTIONAL CENTRAL LIMIT THEOREM

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 fractional functional central limit theorem (FCLT) for partial sums of xt = Δ−dut, where is the fractional integration parameter and ut is weakly dependent. The classical condition is existence of q ≥ 2 and moments of the innovation sequence. When d is close to this moment condition is very strong. Our main result is to show that when and under some relatively weak conditions on ut, the existence of moments is in fact necessary for the FCLT for fractionally integrated processes and that moments are necessary for more general fractional processes. Davidson and de Jong (2000, Econometric Theory 16, 643–666) presented a fractional FCLT where only q > 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.

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, three anonymous referees, and James Davidson for comments and to the Social Sciences and Humanities Research Council of Canada (SSHRC grant 410-2009-0183) and the Center for Research in Econometric Analysis of Time Series, (CREATES, funded by the Danish National Research Foundation) for financial support.

References

REFERENCES

Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Bingham, N.H., Goldie, C.M., & Teugels, J.L. (1989) Regular Variation. Cambridge University Press.Google Scholar
Davidson, J. & de Jong, R.M. (2000) The functional central limit theorem and weak convergence to stochastic integrals II: Fractionally integrated processes. Econometric Theory 16, 643–666.CrossRefGoogle Scholar
Davydov, Y.A. (1970) The invariance principle for stationary processes. Theory of Probability and Its Applications 15, 487–498.Google Scholar
Marinucci, D. & Robinson, P.M. (2000) Weak convergence of multivariate fractional processes. Stochastic Processes and Their Applications 86, 103–120.CrossRefGoogle Scholar
Taqqu, M.S. (1975) Weak convergence to fractional Brownian motion and to the Rosenblatt process. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 31, 287–302.CrossRefGoogle Scholar
Wu, W.B. & Shao, X. (2006) Invariance principles for fractionally integrated nonlinear processes. IMS Lecture Notes—Monograph Series: Recent Developments in Nonparametric Inference and Probability 50, 20–30.CrossRefGoogle Scholar