Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-07T04:29:05.569Z Has data issue: false hasContentIssue false
  • English
  • Français

IMPULSE RESPONSES OF FRACTIONALLY INTEGRATED PROCESSES WITH LONG MEMORY

Published online by Cambridge University Press:  08 April 2010

Uwe Hassler  and
Piotr Kokoszka
Get access Rights & Permissions [Opens in a new window]

Abstract

Fractionally integrated time series, which have become an important modeling tool over the last two decades, are obtained by applying the fractional filter to a weakly dependent (short memory) sequence. Weakly dependent sequences are characterized by absolutely summable impulse response coefficients of their Wold representation. The weights bn decay at the rate nd−1 and are not absolutely summable for long memory models (d > 0). It has been believed that this rate is inherited by the impulse responses of any long memory fractionally integrated model. We show that this conjecture does not hold in such generality, and we establish a simple necessary and sufficient condition for the rate nd−1 to be inherited by fractionally integrated processes.

Type
Brief Report
Information
Econometric Theory , Volume 26 , Issue 6 , December 2010 , pp. 1855 - 1861
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research was partially supported by NSF grants DMS-0804165 and DMS-0931948 at Utah State University. We thank Stanley Williams for suggesting Example 2.1. We thank three referees and Professor P.C.B. Phillips for comments that led to substantive improvements of our results and Professor Giuseppe Cavaliere for efficiently handling this submission.

References

REFERENCES

Baillie, R.T. & Kapetanios, G. (2008) Nonlinear models for strongly dependent processes with financial applications. Journal of Econometrics 147, 60–71.CrossRefGoogle Scholar
Doukhan, P., Oppenheim, G., & Taqqu, M.S. (eds.) (2003) Theory and Applications of Long Range Dependence. Birkhäuser.Google Scholar
Giraitis, L., Robinson, P., & Surgailis, D. (2000) A model for long memory conditional heteroskedasticity. Annals of Applied Probability 10, 1002–1024.Google Scholar
Granger, C.W.J. & Joyeux, R. (1980) An introduction to long-memory time series and fractional differencing. Journal of Time Series Analysis 1, 15–30.Google Scholar
Hosking, J.R.M. (1981) Fractional differencing. Biometrika 68, 165–176.CrossRefGoogle Scholar
Kokoszka, P.S. & Mikosch, T., (1997) The integrated periodogram for long memory processes with finite and infinite variance. Stochastic Processes and Their Applications 66, 55–78.Google Scholar
Kokoszka, P.S. & Taqqu, M.S. (1995) Fractional ARIMA with stable innovations. Stochastic Processes and Their Applications 60, 19–47.Google Scholar
Kokoszka, P.S. & Taqqu, M.S. (1996) Parameter estimation for infinite variance fractional ARIMA. Annals of Statistics 24, 1880–1913.CrossRefGoogle Scholar
Marinucci, D. & Robinson, P.M. (1999) Alternative forms of fractional Brownian motion. Journal of Statistical Planning and Inference 80, 111–122.Google Scholar
Phillips, P.C.B. (2010) Long memory and long run variation. Journal of Econometrics 151, 150–158.Google Scholar
Poskitt, D.S. (2007) Autoregressive approximation in nonstandard situations: The fractionally integrated and non-invertible cases. Annals of the Institute of Statistical Mathematics 59, 697–725.Google Scholar
Pourahmadi, M. (2001) Foundations of Time Series Analysis and Prediction Theory. Wiley.Google Scholar
Robinson, P.M. (ed.) (2003) Time Series with Long Memory. Advanced Texts in Econometrics. Oxford University Press.Google Scholar
Robinson, P.M. (2005) The distance between rival nonstationary fractional processes. Journal of Econometrics 128, 283–300.Google Scholar
Teyssiere, G. & Kirman, A. (eds.) (2006) Long Memory in Economics. Springer-Verlog.Google Scholar
Tolstov, G.P. (1976) Fourier Series. Dover.Google Scholar
Zygmund, A. (1959) Trigonometric Series, vols. 1 and 2. Cambridge University Press.Google Scholar