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Discrete Fourier Transforms of Fractional Processes

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Author Info
Peter C.B. Phillips () (Cowles Foundation, Yale University)

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Abstract

Discrete Fourier transforms (dft's) of fractional processes are studied and an exact representation of the dft is given in terms of the component data. The new representation gives the frequency domain form of the model for a fractional process, and is particularly useful in analyzing the asymptotic behavior of the dft and periodogram in the nonstationary case when the memory parameter d >= 1/2. Various asymptotic approximations are suggested. It is shown that smoothed periodogram spectral estimates remain consistent for frequencies away from the origin in the nonstationary case provided the memory parameter d < 1. When d = 1, the spectral estimates are inconsistent and converge weakly to random variates. Applications of the theory to log periodogram regression and local Whittle estimation of the memory parameter are discussed and some modified versions of these procedures are suggested.

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File URL: http://cowles.econ.yale.edu/P/cd/d12a/d1243.pdf
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Publisher Info
Paper provided by Cowles Foundation, Yale University in its series Cowles Foundation Discussion Papers with number 1243.

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Length: 59 pages
Date of creation: Dec 1999
Date of revision:
Handle: RePEc:cwl:cwldpp:1243

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Postal: Yale University, Box 208281, New Haven, CT 06520-8281 USA
Phone: (203) 432-3702
Fax: (203) 432-6167
Web page: http://cowles.econ.yale.edu/
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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Related research
Keywords: Discrete Fourier transform; fractional Brownian motion; fractional integration; nonstationarity; operator decomposition; semiparametric estimation; Whittle likelihood;

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Find related papers by JEL classification:
C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

  1. Peter C.B. Phillips & Victor Solo, 1989. "Asymptotics for Linear Processes," Cowles Foundation Discussion Papers 932, Cowles Foundation, Yale University. [Downloadable!]
  2. Dean Corbae & Sam Ouliaris & Peter C. B. Phillips, 2002. "Band Spectral Regression with Trending Data," Econometrica, Econometric Society, vol. 70(3), pages 1067-1109, May. [Downloadable!] (restricted)
    Other versions:
  3. Peter C.B. Phillips, 1988. "Spectral Regression for Cointegrated Time Series," Cowles Foundation Discussion Papers 872, Cowles Foundation, Yale University. [Downloadable!]
  4. Peter C.B. Phillips, 1999. "Unit Root Log Periodogram Regression," Cowles Foundation Discussion Papers 1244, Cowles Foundation, Yale University. [Downloadable!]
    Other versions:
  5. Peter C.B. Phillips, 1985. "Fractional Matrix Calculus and the Distribution of Multivariate Tests," Cowles Foundation Discussion Papers 767, Cowles Foundation, Yale University. [Downloadable!]
  6. Gourieroux Christian & Akonom, J., 1988. "Functional limit theorem for fractional processes (a)," CEPREMAP Working Papers (Couverture Orange) 8801, CEPREMAP.
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Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Krüger, Niclas A, 2008. "Climate Variability and Health: Sweden 1751-2004," Working Papers 2008:4, Örebro University, Swedish Business School. [Downloadable!]
  2. Peter C.B. Phillips, 2004. "Challenges of Trending Time Series Econometrics," Cowles Foundation Discussion Papers 1472, Cowles Foundation, Yale University. [Downloadable!]
  3. Katsumi Shimotsu & Peter C.B. Phillips, 2000. "Modified Local Whittle Estimation of the Memory Parameter in the Nonstationary Case," Cowles Foundation Discussion Papers 1265, Cowles Foundation, Yale University. [Downloadable!]
  4. Katsumi Shimotsu & Peter C.B. Phillips, 2002. "Exact Local Whittle Estimation of Fractional Integration," Economics Discussion Papers 535, University of Essex, Department of Economics. [Downloadable!]
    Other versions:
  5. Offer Lieberman & Peter C. B. Phillips, 2006. "Refined Inference on Long Memory in Realized Volatility," Cowles Foundation Discussion Papers 1549, Cowles Foundation, Yale University. [Downloadable!]
    Other versions:
  6. Andersson, Fredrik N. G., 2008. "Bandspectrum Cointegration," Working Papers 2008:18, Lund University, Department of Economics. [Downloadable!]
  7. Aaron Smallwood; Alex Maynard; Mark Wohar, 2005. "The Long and the Short of It: Long Memory Regressors and Predictive Regressions," Computing in Economics and Finance 2005 384, Society for Computational Economics. [Downloadable!]
  8. Alex Maynard & Katsumi Shimotsu, 2007. "Covariance-based orthogonality tests for regressors with unknown persistence," Working Papers 1122, Queen's University, Department of Economics. [Downloadable!]
    Other versions:
  9. Katsumi Shimotsu, 2003. "Exact Local Whittle Estimation of Fractionally Cointegrated Systems," Economics Discussion Papers 570, University of Essex, Department of Economics. [Downloadable!]
  10. Claudio Morana, 2006. "Multivariate modelling of long memory processes with common components," ICER Working Papers 40-2006, ICER - International Centre for Economic Research. [Downloadable!]
  11. Chang Sik Kim & Peter C.B. Phillips, 2006. "Log Periodogram Regression: The Nonstationary Case," Cowles Foundation Discussion Papers 1587, Cowles Foundation, Yale University. [Downloadable!]
  12. Valle e Azevedo, João, 2007. "Exact Limit of the Expected Periodogram in the Unit-Root Case," MPRA Paper 6553, University Library of Munich, Germany. [Downloadable!]
  13. Krüger, Niclas A & Svensson, Mikael, 2008. "Good Times Are Drinking Times: Empirical Evidence on Business Cycles an Alcohol Sales in Sweden 1861-2000," Working Papers 2008:2, Örebro University, Swedish Business School. [Downloadable!]
  14. Alex Maynard & Peter C. B. Phillips, 2001. "Rethinking an old empirical puzzle: econometric evidence on the forward discount anomaly," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 16(6), pages 671-708. [Downloadable!]
  15. T. Di Matteo & T. Aste & Michel M. Dacorogna, 2005. "Long-term memories of developed and emerging markets: Using the scaling analysis to characterize their stage of development," Econometrics 0503004, EconWPA. [Downloadable!]
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  16. Yixiao Sun & Peter C.B. Phillips, 2002. "Nonlinear Log-Periodogram Regression for Perturbed Fractional Processes," Cowles Foundation Discussion Papers 1366, Cowles Foundation, Yale University. [Downloadable!]
    Other versions:
  17. Peter C. B. Phillips, 2006. "Optimal Estimation of Cointegrated Systems with Irrelevant Instruments," Cowles Foundation Discussion Papers 1547, Cowles Foundation, Yale University. [Downloadable!]
  18. Basma Bekdache & Christopher F. Baum, 1999. "A re-evaluation of empirical tests of the Fisher hypothesis," Computing in Economics and Finance 1999 944, Society for Computational Economics, revised 18 Sep 2000. [Downloadable!]
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  19. Laura Mayoral, 2006. "Minimum Distance Estimation of stationary and non-stationary ARFIMA Processes," Economics Working Papers 959, Department of Economics and Business, Universitat Pompeu Fabra. [Downloadable!]
    Other versions:
  20. Katsumi Shimotsu & Peter C.B. Phillips, 2000. "Local Whittle Estimation in Nonstationary and Unit Root Cases," Cowles Foundation Discussion Papers 1266, Cowles Foundation, Yale University, revised Sep 2003. [Downloadable!]
  21. Peter C.B. Phillips, 1999. "Unit Root Log Periodogram Regression," Cowles Foundation Discussion Papers 1244, Cowles Foundation, Yale University. [Downloadable!]
    Other versions:
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