The Annals of Statistics

Exact local Whittle estimation of fractional integration

Katsumi Shimotsu and Peter C. B. Phillips

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Abstract

An exact form of the local Whittle likelihood is studied with the intent of developing a general-purpose estimation procedure for the memory parameter (d) that does not rely on tapering or differencing prefilters. The resulting exact local Whittle estimator is shown to be consistent and to have the same $N(0,\frac{1}{4})$ limit distribution for all values of d if the optimization covers an interval of width less than $\frac{9}{2}$ and the initial value of the process is known.

Article information

Source
Ann. Statist. Volume 33, Number 4 (2005), 1890-1933.

Dates
First available in Project Euclid: 5 August 2005

Permanent link to this document
http://projecteuclid.org/euclid.aos/1123250232

Digital Object Identifier
doi:10.1214/009053605000000309

Mathematical Reviews number (MathSciNet)
MR2166565

Zentralblatt MATH identifier
1081.62069

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Keywords
Discrete Fourier transform fractional integration long memory nonstationarity semiparametric estimation Whittle likelihood

Citation

Shimotsu, Katsumi; Phillips, Peter C. B. Exact local Whittle estimation of fractional integration. Ann. Statist. 33 (2005), no. 4, 1890--1933. doi:10.1214/009053605000000309. http://projecteuclid.org/euclid.aos/1123250232.


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References

  • Geweke, J. and Porter-Hudak, S. (1983). The estimation and application of long memory time series models. J. Time Ser. Anal. 4 221--238.
  • Henry, M. and Robinson, P. M. (1996). Bandwidth choice in Gaussian semiparametric estimation of long range dependence. Athens Conference on Applied Probability and Time Series. Lecture Notes in Statist. 115 220--232. Springer, New York.
  • Hurvich, C. M. and Chen, W. W. (2000). An efficient taper for potentially overdifferenced long-memory time series. J. Time Ser. Anal. 21 155--180.
  • Kim, C. S. and Phillips, P. C. B. (1999). Log periodogram regression: The nonstationary case. Mimeographed, Cowles Foundation, Yale Univ.
  • Künsch, H. (1987). Statistical aspects of self-similar processes. In Proc. First World Congress of the Bernoulli Society (Yu. Prokhorov and V. V. Sazanov, eds.) 1 67--74. VNU Science Press, Utrecht.
  • Marinucci, D. and Robinson, P. M. (1999). Alternative forms of fractional Brownian motion. J. Statist. Plann. Inference 80 111--122.
  • Phillips, P. C. B. (1999). Discrete Fourier transforms of fractional processes. Cowles Foundation Discussion Paper #1243, Yale Univ. Available at cowles.econ.yale.edu.
  • Phillips, P. C. B. and Shimotsu, K. (2004). Local Whittle estimation in nonstationary and unit root cases. Ann. Statist. 32 656--692.
  • Phillips, P. C. B. and Solo, V. (1992). Asymptotics for linear processes. Ann. Statist. 20 971--1001.
  • Robinson, P. M. (1995). Log-periodogram regression of time series with long-range dependence. Ann. Statist. 23 1048--1072.
  • Robinson, P. M. (1995). Gaussian semiparametric estimation of long-range dependence. Ann. Statist. 23 1630--1661.
  • Robinson, P. M. (2005). The distance between rival nonstationary fractional processes. J. Econometrics. To appear.
  • Robinson, P. M. and Marinucci, D. (2001). Narrow-band analysis of nonstationary processes. Ann. Statist. 29 947--986.
  • Shimotsu, K. (2004). Exact local Whittle estimation of fractional integration with unknown mean and time trend. Mimeographed, Queen's Univ., Kingston.
  • Velasco, C. (1999). Gaussian semiparametric estimation of non-stationary time series. J. Time Ser. Anal. 20 87--127.
  • Zygmund, A. (1977). Trigonometric Series. Cambridge Univ. Press.