The estimation and application of long memory time series models. (English) Zbl 0534.62062

Geweke, John; Porter-Hudak, Susan

J. Time Ser. Anal. 4, 221-238 (1983).

Let \(\{\epsilon_ t\}\) be a white noise with a variance \(\sigma^ 2\). Consider the model \((1-B)^ dX_ t=\epsilon_ t\), where \(d\in(-.5,.5)\) and B is the back-shift operator. Then the spectral density of \(\{X_ t\}\) is \(f_ 2(\lambda;d)=(\sigma^ 2/2\pi)\{4 \sin^ 2(\lambda /2)\}^{-d}.\)
Let \(f_ u(\lambda)>0\) be a bounded continuous function on [- \(\pi\),\(\pi]\). A series with the spectral density \(f_ 2(\lambda;d)\) and \(f_ 2(\lambda;d)f_ u(\lambda)\) is called simple integrated series respectively general integrated series (GIS). It is shown that \(\{X_ t\}\) is a GIS iff it is a general fractional Gaussian noise. The authors propose a new estimator \(\hat d\) of the parameter d in GIS, which is based on the linear regression of the lowest frequency ordinates of the log periodogram on a deterministic regressor. The asymptotics of \(\hat d\) is analyzed.
A simulation study indicates that the asymptotic results are applicable in samples of at least 50 observations. For three economic time series the estimated integrated series model provides more reliable out-of- sample forecasts than do the classical procedures.

Reviewer: J.Anděl

MSC:

62M10

Time series, auto-correlation, regression, etc. in statistics (GARCH)

Keywords:

estimation; long memory time series models; fractional differencing; spectral density; simple integrated series; general integrated series; general fractional Gaussian noise; simulation study; asymptotic results; economic time series

References:

[1]	Buck R. C., Advanced Calculus (1978) · Zbl 0385.26002
[2]	DOI: 10.2307/1401322 · Zbl 0101.35604
[3]	Gradshteyn I. S., Tables of Integrals, Series and Products (1980) · Zbl 0521.33001
[4]	Granger C. W. G., J. Time Series Anal. 1 (1) pp 15– (1980)
[5]	DOI: 10.1016/0304-4076(80)90092-5 · Zbl 0466.62108
[6]	DOI: 10.1007/BF00533484 · Zbl 0246.62086
[7]	DOI: 10.1093/biomet/68.1.165 · Zbl 0464.62088
[8]	Hurst H. E., Trans. Amer. Soc. Civil Engrs. 116 pp 770– (1951)
[9]	A. Jonas (1981 ) Long Memory Self Similar Time Series Models. Unpublished Harvard University manuscript.
[10]	Kendall M., The Advanced Theory of Statistics 1, 4. ed. (1977)
[11]	Levinson N., J. Math. Phys. 25 pp 261– (1947)
[12]	DOI: 10.1137/1010093 · Zbl 0179.47801
[13]	Mandelbrot B. B., Water Resources Research 7 pp 543– (1971)
[14]	B. B. Mandelbrot (1973 ) Statistical Methodology for Nonperiodic Cycles: From the Covariance to R/S Analysis. Rev. Econ. Soc. Meas.pp.259 -290 .
[15]	Mcleod A. I., Water Resources Research 14 (3) pp 491– (1978)
[16]	S. Porter-Hudak (1982 ) Long-Term Memory Modelling-A Simplified Spectral Approach . Unpublished University of WisconsinPh.D. Thesis .
[17]	Rozanov Y. A., Stationary Random Processes. (1967) · Zbl 0152.16302
[18]	Whittle P., Biometrika 50 pp 129– (1963) · Zbl 0129.11304