# zbMATH — the first resource for mathematics

The estimation and application of long memory time series models. (English) Zbl 0534.62062
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)
Full Text:
##### References:
 [1] Buck R. C., Advanced Calculus (1978) · Zbl 0385.26002 [2] DOI: 10.2307/1401322 · Zbl 0101.35604 · doi:10.2307/1401322 [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 · doi:10.1016/0304-4076(80)90092-5 [6] DOI: 10.1007/BF00533484 · Zbl 0246.62086 · doi:10.1007/BF00533484 [7] DOI: 10.1093/biomet/68.1.165 · Zbl 0464.62088 · doi:10.1093/biomet/68.1.165 [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 Unpublished Harvard University manuscript.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) · doi:10.1002/sapm1946251261 [12] DOI: 10.1137/1010093 · Zbl 0179.47801 · doi:10.1137/1010093 [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 · doi:10.1093/biomet/50.1-2.129
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.