## 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)
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### References:

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