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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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