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Conditional heteroscedastic time series models. (English) Zbl 0636.62092
The paper investigates models for time series \(Y_ 1,Y_ 2,..\). with time varying conditional variances. The author proposes the new class of conditional heteroscedastic moving average models, CHARMA(p,q,r,s), defined by the observation equation \(\Phi(B)(Y_ t-\mu)= \vartheta(B)a_ t\) and the equation \[ \delta_ t(B)a_ t= \omega_ t[\hat Y_{t-1}(1)-\mu]+ \omega^*_ t(B)(Y_ t-\mu)+e_ t \] for the innovation process \(a_ 1,a_ 2,... \). Here \(\Phi(B)\), \(\vartheta(B)\), \(\delta_ t(B)\), \(\omega^*_ t(B)\) are polynomials of the backshift operator B (the two latter ones with random coefficients), \(e_ t\) is a white noise and \(\hat Y_{t-1}(1)\) is the optimum predictor of \(Y_ t\) based on the past up to time t-1.
The relation between CHARMA and two formerly proposed models (special cases) ARCH(p,q) and RCA(p) is established. The invertibility of the \(Y_ t\) process and the (moment and covariance) properties of the innovation process \(\{a_ t\}\) are investigated. Ordinary least squares estimates for the process parameters are shown to be consistent and asymptotically normally distributed. Model building and an F test for heteroscedasticity are considered. Finally, two applications are given.
Reviewer: H.H.Bock

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
91B84 Economic time series analysis
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