# zbMATH — the first resource for mathematics

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
Full Text: