# zbMATH — the first resource for mathematics

The moment structure of ARCH processes. (English) Zbl 0595.62089
The moment structure of a particular class of nonlinear stochastic processes known as the linear ARCH (autoregressive conditional heteroscedastic) models [cf. R. F. Engle, Econometrica 50, 987-1007 (1982; Zbl 0491.62099)] is derived and applied to test for stationarity and autocorrelation. In terms of a time series $$\eta_ t$$ of independent standard normal variates, the stochastic process is written as $$X_ t=\eta_ t\sqrt{V_ t}$$, where the distribution of the real-valued stochastic process $$X_ t$$ conditioned on the past values $$X_{t-i}$$, $$i\geq 1$$ is assumed to be normal with mean zero and variance $$V_ t$$, where $V_ t=\gamma +\phi_ 1(X^ 2_{t-1}-\gamma)+...+\phi_ p(x^ 2_{t-p}-\gamma).$ Under the regularity conditions $$\gamma >0$$ and $$\phi_ 1+...+\phi_ p<1$$ the series $$\{X^ 2_ t\}$$ is shown to converge and it is shown that the ARCH process $$X_ t$$ is uncorrelated with constant expectation zero and variance $$\gamma$$. The $$X_ t's$$ are uncorrelated but of course not independent. Their dependence is studied by the higher order moments provided the conditions for their existence are satisfied.
Reviewer: J.K.Sengupta

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)