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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

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