This paper studies a broad class of nonnegative
ARCH(∞) models. Sufficient conditions for the existence
of a stationary solution are established and an explicit
representation of the solution as a Volterra type series
is found. Under our assumptions, the covariance function
can decay slowly like a power function, falling just short
of the long memory structure. A moving average representation
in martingale differences is established, and the central
limit theorem is proved.
This paper studies the change-point problem and the cross-covariance function for ARCH models. Bounds for the cross-covariance function are derived and explicit formulae are obtained in special cases. Consistency of a CUSUM type change-point estimator is proved and its rate of convergence is established. A Ha Âjek±Re Ânyi type inequality is also proved. Results are obtained under weak moment assumptions.
We consider the long memory and leverage properties of a model for the conditional variance of an observable stationary sequence, where the conditional variance is the square of an inhomogeneous linear combination of past values of the observable sequence, with square summable weights. This model, which we call linear ARCH (LARCH), specializes to the asymmetric ARCH model of Engle (1990), and to a version of the quadratic ARCH model of Sentana (1995), these authors having discussed leverage potential in such models. The model which we consider was suggested by Robinson (1991), for use as a possibly long memory conditionally heteroscedastic alternative to i.i.d. behaviour, and further studied by Giraitis, Robinson and Surgailis (2000), who showed that integer powers, of degree at least 2, can have long memory autocorrelation. We establish conditions under which the cross-autovariance function between volatility and levels decays in the manner of moving average weights of long memory processes. We also establish a leverage property and conditions for finiteness of third and higher moments.
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