A nonlinear model for long-memory conditional heteroscedasticity*

We discuss a class of conditionally heteroscedastic time series models satisfying the equation rt = ζtσt, where ζt are standardized i.i.d. r.v.s, and the conditional standard deviation σt is a nonlinear function Q of inhomogeneous linear combination of past values rs, s < t, with coefficients bj . The existence of stationary solution rt with finite pth moment, 0 < p < ∞ is obtained under some conditions on Q, bj and the pth moment of ζ0. Weak dependence properties of rt are studied, including the invariance principle for partial sums of Lipschitz functions of rt. In the case where Q is the square root of a quadratic polynomial, we prove that rt can exhibit a leverage effect and long memory in the sense that the squared process rt2 has long-memory autocorrelation and its normalized partial-sum process converges to a fractional Brownian motion.