A nonlinear model for long-memory conditional heteroscedasticity

We discuss a class of conditionally heteroscedastic time series models satisfying the equation r (t) = zeta (t) sigma (t) , where zeta (t) are standardized i.i.d. r.v.s, and the conditional standard deviation sigma (t) is a nonlinear function Q of inhomogeneous linear combination of past values r (s) , s < t, with coefficients b (j) . The existence of stationary solution rt with finite pth moment, 0 < p < a is obtained under some conditions on Q, b (j) and the pth moment of zeta (0). Weak dependence properties of r (t) are studied, including the invariance principle for partial sums of Lipschitz functions of r (t) . In the case where Q is the square root of a quadratic polynomial, we prove that r (t) can exhibit a leverage effect and long memory in the sense that the squared process r (t) (2) has long-memory autocorrelation and its normalized partial-sum process converges to a fractional Brownian motion.