A generalized nonlinear model for long memory conditional heteroscedasticity

Abstract

We study the existence and properties of stationary solution of ARCH-type equation $r_t={\zeta }_t{\sigma }_t$, where ${\zeta }_t$ are standardized i.i.d. r.v.'s and the conditional variance satisfies an AR(1) equation ${\sigma }_t^2=Q^2(a+{\sum }_{j=1}^{\infty }{b}_j{r}_{t-j})+\gamma {\sigma }_{t-1}^2$ with a Lipschitz function $Q(x)$ and real parameters $a,\gamma ,b_j$. The paper extends the model and the results in Doukhan et al. (2015) from the case $\gamma =0$ to the case $0<\gamma <1$. We also obtain a new condition for the existence of higher moments of $r_t$ which does not include the Rosenthal constant. In particular case when $Q$ is the square root of a quadratic polynomial, we prove that $r_t$ can exhibit a leverage effect and long memory. We also present simulated trajectories and histograms of marginal density of ${\sigma }_t$ for different values of $\gamma$.