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. random variables and the conditional variance satisfies an AR(1) equation ${\sigma }_t^2=Q^2(a+\sum _{j=1}^{\mathrm{\infty }}{b}_j{r}_{t-j})+\gamma {\sigma }_{t-1}^2$ with a Lipschitz function $Q\left(x\right)$ and real parameters $a,\gamma ,b_j$. The paper extends the model and the results in Doukhan, Grublytė, and Surgailis [A nonlinear model for long memory conditional heteroscedasticity. Lithuanian Math J. 2016;56:164–188] 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 the 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 γ.

KEYWORDS:

• Asymmetric ARCH model
• LARCH model
• leverage
• long memory

Disclosure statement

No potential conflict of interest was reported by the authors.