A generalized nonlinear model for long memory conditional heteroscedasticity

TL;DR

This paper introduces a generalized nonlinear ARCH-type model with long memory properties, extending previous work to include a parameter for persistence, and analyzes conditions for stationarity, moments, and leverage effects.

Contribution

It extends the nonlinear ARCH model to include a persistence parameter, providing new conditions for stationarity, moments, and long memory behavior.

Findings

Model exhibits long memory and leverage effects.

New conditions for higher moments without Rosenthal constant.

Simulation results illustrate long memory and density behaviors.

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 $\sigma^2_t = Q^2\big(a + \sum_{j=1}^\infty b_j r_{t-j}\big) + \gamma \sigma^2_{t-1}$ 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 $\gamma$.