A nonlinear model for long memory conditional heteroscedasticity

TL;DR

This paper introduces a nonlinear conditional heteroscedastic model for time series, demonstrating its ability to exhibit long memory and leverage effects, with theoretical results on stationarity, dependence, and convergence to fractional Brownian motion.

Contribution

It develops a new nonlinear model for heteroscedastic time series and proves conditions for stationarity, long memory, and leverage effects, extending existing models.

Findings

Existence of stationary solutions with finite moments under certain conditions.

The model can exhibit long memory in squared returns.

Normalized partial sums converge to fractional Brownian motion.

Abstract

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 $r_t$ with finite $p$th moment, $0< p < \infty $ is obtained under some conditions on $Q, b_j$ and $p$th 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 of quadratic $Q^2$, we prove that $r_t$ can exhibit a leverage effect and long memory, in the sense that the squared process $r^2_t$ has long memory autocorrelation and its normalized partial sums process converges to a fractional Brownian motion.