Chaos in Fractionally Integrated Generalized Autoregressive Conditional Heteroskedastic Processes

TL;DR

This paper investigates the chaotic properties of FIGARCH processes used in financial modeling, finding evidence that they are not deterministic chaotic systems based on Lyapunov exponents and other nonlinear analysis tools.

Contribution

It provides a detailed nonlinear analysis of FIGARCH (p,d,q) processes, revealing their non-chaotic nature through various chaos detection methods.

Findings

Maximal Lyapunov exponents are negative.

FIGARCH processes are not deterministic chaotic.

Analysis includes mutual information, correlation dimensions, and FNNs.

Abstract

Fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) arises in modeling of financial time series. FIGARCH is essentially governed by a system of nonlinear stochastic difference equations ${u_t}$ = ${z_t}$ $(1-\sum\limits_{j=1}^q \beta_j L^j)\sigma_{t}^2 = \omega+(1-\sum\limits_{j=1}^q \beta_j L^j - (\sum\limits_{k=1}^p \varphi_k L^k) (1-L)^d) u_t^2$, where $\omega\in$ R, and $\beta_j\in$ R are constant parameters, $\{u_t\}_{{t\in}^+}$ and $\{\sigma_t\}_{{t\in}^+}$ are the discrete time real valued stochastic processes which represent FIGARCH (p,d,q) and stochastic volatility, respectively. Moreover, L is the backward shift operator, i.e. $L^d u_t \equiv u_{t-d}$ (d is the fractional differencing parameter 0$<$d$<$1). In this work, we have studied the chaoticity properties of FIGARCH (p,d,q) processes by computing mutual information, correlation dimensions, FNNs (False Nearest Neighbour), the Lyapunov exponents, and for both the stochastic difference equation given above and for the financial time series. We have observed that maximal Lyapunov exponents are negative, therefore, it can be suggested that FIGARCH (p,d,q) is not deterministic chaotic process.