Theoretical Results on FIEGARCH Processes

TL;DR

This paper provides a comprehensive theoretical analysis of FIEGARCH processes, detailing their properties, conditions for stationarity, and relationships with ARFIMA models, along with simulation and empirical forecasting results.

Contribution

It establishes new theoretical results on the properties, existence, and spectral structure of FIEGARCH processes, linking them to ARFIMA models and providing forecasting formulas.

Findings

FIEGARCH processes are related to ARFIMA$(q,d,0)$ under mild conditions.

Derived explicit formulas for kurtosis and asymmetry measures.

Monte Carlo simulations and empirical forecasts demonstrate model performance.

Abstract

Here we present a theoretical study on the main properties of Fractionally Integrated Exponential Generalized Autoregressive Conditional Heteroskedastic (FIEGARCH) processes. We analyze the conditions for the existence, the invertibility, the stationarity and the ergodicity of these processes. We prove that, if $\{X_t\}_{t \in \mathds{Z}}$ is a FIEGARCH$(p,d,q)$ process then, under mild conditions, $\{\ln(X_t^2)\}_{t\in\mathds{Z}}$ is an ARFIMA$(q,d,0)$, that is, an autoregressive fractionally integrated moving average process. The convergence order for the polynomial coefficients that describes the volatility is presented and results related to the spectral representation and to the covariance structure of both processes $\{\ln(X_t^2)\}_{t\in\mathds{Z}}$ and $\ {\ln(\sigma_t^2)\}_{t\in\mathds{Z}}$ are also discussed. Expressions for the kurtosis and the asymmetry measures for any stationary FIEGARCH$(p,d,q)$ process are also derived. The $h$-step ahead forecast for the processes $\{X_t\}_{t \in \mathds{Z}}$, $\{\ln(\sigma_t^2)\}_{t\in\mathds{Z}}$ and $\{\ln(X_t^2)\}_{t\in\mathds{Z}}$ are given with their respective mean square error forecast. The work also presents a Monte Carlo simulation study showing how to generate, estimate and forecast based on six different FIEGARCH models. The forecasting performance of six models belonging to the class of autoregressive conditional heteroskedastic models (namely, ARCH-type models) and radial basis models is compared through an empirical application to Brazilian stock market exchange index.