  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.
