Mathematics – Statistics Theory
Scientific paper
2012-01-19
Mathematics
Statistics Theory
Scientific paper
Here we present a theoretical study on the main properties of 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$(q,d,0)$ 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 five different FIEGARCH models. The forecasting performance of ARCH-type models is compared through an empirical application to Brazilian stock market exchange index.
Lopes Sílvia R. C.
Prass Taiane S.
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