Introduction

AutoRegressive Conditional Heteroscedasticity (ARCH) models, introduced by Engle, have become a standard tool in analyzing economic and financial time series to describe the variance of the error terms. There has been extensive literature on the subject including many review articles and book chapters. Many extensions of ARCH model have been proposed to overcome certain type of problems in the past three decades, for instance, the GARCH model, the nonlinear-ARCH model and the EGARCH model. Denote $X_t$ as the error term. The classical ARCH model of order $p$ is defined as follows:

$\begin{equation} X_t=\sigma_t Z_t, \qquad \sigma^2_t=a_0+\sum_{j=1}^{p}a_j X^2_{t-j}, \ t=1,2,\cdots,N, \end{equation}$

where $Z_t$ is a White noise process with mean 0 and variance 1, and $a_0>0, a_i \geq 0, i=1,2, \cdots, p$ .