Econofisica: Especificación del modelo GARCH

Suppose one wishes to model a time series using an ARCH process. Let

~\epsilon_t~

denote the error terms (return residuals, with respect to a mean process), i.e. the series terms. These

~\epsilon_t~

are split into a stochastic piece

z_t

and a time-dependent standard deviation

\sigma_t

characterizing the typical size of the terms so that

~\epsilon_t=\sigma_t z_t ~

The random variable

z_t

is a strong white noise process. The series

\sigma_t^2

is modelled by

\sigma_t^2=\alpha_0+\alpha_1 \epsilon_{t-1}^2+\cdots+\alpha_q \epsilon_{t-q}^2 = \alpha_0 + \sum_{i=1}^q \alpha_{i} \epsilon_{t-i}^2

where

~\alpha_0>0~

and

\alpha_i\ge 0,~i>0

Econofisica