Financial Mathematics, Volatility and Covariance Modelling

13The log-GARCH model via ARMA representations

Genaro Sucarrat

1. Introduction

The starting point of Engle’s (1982) Autoregressive Conditional Heteroskedasticity (ARCH) class of models is

$$y_t={\mu }_t+{ϵ}_t,{ϵ}_t={\sigma }_t{\eta }_t,{\sigma }_t>0,{\eta }_t\sim iid(0,1),$$

where y_t denotes the variable of interest (e.g., financial return), μ_t is the mean specification (e.g., an AR-X model) or simply zero, ϵ_t is the error term or mean-corrected variable of interest, σ_t is the conditional standard deviation or volatility, and η_t is an innovation with mean zero and unit variance. Arguably, the most common specification of σ_t is the first-order Generalized ARCH (GARCH) model of Bollerslev (1986):

$$\begin{array}{c}\hfill {\sigma }_t^2=\omega +{\alpha }_1{ϵ}_{t-1}^2+{\beta }_1{\sigma }_{t-1}^2,\omega >0,{\alpha }_1,{\beta }_1\ge 0.\end{array}$$

(1)

Usually, this model is referred ...