Volatility Models: ARCH(1) & GARCH(1,1)

The ARCH(1) model, introduced by Engle in 1982, posits that the variance of the error at time t depends on the squares of past errors. In other words, current volatility is conditioned by previous shocks. The relationship is defined as follows: $$ \begin{aligned} & r_t = \mu + \varepsilon_t,\\ & \varepsilon_t = \sigma_t z_t,\quad z_t \sim \mathcal{N}(0,1),\\ & \sigma_t^{2} = \alpha_0 + \alpha_1\,\varepsilon_{t-1}^{2} \end{aligned} $$ The assumptions for the ARCH(1) model include positivity, where \(\alpha_0 > 0\) and \(\alpha_1 \ge 0\), and stationarity, requiring that \(\alpha_1 < 1\). The GARCH(1,1) model, proposed by Bollerslev in 1986, generalizes the ARCH model by introducing a dependence of the conditional variance on past shocks and past volatility. The relationship is expressed as: $$ \begin{aligned} & r_t = \mu + \varepsilon_t,\quad \varepsilon_t|\Omega_{t-1} \sim \mathcal{N}(0,\sigma_t^2),\\ & \sigma_t^{2} = \omega + \alpha_1\,\varepsilon_{t-1}^{2} + \beta_1\,\sigma_{t-1}^{2} \end{aligned} $$ For the GARCH(1,1) model, the assumptions include positivity, where \(\omega > 0\), \(\alpha_1 \ge 0\), and \(\beta_1 \ge 0\), along with the stationarity condition that \(\alpha_1 + \beta_1 < 1\).