Standard GARCH(1,1) model with external regressors

Question

I have a queastion how does a standard GARCH(1,1) model with external regressors in mean and variance euqations look like ?

I know that standard GARCH(1,1) model without external regressors has the following form: \begin{equation*} r_t = \mu + \epsilon_t \end{equation*} \begin{equation*} \epsilon_t = \sigma_t z_t \end{equation*} \begin{equation*} \sigma_t^2 = \omega + \alpha_1 \sigma_{t-1}^2 + \beta_1 \epsilon_{t-1}^2 \end{equation*} \begin{equation*} z_t \sim N(0, 1). \end{equation*} But where should I include mentioned external regressors in mean and variance ? And how does the model look then ?

Thank you in advance for your help.

Answer 1

An external regressor in the mean specification can be added to the mean specification, i.e. $$r_t = \mu + \varepsilon_t + \theta x_t $$.

An external regressor in the variance specification can be added to the variance specification, i.e. $$\sigma^2_t = \omega + \alpha \sigma_{t-1}^2 + \beta \varepsilon_{t-1}^2 + \theta x_{t}$$