Adding External Regressors to GARCH Mean and Variance Equations

Summary

The document explains where an external variable can enter a standard GARCH(1,1) model. A regressor in the mean equation adds a term proportional to that variable alongside the baseline return level and innovation. A regressor in the variance equation adds a term to the conditional variance specification alongside its constant and lagged variance and shock components.

The response gives the general placement of a regressor in each equation, making clear that mean and variance effects are modeled separately. It does not discuss parameter estimation, statistical testing, constraints needed to keep conditional variance positive, or how to select contemporaneous versus lagged regressors. No empirical example or performance evidence is supplied, so the equations serve as a basic specification guide rather than a complete modeling procedure.

Key ideas

• A regressor can enter the conditional mean as an additive term with its own coefficient.
• A regressor can enter the conditional variance as an additive term with its own coefficient.
• Mean and variance regressors affect distinct parts of a GARCH specification.
• The basic equations do not address estimation, variance constraints, or regressor selection.

Tags

# Standard GARCH(1,1) model with external regressors


# Standard GARCH(1,1) model with external regressors












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 by Markus Hauschel (score 2)

https://quant.stackexchange.com/a/47574

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. $$\sigma^2_t = \omega + \alpha \sigma_{t-1}^2 + \beta \varepsilon_{t-1}^2 + \theta x_{t}$$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.