How the Conditional Mean Model Affects GARCH Volatility Estimates
Summary
The document explains how the return forecasting model relates to a GARCH volatility model. It distinguishes the conditional mean of returns from the residual process: returns equal their model-implied conditional mean plus an error, and the error is represented as a time-varying volatility scale multiplied by a standardized random shock. The GARCH variance equation uses lagged squared errors and lagged conditional variance to model changing volatility.
If researchers use different conditional mean specifications, such as a lagged return or an external predictor, they obtain different residual series. Those residuals feed into the conditional variance model, so estimated GARCH parameters can change with the chosen mean model. The answer establishes this dependence conceptually, but offers no empirical comparison or rule for choosing among mean specifications. The quality of volatility estimates therefore depends in part on how well the conditional mean is specified, as well as on the variance model and distributional assumptions.
Key ideas
- A GARCH model describes conditional variance using past residual information and past conditional variance.
- The residual is defined relative to the chosen conditional mean model.
- Alternative mean specifications produce different residuals and can therefore change the estimated variance parameters.
- The document explains the dependency but gives no empirical procedure for selecting the best mean model.
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# Are GARCH models dependent on the returns forecasting model?
# Are GARCH models dependent on the returns forecasting model?
Hi Quantitative Fiance Stack Exchange,
It's my first go at GARCH models so please give me a chance with my phrasing.
I understand that GARCH models are used to forecast volatility. The GARCH(1,1) takes the form:
$$\sigma^2_t=\alpha+\beta_1\epsilon_{t-1}+\beta_2\sigma^2_{t-1}$$
I understand the lagged term $\sigma^2_{t-1}$ makes up the AR part of GARCH. However, I also understand the error term $\epsilon_{t-1}$ is dependent on the forecasting model. Consider, forecasting returns using one of the two models:
$$\hat{y_t}=\gamma\cdot y_{t-1}+\epsilon_t$$
and
$$\hat{y_t}=\theta\cdot x_{t-1}+\epsilon_t$$
Each model gives a different error term, which I believe is calculated as $\epsilon_t=y_t-\hat{y_t}$. So for the above models, error terms are $\epsilon_t=y_t-\gamma\cdot y_{t-1}$ and $\epsilon_t=y_t-\theta\cdot x_{t-1}$
Hence, is my understanding correct that calculating $\beta_1$ and $\beta_2$ of the GARCH(1,1) model depends on which forecasting model we're using?
Thank you for the help, Donny
## Answer by Richard Hardy (score 3)
https://quant.stackexchange.com/a/31012
> I also understand the error term $\varepsilon_{t-1}$ is dependent on the forecasting model.
Yes, it is. The error term $\varepsilon_t$ in the GARCH model is coming from the full distributional model of $y_t$. The full model is $$ \begin{aligned} y_t &= \mu_t + \varepsilon_t, \\ \varepsilon_t &= \sigma_t \xi_t, \\ \sigma_t^2 &= \omega + \alpha_1 \varepsilon_{t-1}^2 + \beta_1 \sigma_{t-1}^2, \\ \xi_t &\sim i.i.d(0,1), \end{aligned} $$ where $\mu_t$ is the conditional mean of $y_t$, $\sigma_t^2$ is the conditional variance of $y_t$ and $d$ is some probability distribution with zero mean and unit variance.
If you are not sure which conditional mean model is best for $y_t$, you may end up with a few alternative models characterized by the conditional means $\mu_{1,t}, \mu_{2,t}, \dots$. The the corresponding error terms will differ across the models and will be $\varepsilon_{1,t} = y_t-\mu_{1,t}, \varepsilon_{2,t} = y_t-\mu_{2,t}, \dots$. This will affect the parameter estimates of the conditional variance model, just as you said.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.