Combining Mean and GARCH Models for Return Inference
Summary
The document explains how to use GARCH when modeling returns with conditional heteroscedasticity. An ARCH or GARCH specification describes the changing variance of the residuals; it must be paired with a model for the conditional mean. That mean may be a constant average or a time-series model such as ARIMA, with the volatility process fitted to its residuals.
The preferred approach in principle is joint maximum-likelihood estimation of the mean and volatility parameters. The resulting standard errors can be derived from the Fisher information matrix, enabling inference that accounts for the modeled conditional variance. When the joint model has many parameters and they cannot be estimated precisely, a two-stage procedure—first fitting the mean, then fitting GARCH to the residuals—may be preferable. The document gives general modeling guidance, not a worked dataset or diagnostics. Appropriate model choice and parameter restrictions still matter, and the fit should reflect the time series under study.
Key ideas
- A GARCH model for variance is paired with a model for the return series’ conditional mean.
- The mean can be a constant or a time-series model such as ARIMA.
- GARCH models the residual volatility after accounting for the chosen mean specification.
- Joint maximum-likelihood estimation can provide standard errors based on the full model.
- A two-stage estimation may be useful when joint estimation involves too many imprecise parameters.
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# The use of GARCH
# The use of GARCH
I have a conceptual question that I haven't managed to grasp yet and is most likely a econometrics 101 question by here it goes:
If we estimate a GARCH model for a time series, how do we then use this in my model for the returns? For example; I have the return data of an index. I know that I have volatility clustering in this data. I find a suitable GARCH model for the volatility (variance). Now, if I model the returns an a suitable model, i.e. a regression model, and look at the coefficients and the p-values that it spits out, these values are still based on the regular OLS assumptions right? How do I make use of the GARCH in this model so that I can get coefficients and p-values that have accounted for the conditional heteroscedastic variances in the time series?
## Answer by Kumar (score 1, accepted)
https://quant.stackexchange.com/a/17086
You first fit a ARIMA model to the returns data and then a GARCH model to the residuals.
## Answer by pbr142 (score 5)
https://quant.stackexchange.com/a/18078
Any ARCH type model always requires an additional model for the mean of the time series. If nothing is said about the mean model, then usually is simply a time average plus residual. So, if $y_t$ is your stationary time series, the mean model would be $$ y_t = \bar{y} + \epsilon_t $$ where $\bar{y}$ is the average value of $y_t$. And then $\epsilon_t$ would be assumed to follow another time series model, such as GARCH(p,q): $$\epsilon_t = \sigma_t z_t$$ $$ \sigma_t^2 = \alpha_0 + \sum_{i=1}^{q} \alpha_i \epsilon_{t-i} + \sum_{j=1}^{p} \beta_i \sigma_{t-i}^2$$ You can, however, use any mean model for the original time series $y_t$ (such as ARIMA) and then specify the model for the residuals via GARCH; subject to some parameter restrictions.
A common question in this regard is how to estimate the model. In principle, a joint estimation via Maximum Likelihood of both the mean and volatility model simultaneously is preferable. Depending on the model, this may lead to a large number of free parameters which cannot be estimated precisely. In such cases, it is sometimes preferable to first estimate the mean model and then estimate the volatility model from the residuals.
If you use joint estimation via maximum likelihood, you can get correct standard errors in the normal way via the Fisher-Information matrix. Any statistical software should produce these by default.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.