# 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?

You first fit a ARIMA model to the returns data and then a GARCH model to the residuals.

• Your answer is too short, @user2142 , it is good for a comment but if you want to post it as a question, please ellaborate. – lehalle Mar 25 '15 at 6:24
• @lehalle, does it not directly address the question and answer it with which steps to take? – Matthias Wolf Mar 25 '15 at 7:49
• @MattWolf it would be great to read something like: the ARIMA on the returns will provide you this feature [biblio] but you have remaining heteroskedasticity on the residuals, use a GARCH to capture it [biblio again]. By the way, are you sure that if you generate your data by GARCH+ARIMA the parameters estimated that way will be unbiased ;{)} ? – lehalle Mar 26 '15 at 16:41
• @lehalle, sure that would be nice to have, maybe you could write up an answer that includes all that? I am sure the community will attach a fair value to your answer and also assess a relative fair value to this answer as well. – Matthias Wolf Mar 27 '15 at 4:48
• I try to focus my answers on market microstructure, @MattWolf , my plan is not to write tutorials on time series here... Let's hope we have another user who will contribute on this... – lehalle Mar 27 '15 at 6:11

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.