# Can GARCH volatility simulations generally be applied to return-modelling models?

This may be a naive question, but I still hope some discussion can elucidate a (so far) totally nebulous point for me.

I've recently learned that GARCH models can give one simulations of volatilities over a chosen future horizon, and understandably this has substantial applications for estimating covariance matrices and option pricing. However, can GARCH simulated volatilities be ad hoc inserted into non-GARCH econometric models, to improve them by capturing volatility clustering?

For example, if I estimate both an ARIMA(p,q,d) and GARCH(p,d) model on the same data, could they complement each other (possibly within a new model)?

If the following is generally possible, could someone with more experience describe how and why? Thank you enormously in advance.

In general, if you have a model of relation between $y$ and $x$ whereby the relation is not perfect but measured with errors:

$$y_t = f(x_t) + \varepsilon_t,$$

where errors $\varepsilon$ are assumed to be additive but need not be, you are free to choose the distribution of these errors to better fit the reality. That is where GARCH enters as a great alternative to the i.i.d. case! Note that the dependence structure above subsumes a broad range of models, e.g. linear regressions with heteroskedastic errors:

$$y_t = \alpha + \beta x_t + \varepsilon_t,$$ $$\varepsilon_t \sim N(0,\sigma_t),$$ $$\sigma_t^2 = \omega + \theta_1 \varepsilon_{t-1}^2 + \theta_2 \sigma_{t-1}^2,$$

whereby the equations are estimated simultaneously, leading to improved inference.

I have not seen ARIMA models with GARCH errors, but cannot readily think of a reason for them not to be.

• I was suspecting utilizing a GARCH model's predictions could be done via a return model's error terms, but wouldn't this ruin the model's statistical tests? I understand linear regression can handle heteroskedasticity to some extent when using robust errors, but it seems to be asking a lot more to use error terms from GARCH. I understand this may be a more mathematical subject, so would it be possible to link a reference I could use on this subject? – Coolio2654 Jul 18 '18 at 16:41
• Google "ols with garch errors", there's a lot on the subject out there. GARCH errors in the OLS setting may or may not give you advantage over the robust covariance approach, dependent on the assumptions you make, whereby the standard concerns about which model is "more correct" apply. Still, the former will surely give you a nice model to simulate residuals from. – Igor Pozdeev Jul 19 '18 at 6:04
• Regarding I have not seen ARIMA models with GARCH errors, but cannot readily think of a reason for them not to be., ARIMA models with GARCH errors are so standard that statistical or econometric software often includes ARIMA conditional mean as an option in GARCH models (fGarch and rugarch in R are just a couple of many examples). – Richard Hardy Sep 2 '18 at 7:49