Combining GARCH Volatility with Regression and ARIMA Models
Summary
The document explains how GARCH can be used to model changing error variance within a broader model for the relationship between an outcome and its predictors. Instead of assuming independent, identically distributed errors, a regression can allow its residual variance to depend on past squared residuals and past variance. The response notes that estimating the relationship and variance equations together can improve inference when errors are heteroskedastic.
It also considers pairing GARCH with an ARIMA model, suggesting that ARIMA with GARCH errors is plausible, though the respondent had not encountered an example and gives no derivation or empirical comparison. The discussion is conceptual and brief: it does not specify model selection, estimation details, forecast evaluation, or conditions under which the added volatility structure helps. GARCH simulated volatility is therefore not presented as a plug-in series; it enters through a specified error distribution within the model.
Key ideas
- GARCH can describe time-varying variance in the errors of a regression or other relationship model.
- Jointly estimating the mean relationship and conditional variance can improve inference when errors are heteroskedastic.
- The discussion treats ARIMA models with GARCH errors as a plausible combination.
- No empirical results or implementation guidance are provided, so the benefit must be checked for the data and model at hand.
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Full text
# Can GARCH volatility simulations generally be applied to return-modelling models?
# 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.
## Answer by Igor Pozdeev (score 2)
https://quant.stackexchange.com/a/40818
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.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.