Fitting ARMA and GARCH Models for Return Forecasting

Summary

The document outlines a workflow for modeling returns with a combined ARMA-GARCH specification. ARMA terms describe the conditional mean using past returns and shocks, while GARCH terms model time-varying conditional variance from earlier variance and squared shocks. This combination lets the volatility process affect the uncertainty assigned to returns, even when the main forecast target is the conditional mean.

The suggested workflow is to check stationarity and heteroscedasticity, choose ARMA orders using information criteria, and then fit a GARCH model, with a likelihood-ratio test also mentioned. The response proposes alternative error distributions, including Student-t and GED, and asymmetric volatility variants such as T-GARCH or EGARCH when warranted. It recommends residual diagnostics, including checks for remaining heteroscedasticity and distributional fit. The advice is a general modeling outline; it provides no data example or evidence that sequential estimation is always equivalent to joint estimation, so model choice and diagnostics remain important.

Key ideas

• ARMA terms model conditional return means, while GARCH terms model conditional variance.
• The proposed workflow checks stationarity and heteroscedasticity before fitting the models.
• Information criteria can guide ARMA order selection before estimating GARCH dynamics.
• Alternative error distributions and asymmetric GARCH variants can address non-normality or leverage effects.
• Residual diagnostics are needed to assess remaining variance patterns and distributional fit.

Tags

# Can ARMA and GARCH models be estimated separately in ARMA/GARCH？


# Can ARMA and GARCH models be estimated separately in ARMA/GARCH？












Can I use the residuals of the ARMA model to build a GARCH model(with Zero mean)? If so, does this mean that this GARCH model(with Zero mean) has no effect on ARMA's estimates. For example, if I want to use ARMA models to predict returns, then GARCH (with Zero mean) will not help, unless both models estimate at the same time. Thanks.

## Answer by Alex (score 1)

https://quant.stackexchange.com/a/51867

You can combine AR(I)MA and GARCH models. For instance, a (Gaussian) ARMA(1,1)-GARCH(1,1) model would read as \begin{align*} r_t &= c + ar_{t-1} + b\epsilon_{t-1} + \epsilon_t, \\ \sigma^2_t &= \omega + \alpha \sigma_{t-1}^2 + \beta \epsilon_{t-1}^2 \end{align*} where $\epsilon_t\mid\mathcal{F}_{t-1}\sim N(0,\sigma_t^2)$.

You should first test for stationary returns and heteroscedastic variance.Using information criteria like AIC, BIC and HQIC you can first find the optimal ARMA parameters and then continue with fitting the GARCH model. A likelihood ratio test may be useful as well. You can improve the fit by using a Student's $t$-distribution or a GED distribution or by using T-GARCH or EGARCH models to allow for asymmetry. Don't forget diagnostics like checkinng the final residuals (homoscedastic, QQ-plot etc.)

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.