# 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.

• This has been discussed multiple times at Cross Validated. Some relevant threads are here, especially this one. – Richard Hardy Apr 1 '20 at 12:16

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.)
• I'm not familiar with fitting arima-garch but wouldn't the two equations have to be estimated simultaneously since $\epsilon_{t-1}$ and $\sigma^2_t$ are related ? Thanks. – mark leeds Mar 28 '20 at 12:07
• Thanks. But just to make sure I understand: you estimate the $r_{t}$ equation first to get the c, a, b and estimates $\hat\sigma^2$ and $\hat\epsilon_t$. Then you use those estimates in the second equation in order to estimate only $\alpha$ and $\beta$ ? Is that right. – mark leeds Mar 28 '20 at 17:21