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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.

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

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  • $\begingroup$ 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. $\endgroup$ – mark leeds Mar 28 at 12:07
  • $\begingroup$ They can be fitted simultaneously. But the number of possible models quickly increases. To simplify, you can do it in two steps, as well. The latter results as trade-off between accuracy and computation time $\endgroup$ – Alex Mar 28 at 12:31
  • $\begingroup$ The difference lies in how the maximum likelihood estimator will penalize different observations. In particular, a GARCH model can allow for temporarily large deviations from the mean to not be a big deal by letting the associated conditional variance be large. $\endgroup$ – Stéphane Mar 28 at 15:35
  • $\begingroup$ 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. $\endgroup$ – mark leeds Mar 28 at 17:21
  • $\begingroup$ Oh, the reason I ask is because doing it that way seems inefficient in that information is not being used that's there. But I would imagine that simultaneous estimation is way more complicated. Also, I meant to say that I don't understand what you mean by "the number of possible models quickly increases". It's still just one model ? Thanks. $\endgroup$ – mark leeds Mar 28 at 17:23

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