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I need some help with fitting an ARMA-GARCH model. I'm analyzing a daily time series. I don't understand how the order of the GARCH impacts ACF or PACF plots. I mean: what is the difference I should notice in the ACF plot when I'm using GARCH(1,1)+ARMA(1,1) instead of GARCH(2,1)+ARMA(1,1)? What 'part' of the ACF plot should change when I modify the GARCH or/and ARMA orders?

Richard Hardy
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delly.s
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2 Answers2

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An ARMA(p,q)-GARCH(r,s) model looks like this: \begin{aligned} x_t &= \mu_t + u_t, \\ \mu_t &= c + \varphi_1 x_{t-1} + \dots + \varphi_p x_{t-p} + \theta_1 u_{t-1} + \dots + \theta_q u_{t-q} \quad \text{("ARMA")}, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dots + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dots + \beta_r \sigma_{t-r}^2 \quad \text{("GARCH")}, \\ \varepsilon_t &\sim i.i.D(0,1), \end{aligned} where $D$ is some probability distribution with zero mean and unit variance.

The ACF/PACF of the original series $x_t$ would mostly be determined by the ARMA part, not the GARCH part. The GARCH part implies nonconstant variance of $u_t$, so you have an ARMA model with the noise part being a bit ill-behaved (not i.i.d.). This might blur the ACF/PACF a little bit, but the general patterns would be the same.

If you are interested in selecting and fitting an ARMA-GARCH model, then you would probably also look at the ACF/PACF plots for the model's residuals with the aim of model diagnostics. You may look at the raw standardized residuals $\hat\varepsilon_t$. If you change something in the equation for $\sigma_t^2$ (the GARCH part of ARMA-GARCH), you may not notice much difference in their ACF/PACF plots. However, you should notice some changes in the ACF/PACF plots of squared standardized residuals $\hat\varepsilon_t^2$.

Richard Hardy
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As explained by @Richard Hardy above. The time series, $x_t$ is first decomposed into the conditional mean process $\mu{_t}$ and residual/error part $u_t$. The conditional mean process can be modelled by zero, constant or ARMA. The residual $u_t$ by GARCH. Actually, you can start by assuming a constant or a zero conditional mean process (instead of ARMA) if you are using financial returns time series. Then subtract the fitted conditional mean process, $\mu_t$ from the original series, $x_t$. Then square this residual term and plot it to get a sense of the variance (squared volatility) in the series. Let's call this series $u_t{^2}$. The GARCH model is trying to model this new series. The ACF/PACF of $u_t{^2}$ will help to choose the order of $\alpha$ and $\beta$

Note the stochastic nature of the conditional variance term, $\sigma_t$ due to the random variable, $\epsilon{_t}$ term.

Jose_Peeterson
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    you can start by assuming ARMA to be a constant or a zero mean process This sounds contradictory, as ARMA is nonconstant. Did you perhaps mean $\mu_t$ instead of ARMA? Regarding your question, simulataneous estimation is more efficient and gives correct standard errors and such. Stepwise estimation ignores the uncertainty about the first-stage conditional mean when fitting the second-stage zero-mean GARCH model. See e.g. this for some more detail. – Richard Hardy Dec 03 '23 at 07:24
  • @RichardHardy I have corrected above. – Jose_Peeterson Dec 03 '23 at 08:55
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    Now that I have answered your question, you might wish to remove it from your answer (and perhaps replace it with what I responded). – Richard Hardy Dec 03 '23 at 15:08