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I am fairly new to time-series analysis, I am trying to learn ARCH/GARCH models.

My understanding is that ARCH/GARCH models try to predict the residuals (difference between an observed value from DGP and a predicted value from a model fit earlier), as illustrated in this video.

However it is sometimes discussed like they predict the time series value itself instead of the residuals.
For example, in this video and also this slide (page 7) (Shouldn't the $r_{t}$ be a residual instead of the return value itself?).

Why are they discussed differently? I am confused.

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  • $\begingroup$ In a GARCH model (as in many other models), residuals are differences between the actual values and the modelled conditional means, not predictions. It is quite common to use the conditional mean as prediction, and when the two coincide, we get what you say: difference between an observed value from DGP and a predicted value from a model fit earlier. But this is a coincidence, not a fundamental fact. $\endgroup$
    – Richard Hardy
    Commented Apr 25, 2021 at 8:01
  • $\begingroup$ Really appreciate your reply. In what kind of situation do you NOT use the conditional mean as prediction (Do you have any exmaples?)? $\endgroup$
    – koyamashinji
    Commented Apr 25, 2021 at 9:31
  • $\begingroup$ Conditional mean is an optimal prediction under square loss. Under other loss function, other distributional characteristics are optimal. E.g. under absolute loss, the optimal prediction is the conditional median. Under asymetric absolute loss (a.k.a. quantile loss, pinball loss, tick loss), the optimal prediction is the conditional quantile of the distribution. $\endgroup$
    – Richard Hardy
    Commented Apr 25, 2021 at 10:41

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A GARCH model generally is

\begin{aligned} r_t &= \mu_t+\varepsilon_t, \\ \mu_t &= \dots, \\ \varepsilon_t &= \sigma_t z_t, \\ \sigma_t^2 &= \omega + \alpha_1 \varepsilon_{t-1}^2 + \dotsc + \alpha_s \varepsilon_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dotsc + \beta_r \sigma_{t-r}^2, \\ z_t &\sim i.i.D(0,1), \\ \end{aligned}

where $\mu_t$ is some function of past information (maybe a constant, maybe some ARMA terms, maybe something else) representing the conditional mean of $r_t$, and $D$ is some distribution with zero mean and unit variance. The dependent variable of and the only necessary input into the model is $r_t$, and the model is that of the conditional distribution of $r_t$ (conditioned on past values of $r_t$ and possibly other variables). What makes the model a GARCH model is the particular form of the specification of conditional variance, $\sigma_t^2$.

Regarding your question, if $\mu_t=0$ then $r_t=\varepsilon_t$, and GARCH seems to model the conditional variance of $r_t$ "directly". If $\mu_t\neq 0$, then $r_t\neq\varepsilon_t$, and GARCH seems to model the conditional variance of $\varepsilon_t$ "directly" but of $r_t$ only "indirectly". However, this is a false dichotomy. Since we are conditioning on all past information, the conditional variance of $r_t$ equals the conditional variance of $\varepsilon_t$ by definition, so it is hard to make a mathematical distinction between them.

For more details, see "What is the difference between GARCH and ARMA?".

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