Why are ARCH/GARCH discussed like they predict the time series value itself instead of residuals? 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.
 A: 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?".
