# How is modeling the time series error/variance, e.g. ARCH or GARCH models, different from modeling time varying forecast intervals?

I'm having a hard time understanding the intuitive difference between modeling the volatility or variance of a time series as it is done in ARCH and GARCH models:

$$Y_t = c+\epsilon_t+\phi_1Y_{t-1}+...+\phi_p Y_{t-p}+ \epsilon_t$$ and then we model $$\epsilon_t$$ as: $$\epsilon_t=\sigma_tz_t$$ with $$z_t$$ white noise, and $$\sigma_t$$ is modeled itself as its own time series process:

$$\sigma_{t+h}=f(\sigma_t,\sigma_{t-1},\sigma_{t-2},...,\epsilon_t,\epsilon_{t-1},\epsilon_{t-2},...)$$

On one hand.

On the other hand, using a sophisticated mean time series model where the forecast intervals are estimated "properly" as opposed to using some simplifying assumptions, so that there width is time dependent, e.g. when using FB Prophet and simulating the intervals using MCMC to get intervals that vary with the seasonality.

Or better still, forecasting full densities, so that the entire shape of the distribution is time dependent, and therefore $$\epsilon_t$$ and $$\sigma_t$$ are time dependent as well.

It seems to me that although the math and the terminology are different, whether we model the variance as its own explicit time series, or whether we just make sure that we model the forecast intervals as best as we can (using MCMC or by forecasting full densities, parametric or non parametric), the end result is the same: A time dependent uncertainty that is conditional on the previous uncertainty estimates.

What am I missing? Why is volatility modeling considered a separate topic from simulating forecast intervals or modeling full forecast densities?

• Does this answer your question? What is the difference between GARCH and ARMA?. Perhaps your last paragraph is a distinct question that can be posted as such, while the preceding paragraphs would be answered by the linked thread? – Richard Hardy Jun 9 '20 at 9:20
• Thanks @RichardHardy. Let me re-write this question instead of posting another one all together. – Akaike's Children Jun 9 '20 at 9:29
• Yes, that sounds smart. – Richard Hardy Jun 9 '20 at 9:56
• What do you think about my answer? Do you need any further clarification? – Richard Hardy Jun 14 '20 at 9:32

GARCH models the entire distribution of a time series with (potentially) time-varying mean, time-varying variance but constant distributional features otherwise (e.g. time-constant higher moments once time variation in the first two moments is accounted for). The distribution of $$y_t$$ modelled by GARCH is moving up and down due to the conditional mean equation (unless the equation specifies a constant conditional mean) and its spread increases and decreases due to the conditional variance equation; see the direct expressions of the density of $$y_t$$ under GARCH and ARMA in my answer to the question "What is the difference between GARCH and ARMA?". You can also include seasonal and other effects into the equation of $$\sigma^2_t$$ in the GARCH model, thereby allowing richer dynamics depending on external regressors.