Is a (G)ARCH model a regression model? I am wondering how to distinguish between the terms 'regression model' and '(G)ARCH model':


*

*Is a GARCH-model a (special) kind of a regression model? 

*What are the major groups of econometric methodologies (time series analysis, linear models,...) and to which of them belong the two methods?   

 A: *

*Regression model is very general. In econometrics, regressions is used to study time series, and the model goes under the name of ARMA. When you think that heteroscedasticity is present in the terms of the time series regression you use a GARCH(p,q) model. 
The answer here by Fg Nu explains that properly What is the difference between GARCH and ARMA?

*I'm not an econometrician, so I cannot help you properly here, did you try to read this wikipedia article? https://en.wikipedia.org/wiki/Methodology_of_econometrics
A: Interesting question. I think, for all practical purposes, the answer is "No." I think that if you added exogenous variables to your GARCH model, then I think the answer would be "yes." But it's about the meaning of a word, so there is room for debate and I doubt we'll see many proofs here.
The origin of the term "regression" is about a phenomenon rather than a technique. The phenomenon itself relates draws from a distribution to a key feature of the distribution: the mean. In the wild, "regression" is associated with linking such features of a probability distribution to external variables, the most common of these being the mean (leading to the conditional expectation function), but quantiles can also be used and the result is still called "regression." The variance is also a feature of a probability distribution, and modeling the variance as a function of external variables would also be "regression," in my opinion.
The reason why I don't think GARCH models are regression models is that they are describing the variance without linking it to other variables. A counterargument might include the point that OLS regression procedures can be used in their estimation. That's an awkward position to be in, no doubt, and if I commit to this line of reasoning, then AR models are not regression models either. Sure, they can be fit using basic OLS regression against lagged responses, but the purpose OLS in this case is to get a dynamic reduced form of a model that is actually based on an infinite number of random shocks. The model describes the variability of the series, but not in a way that is linked to other exogenous factors.
BTW, I'm going for partial credit here; I can't help with the major groups of econometric methodologies.
A: It's a regression model.
AR in GARCH means auto regressive.
It's special in that the AR (auto regressive) means it regress on itself or in time series case it regress on it's past values.
The other answer is flat out wrong. Just because it doesn't use other predictors/covariates doesn't mean it's not regression. Nor the fact that it regress on itself means it loses it regression status.
GARCH, at least where I've learned it, falls under time series analysis. More specifically it is accounting for risk/volatility or in general statistic domain variance/std deviation. Within the framework of econometric, GARCH is modeling variance/std dev and is often use in tandem with AR/ARIMA/ARMA which model the mean/expectation. The reason why we care about modeling variance/std dev is because it is something econometric cares about, risk and volatility.
