My background is signal processing and I am fairly new to (financial) time series analysis. I was reading the article about autoregressive conditional heteroskedasticity (ARCH) models on Wikipedia.
https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity
I am confused about what AR (I know it means autoregressive :-) ) refers to in an ARCH model. There are two possibilities in my view:
1.) The process of interest (and not its volatility), e.g., log-returns, is assumed to follow an AR-model: $$ y_t = a_0 + \sum_{i=1}^p y_{t-i} a_i $$
2.) We assume that the volatility of the process we want to model (e.g., the above mentioned log-returns) follows an AR process.
The introduction section in Wikipedia seems to support my first hypothesis: "The ARCH model is appropriate when the error variance in a time series follows an autoregressive (AR) model..."
What confuses me though is this formula:
$$\sigma_t^2 = \alpha_0 + \sum_{i=1}^q \alpha_i \epsilon_{t-i}^2$$
I don't see the "autoregression" here... Wouldn't and AR be something like:
$$\sigma_t^2 = \alpha_0 + \sum_{i=1}^q \alpha_i \sigma_{t-i}^2$$.
This would then be a special case what is referred to as GARCH model.