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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.

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    $\begingroup$ The line just above your second to last quote from the Wiki article says $\epsilon_t=\sigma_t z_t$ $\endgroup$ – Raskolnikov May 6 '18 at 11:20
  • $\begingroup$ Yes. I'm not sure though what you want to say by that. Do you mean that this implies that the second moments are the same? $$E[\epsilon_t^2] = E[\sigma_t^2 z_t^2] = E[\sigma_t^2].$$ Is the ARCH model not a generative model? $\endgroup$ – Effesian May 6 '18 at 16:24
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In the following, what you call sigma is called h. So because you can write e(t) as a function of h(t) and a random variable z(t), when you write h(t) as a function of the past of e(t) you are equivalently writing out an autoregressive process for the error.

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  • $\begingroup$ Wouldn't you need to regress on $h_{t-1}, h_{t-2}, ...$ to have the variance be an AR process? $\endgroup$ – Effesian May 7 '18 at 21:04

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