Is an ARCH model regressed over previous time series values or their residuals?

Question

Suppose we have a time series $\{y_t\}$ which we would like to model using an ARCH or GARCH model. That is we assume the time series can be written in the form $$y_t = \mu_t + \epsilon_t$$ where $\mu_t$ is the conditional mean at time $t$ which can be estimated by, for example, an ARMA model and $\epsilon_t \sim \mathcal{D}(0, \sigma_t^2)$ with $\mathcal{D}$ an arbitrary distribution.

The point of both the ARCH and GARCH models is to model the conditional variance $\sigma_t^2$. If we use an ARCH model then the idea is to model $\sigma_t^2$ by using an AR model. This would mean that, for an ARCH(1) model: $$\sigma_t = \alpha_0 + \alpha_1 y_{t-1}. \tag{1}$$ However other sources, such as on Wikipedia, state that we do not regress against the lagged $y_t$ terms but instead the lagged residuals $\epsilon_t$: $$\sigma_t = \alpha_0 + \alpha_1 \epsilon_{t-1} \tag{2}.$$ For either model the parameters can be found using OLS.

I have two questions about the above:

1. Which is the correct ARCH model, (1) or (2)? In other words, should we regress on the previous time series values $y_t$ or the residuals $\epsilon_t$?
2. The $\epsilon_t$ are innovations that are unobservable. I've read that in practice if we use model (2) then then we can estimate $\epsilon_{t-1}$ as $\hat{y}_{t-1} - y_{t-1}$. Is this true, and if so, what justifies this?

Answer 1

1. Neither $(1)$ nor $(2)$ is correct (and Wikipedia does not seem to suggest $(2)$). The correct equation for ARCH(1) is $$ \sigma_t^2 = \alpha + \alpha_1 \epsilon_{t-1}^2 \tag{3}. $$ As you can see, we use residuals/innovations, not the raw values here.

2. Ideally, you would estimate the conditional mean model and the conditional variance model simultaneously. This is what the modern software implementations of ARCH and GARCH would do. If you want to do it yourself and do not mind losing some efficiency, you can do it stepwise. Then you could indeed estimate the conditional mean model first, obtain residuals from it and then use them to estimate the conditional variance equation.