I'm new to ARCH models, and I have a question about the correct notation for expressing the conditional expectation of the return at time $t(r_t)$ given the information available up to time t-1. I'd appreciate some guidance on which of the two versions below is the correct way to write this, or if both versions are acceptable.
Given the following formulas and equations from an ARCH(1) model:
\begin{equation} r_t = \sigma_t e_t \end{equation}
\begin{equation} e_t \sim \text{white noise}(0, 1) \end{equation}
\begin{equation} \sigma_t = \sqrt{\omega + \alpha_1 r_{t-1}^2} \end{equation}
\begin{equation} E(r_t|r_{t-1}, r_{t-2}, \ldots) = E(\sigma_t e_t|r_{t-1}, r_{t-2}, \ldots) \end{equation}
\begin{equation} = \sigma_t E(e_t|r_{t-1}, r_{t-2}, \ldots) \end{equation}
\begin{equation} = \sigma_t * 0 = 0 \end{equation}
Version 1: \begin{align*} E(r_t) &= E_{r_{t-1}} E_{r_{t-2}} \cdots \left[ E(r_t | r_{t-1}, r_{t-2}, \ldots) \right] \\ &= E_{r_{t-1}} E_{r_{t-2}} \cdots E[0] \\ &= 0 \end{align*}
Version 2: \begin{align*} E(r_t) &= E[E(r_t|r_{t-1}, r_{t-2}, \ldots)] \\ &= E[0] \\ &= 0 \end{align*}
Since this question is more related to probability theory than time series analysis, I thought it would be appropriate to post it here. Can someone please point out which of the two versions is the correct one, or if both versions are correct?
Thank you guys in advance and this awesome platform!
