I have the following AR(1): $y_{t}=\phi y_{t-1}+\epsilon_{t}$, where $|\phi|<1$. $\epsilon_{t}$ is assumed to be a margintale difference process, such that $\mathbb{E}[\epsilon_{t}|I_{t-1}]=0$. ($I_{t}$ is an information set containing all realisations of $y$, and $\epsilon$, in the periods up to, and including, $t$). I am trying to find $\mathbb{E}[y_{t-1}\epsilon_{t}]$. Is the following correct?
$\mathbb{E}[y_{t-1}\epsilon_{t}]=\mathbb{E}[\mathbb{E}[y_{t-1}\epsilon_{t}|y_{t-1}]]=\mathbb{E}[y_{t-1}\mathbb{E}[\epsilon_{t}|y_{t-1}]]=0$ (since, by the assumption that $\epsilon_{t}$ is an MDS, the inner conditional expectation is $0$). The first equality follows by LIE.
Thank you.