Independence between raw and standardized innovations and variance in an ARCH model

Let $\epsilon_n$ denote a real-valued discrete-time stochastic process of residuals, the ARCH($p$) specification is given by

$$\label{1.1} \epsilon_n=Z_n\sqrt{\sigma_n}$$ $$\label{1.2} \sigma_n=\alpha_0+\sum \limits_{i=1}^p\alpha_i\epsilon_{n-i}^2\,,$$

where $\alpha_0, \alpha_1, ...,\alpha_p$ are scalar parameters to be estimated, $\mu_n$ is the fitted model. $Z_n$, are a sequence of independent, identically distributed random variables with mean zero and variance one.

Now, is there any result which shows that $Z_n$ is independent of $\epsilon_n$ or $\sigma_n$?

How can that be? $Z_n$ is not independent of $\epsilon_n$ since $Z_n=\frac{\epsilon_n}{\sqrt{\sigma_n}}$ and $Z_n$ is not independent of $\sigma_n$ since $Z_n=\frac{\epsilon_n}{\sqrt{\sigma_n}}$. $Z_n$ is a function of both $\epsilon_n$ and $\sigma_n$, so it cannot be independent of them. Change $\epsilon_n$ and keep $\sigma_n$ fixed, and $Z_n$ will change. Change $\sigma_n$ and keep $\epsilon_n$ fixed, and $Z_n$ will change.
• So if we have $E(\sigma_n^2 \eta_n^2)$ this is not equivalent to $E(\sigma_n^2)$ – Anna Feb 21 '18 at 21:14
• @Anna, What is $\eta_n^2$ – Richard Hardy Feb 22 '18 at 5:41
• Sorry I meant $Z_n^2$, where $E(Z_n^2)$=1 – Anna Feb 22 '18 at 9:16
• @Anna, By $\sigma^2$ do you mean the fourth moment? Because you used $\sigma$ for the second moment in your question. Also, this is actually a new question, so consider posting it as such. The answer to it is not implied by the original question or the original answer. I.e. independence or lack thereof is not informative about the equality vs. inequality there. – Richard Hardy Feb 22 '18 at 10:28