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

Question

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

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

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 $\sigma_n$?

Answer 1

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.

Perhaps you are thinking in terms of what comes first and what generates what, but that is irrelevant when we talk about independence in the statistical sense of the word.