# 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.

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.

• So if we have $E(\sigma_n^2 \eta_n^2)$ this is not equivalent to $E(\sigma_n^2)$
– Anna
Commented Feb 21, 2018 at 21:14
• @Anna, What is $\eta_n^2$ Commented Feb 22, 2018 at 5:41
• Sorry I meant $Z_n^2$, where $E(Z_n^2)$=1
– Anna
Commented Feb 22, 2018 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. Commented Feb 22, 2018 at 10:28
• I edited the question, I think the same answer applies right?
– Anna
Commented Feb 22, 2018 at 10:48