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Let $a_t=\sigma_t\epsilon_t$ and let $a_t$ be modeled e.g. by an ARCH(1) model, that is $a_t^2=\alpha_0+\alpha_1 a_{t-1}^2$.

Are $\sigma_t$ and $\epsilon_t$ independent?

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  • $\sigma_t$ can be expressed as a deterministic function of an infinite amount of lags of $\varepsilon_t$ (by recursively substituting for lags of $a_{t-1}^2$ in the equation $a_t=\alpha_0+\alpha_1 a_{t-1}^2$).
  • $\varepsilon_t$ are i.i.d., thus independent.
  • A function of of some variables that are independent of another variable will be independent of that variable.

Hence, $\sigma_t$ and $\varepsilon_t$ are independent.

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