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Given a time series of $u_i$ returns where i=1 to t. $\sigma_i$ is calculated from GARCH(1,1) as $\sigma_i^2=w+\alpha u_{i-1}^2 +\beta \sigma_{i-1}^2$ . What is the mathematical basis to say that $u_i^2/\sigma_i^2$ will exhibit little auto-correlation in the series? Hull's book Options, Futures and Other Derivative is an excellent reference. In Hull 6th Ed p470, "How Good is the Model?" he states that "If a GARCH model is working well, it should remove the auto-correlation. We can test whether it has done so by considering the auto-correlation structure for the variables $u_i^2/\sigma_i^2$. If these show very little auto-correlation out model for $\sigma_i$ has succeeded in explaining auto-correlation in the $u_i^2$". Max Log-Likelihood Estimation for variance ends with maximize $$ -m \space ln(v) -\sum_{i=1}^{t} u_i^2/v_i $$ where $v_i$ is variance= $\sigma_i^2$. This function does not really mean $u_i^2/v_i$ being minimized, because $-ln(v_i)$ gets larger and so does $u_i^2/v_i$ as $v_i$ gets smaller. However, it makes intuitive sense that dividing $u_t$ return by its volatility(instant or regime) volatility, explains away volatility related component of the time series. I am looking for a mathematical or logical explanation of this.

I think Hull is not very accurate here as the time series may have trends etc, also, there are better approaches to finding i.i.d from the times series than using $u_i^2/\sigma_i^2$ alone. I particularly like Filtering Historical Simulation- Backtest Analysis by Barone-Adesi(2000) FHS

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In a nutshell, you model the variance process of a time series $u_{i}$ with GARCH(1,1). Return time series have small absolute value, so $u_{i}^{2}$ is a good proxy for $\left(u_{i}-\overline{u}\right)^{2}$ as a variance estimator. Therefore $\frac{u_{i}^{2}}{\sigma_{i}^{2}}$ is a good proxy for a white noise time series if the GARCH(1,1) model was right and you explained all auto-correlation in the initial time series by it. –  Marco Breitig May 27 at 16:28

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