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I've always wondered why do one use squared or absolute returns to determine if volatility modeling is required for the return series? We understand that there are various tests for its autocorrelation and conditional heteroskedasticity. However, I don't quite grasp the concept behind it. Can anyone kindly explain what's the statistical intuition behind using squared/abs returns to determine if vol representation is needed? Thank you.

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To simplify, consider the errors rather than the returns. The variance is effectively the average of the squared errors, while absolute deviation is the average of the absolute errors. So plotting the squared errors or absolute errors over time could give an indication of whether the variance or absolute deviation is constant over time. Since variance is more commonly the practical focus, one approach would be to simply regress the squared errors on p of its lags. This is the ARCH(p) model. GARCH(p,q) introduces an additional term, which has the effect of reducing the need of p to be large.

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Simple...because you are interested in deviations from a metric, and not whether it deviates above or below. The very definition of volatility is a "measure of deviation". Squaring returns or using the absolute values just eases the calculation to arrive at a deviation measure. Otherwise volatility would have to be calculated in other ways as positive and negative returns would introduce side effects that will affect the volatility computation.

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Also, often we can assume the average of short-term returns in the long run to be zero, the historic volatility is equal to $\hat{\sigma_T^2}=\frac{\sum_{i=1}^T{r_i^2}}{T-1}$. Sp to study the volatility process we therefore study the squared return process, which is a good proxy.

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