Why Squared or Absolute Returns Reveal Volatility Clustering
Summary
The document explains why volatility analysis often examines squared or absolute returns rather than signed returns. Volatility concerns the size of deviations, regardless of whether returns are positive or negative; squaring or taking absolute values removes the direction while retaining a measure of magnitude. Squared errors are closely related to variance, while absolute errors track absolute deviation. When short-horizon average returns are near zero, squared returns can serve as a proxy for studying changes in variance over time.
Patterns or autocorrelation in these transformed observations can indicate that return variability is not constant. Regressing squared errors on their own lags gives the ARCH(p) setup; GARCH adds lagged conditional variance terms, which can reduce the need for many squared-error lags. These transformations are diagnostic and modeling tools, not proof by themselves that a volatility model is appropriate. The account also offers no data example or discussion of the assumptions behind the proxy.
Key ideas
- Volatility measures the magnitude of deviations rather than their direction.
- Squaring or taking absolute values prevents positive and negative returns from cancelling each other.
- Squared errors relate to variance, while absolute errors measure absolute deviation.
- Autocorrelation in squared errors motivates ARCH models, and GARCH adds lagged conditional variance terms.
- Squared returns are a proxy for variance when short-term average returns are assumed to be near zero.
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# Squared and Absolute Returns
# Squared and Absolute Returns
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.
## Answer by John (score 7)
https://quant.stackexchange.com/a/7003
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.
## Answer by Matt Wolf (score 5)
https://quant.stackexchange.com/a/7007
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.
## Answer by Mh Aztec (score 1)
https://quant.stackexchange.com/a/32935
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.