Why Returns Can Be Uncorrelated While Squared Returns Cluster
Summary
Returns and squared returns describe different features of a time series. Returns measure direction and can have little serial correlation even when their magnitudes are dependent over time. Thus, the absence of correlation in returns does not imply that large and small moves arrive independently.
The discussion illustrates this distinction with a sequence of values whose signs can vary while their squares remain constant. It also separates autocorrelation from heteroskedasticity: autocorrelation concerns dependence across observations, whereas heteroskedasticity concerns changing variance. Volatility clustering is a common pattern in which large or small return magnitudes persist, but heteroskedasticity alone does not establish that variance is autocorrelated. The examples are conceptual rather than empirical, and the brief answers do not specify a statistical model or tests for diagnosing these properties.
Key ideas
- Returns can show little serial correlation while their magnitudes or squared values remain dependent.
- Autocorrelation in returns and autocorrelation in squared returns are distinct properties.
- Heteroskedasticity means variance changes over time, while autocorrelation describes dependence across time.
- Volatility clustering is one pattern of changing variance, not a necessary consequence of every heteroskedastic process.
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Full text
# Time Varying Volatility
# Time Varying Volatility
If stock returns ($r_t$) are not auto correlated why is that the squared term of the returns (volatility) exhibit serial correlation? Does heteroskedacity, by its nature, imply that time varying volatility is autocorrelated and therefore volatility today is influenced by its lagged values?
## Answer by vonjd (score 3)
https://quant.stackexchange.com/a/7848
To your first question: The first and the second moment are independent, so even if returns are not autocorrelated the size of returns can be.
To your second question: Autocorrelated volatility implies volatility not being constant but varying, which is heteroscedasticity or volatility clustering.
## Answer by 4pie0 (score 1)
https://quant.stackexchange.com/a/7851
> The first and the second moment are independent, so even if returns are not autocorrelated the size of returns can be.
of course. Example: generate path of variable with binomial distribution that takes value of 1 or -1, that is
$\sum_{i=1}^n{} x_i,x_i=\{1,-1\}$
now you can generate another path $\sum_{i=1}^n{} x'_i,x'_i=\{1,-1\}$ choosing values of $x'$ with freedom, and this way paths will be correlated or not, and squares of it will be correlated all the time, as two constant series of 1,1,1,...,1
regarding you second question
> Does Autocorrelation imply Heteroskedacity?
No. These are two independent concepts. When you take matrix of varince-covariance of time series homoscedasticity is related to the elements on a diagonal, still with assumption that pairwise coefficients are 0. If they are not 0 then there is autocorrelation.
see here for details on thisShown 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.