# Testing a single time-series for changing variance structure (Heteroscedasticity and Volatility Clustering)

I would like to assess a single time-series for a changing variance structure that might be leading to spurious variance estimates when that time-series is used in regression.

In my head two terms come to mind: (1) Heteroscedasticity and (2) Volatility Clustering

Here are my related questions on this:

1. (True/False) Heteroscedasticity is related to non-constant variance of the errors while Volatility Clustering is related to non-constant variance of the levels of the time-series. As such Heteroscedasticity can only be tested for using residuals from a regression. Variance Clustering on the other hand could be tested for on the levels themselves without the need for regression.
2. How could I test an individual time-series for Volatility Clustering?
3. Can Volatility Clustering lead to spurious regressions? If so is this effect enacted through inducing Heteroscedasticity in the regression?

1. I would say, heteroskedasticity can characterize original series, too. Suppose the variance of the original series was $\sigma_1^2$ from time $t_1$ to $t_k$ and turned to $\sigma_2^2$ since $t_{k+1}$. I would call that heteroskedasticity. Conversely, volatility clustering can characterize model residuals as well as the original series.
3. The question is quite broad; let us consider the base case. Recall that OLS estimates are consistent (even) under heteroskedasticity. Consider two variables $x$ and $y$ that are linearly independent and mean-stationary but one or both of them have volatility clustering. The OLS estimate of the slope coefficient from a regression of $y$ on $x$ should (still) converge to zero as sample size increases. Aside from the base case, I cannot quickly come up with a situation where volatility clustering alone would yield spurious correlation between two linearly independent series (I do not claim it is impossible, but I do not see how it could work).
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