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It is known that the historical series of financial returns are characterized by the so-called volatility clustering. Suppose we approximate the number of two-type clusters, namely the high and low volatility cluster.
Which method would you use to compare if a time series of financial returns has more "clusterized volatility" than another?
On the basis of your knowledge or your studies, would you be able to indicate which stocks or commodities present this feature more clearly?
You can model the return as a Hidden Markov Model (HMM) with several volatility states. The number of states can be chosen based on an objective model selection criterion, like AIC or BIC. For any particular number of volatility states, the model can be estimated using standard HMM algorithms. Then
1) the "volatility persistence" can be measured as the probability that the Markov chain describing volatility stays in the same state tomorrow;
2) the "cluster effect" can be measured as the R-square in regressing $(\rm{Return} - \mu)^2$ on the estimated level of $\rm{Volatility}^2$.
