Volatility dynamics can carry information about temporal relationships that is not apparent in the conditional mean. Extracting this information is difficult because the persistence of each variable can obscure its dependence on other variables. This paper presents QS-TCN, a method for ordering multivariate time series that combines statistical estimates of volatility memory with temporal convolutional learning. Persistence estimates obtained by quasi-maximum likelihood and temporal modes extracted by the Matrix Pencil method define structured filters within the network. Together with learned filters, they represent the history of the target and support comparisons of directional evidence between variables. These comparisons are combined into a global order of precedence, which does not identify a causal graph. The method is evaluated on synthetic systems with linear and nonlinear volatility interactions and on the Finance and fMRI benchmarks, whose reference relations are known. Across the four synthetic volatility regimes studied, QS-TCN reaches a higher mean ordering quality than an external lead--lag ordering and its squared-input adaptation, although the gain over the adaptation is uncertain in two regimes. Applications to currency, gold and cryptocurrency markets are reported descriptively.

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