This paper shows how to decompose weakly stationary time series into the sum, across time scales, of uncorrelated components associated with different degrees of persistence. In particular, we provide an Extended Wold Decomposition based on an isometric scaling operator that makes averages of process innovations. Thanks to the uncorrelatedness of components, our representation of a time series naturally induces a persistence-based variance decomposition of any weakly stationary process. We provide two applications to show how the tools developed in this paper can shed new light on the determinants of the variability of economic and financial time series.

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