Timescale Analysis with an Entropy-Based Shift-Invariant Discrete Wavelet Transform

This paper presents an invariant discrete wavelet transform that enables point-to-point (aligned) comparison among all scales, contains no phase shifts, relaxes the strict assumption of a dyadic-length time series, deals effectively with boundary effects and is asymptotically efficient. It also introduces a new entropy-based methodology for the determination of the optimal level of the multiresolution decomposition, as opposed to subjective or ad-hoc approaches used hitherto. As an empirical application, the paper relies on wavelet analysis to reveal the complex dynamics across different timescales for one of the most widely traded foreign exchange rates, namely the Great Britain Pound. The examined period covers the global financial crisis and the Eurozone debt crisis. The timescale analysis attempts to explore the micro-dynamics of across-scale heterogeneity in the second moment (volatility) on the basis of market agent behavior with different trading preferences and information flows across scales. New stylized properties emerge in the volatility structure and the implications for the flow of information across scales are inferred.