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

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Abstract

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.

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Notes

  1. Quadrature mirror filters are used in engineering for perfect reconstruction of a signal without aliasing effects.

  2. The modulus operator is required in order to deal with the boundary of a finite length vector of observations.

  3. The energy of a vector—proportionate to variance—is defined as the sum of its squared coefficients.

  4. It can be also proven via matrix operations: \(\left\| \mathbf{y} \right\| ^{2}=\mathbf{y}^{T}\mathbf{y}=\left( {\mathbf{Ww}} \right) ^{T}\mathbf{Ww}=\mathbf{w}^{T}\mathbf{W}^{T}\mathbf{Ww}=\mathbf{w}^{T}\mathbf{w}=\left\| \mathbf{w} \right\| ^{2}\).

  5. Shifting a signal simply means delaying its start in the real-axis. In mathematical terms, delaying a function is represented by \(f\left( {t-d} \right) \).

  6. The Shannon entropy criterion shows a downward trend until a minimum value—corresponding to a “threshold” scale level—is reached and then it begins to rise revealing that further signal decomposition “contains” redundant information. The maximum level of decomposition tried in this study is ten, based on the appropriate “translation” of the wavelet scales into economic time horizons.

  7. For instance, if the data appear to be constructed of piecewise linear functions, then the Haar wavelet may be the most appropriate choice, while if the data is fairly smooth, then a longer filter such as the Daubechies asymmetric wavelet filter may be desired.

  8. In the empirical part, alternative choices of wavelet classes were also applied, but the results were very robust to such changes and the current selection appeared to be the most balanced.

  9. Due to the nature of volatility, it is assumed that there is no time trend in the series in the long run (Nikkinen et al. 2006). However, the unit root tests were also performed with a time trend and the results remain unchanged. Moreover, the test results are generally not sensitive to the number of lags used.

  10. Thereafter the notation \(\mathbf{D}_j \) (and not the \({\tilde{\mathbf{D}}}_j \) used in Sect. 3) corresponds to the SIDWT details, to enhance readability.

  11. In empirical applications, quarterly, semi-annual or yearly volatility is not interesting for the economic analysis of high-frequency (daily) FX series, nor “traded” in currency markets, as opposed to daily or weekly volatility. However, the analysis of the returns up to yearly variations can be very useful in detecting FX market linkages with macroeconomic fundamentals (e.g., GDP, CPI, Interest rates) or in producing multi-step ahead price forecasts.

  12. The structural changes in the wavelet approximations mentioned throughout this section have also been tested with the Chow’ and CUSUM tests as well as with the Bai and Perron (2003) and Zivot and Andrews (1992) tests. For the details, the switching regimes have been verified via a Markov-switching model with two regimes, using an AR(1) specification in each regime.

  13. The first Greek credit rating cut by Fitch corresponds to obs. 2,851 in \(\text{ P }_{\mathrm{T}}\).

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Acknowledgments

This research is supported by the Marie Curie Fellowship (FP7-PEOPLE-2011-CIG, No. 303854) under the 7\({\mathrm{th}}\) European Community Framework Programme. I am grateful to the Editor Hans Amman for valuable comments. The usual disclaimers apply.

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Correspondence to Stelios D. Bekiros.

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Bekiros, S.D. Timescale Analysis with an Entropy-Based Shift-Invariant Discrete Wavelet Transform. Comput Econ 44, 231–251 (2014). https://doi.org/10.1007/s10614-013-9381-z

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