Wavelet Applications in Economics and Finance by Marco Gallegati & Willi Semmler

Author:Marco Gallegati & Willi Semmler
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

3 Estimation Techniques

Time series analysis and standard econometric methods cannot account for changes in frequency behavior. We use wavelets as a time–frequency analysis that provides information about the frequency behavior of time series at a given point in time. Wavelet analysis estimates the frequency structure of a time series (forward premium and exchange rate changes). In addition to that it keeps the information when an event of the time series takes place. Wavelet analysis can be understood as a rotation in the function space. The basis functions used in that transformation are wavelets which have finite support on the time axis, i.e. are small waves. For the purpose of transforming the time series, the basis function (wavelet) is dilated, or compressed, to capture frequency behavior, and is shifted along the time axis to capture the date when a certain event takes place. This is how it is possible for a disturbance to be of influence for certain frequencies, or finite time periods only. The result is a representation of the time series in the time and frequency domain. The wavelet approach can allow an analysis of processes whose behaviors differ across scales, i.e. depict different behavior with regards to different time horizons. This is most likely the case for the (forward) currency exchange market due to the aforementioned reasons.

For the purpose of allowing different behavior for different time horizons, the variables exchange rate change and forward premia are decomposed into their time-scale components applying the maximal overlap discrete wavelet transform (MODWT). This procedure allows for any length of a time series and is able to get robust estimators. Wavelets (ψ j,k and φ J,k ) when multiplied with their respective coefficients at a certain level “j” or “J” are called atoms Dj,k and SJ,k (i.e. dj,k * ψ j,k  = Dj,k and sJ,k * φ J,k  = SJ,k) with ψ j,k and φ J,k being the wavelet and scaling functions at level “j” or “J” and “k” indicating the location of the wavelet on the time axis. The sums of all atoms, SJ,k(t) and Dj,k(t), over all locations on the time axis k = 1, …, at a certain level “j” or “J” are called crystals and are given by Eqs. (3) and (4).

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