Analysis of Financial Time-Series Using Fourier and Wavelet Methods

This paper presents a set of tools, which allow gathering information about the frequency components of a time-series. We focus on the concepts rather than giving too much weight to mathematical technicalities. In a first step, we discuss spectral analysis and filtering methods. Spectral analysis can be used to identify and to quantify the different frequency components of a data series. Filters permit to capture specific components (e.g. trends, cycles, seasonalities) of the original time-series. Both spectral analysis and standard filtering methods have two main drawbacks: (i) they impose strong restrictions regarding the possible processes underlying the dynamics of the series (e.g. stationarity), and, (ii) they lead to a pure frequency-domain representation of the data, i.e. all information from the time-domain representation is lost in the operation. In a second step, we introduce wavelets, which are relatively new tools in economics and finance. They take their roots from filtering methods and Fourier analysis. But they overcome most of the limitations of these two methods. Indeed their principal advantages are the following: (1) they combine information from both time-domain and frequency-domain and, (2) they are also very flexible and do not make strong assumptions concerning the data generating process for the series under investigation.