Abstract
This chapter presents a set of tools, which allow gathering information about the frequency components of a time series. 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 overcome most of the limitations of these two methods. Their principal advantages derive from (i) combined information from both time domain and frequency domain and (ii) their flexibility as they do not make strong assumptions concerning the data-generating process for the series under investigation.
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Notes
- 1.
The discrete version of the Fourier transform is used because the time series is recorded at discrete-time intervals.
- 2.
The jth autocovariance of x(t) is given by γ j = E[(x(t) − μ)(x(t − j) − μ)], where μ denotes the expected value of x(t).
- 3.
As an example, let us consider an economic variable, whose evolution is fully determined by the state of the economy. A complete business cycle lasts on average 36 months and therefore f = 1/36 months.
- 4.
See also Gençay et al. (2002) who uses a similar example.
- 5.
In full generality, the phase angle can be computed as \( \theta (f)= \arctan \left(\frac{ Im\left[H(f)\right]}{\mathrm{Re}\left[H(f)\right]}\right), \) where Im[H(f)] and Re[H(f)] are, respectively, the imaginary part and the real part of H(F).
- 6.
- 7.
- 8.
The Morlet wavelet is actually similar to a sin curve modulated by a Gaussian envelope.
Fig. 19.9 
The Morlet wavelet and the sin function
- 9.
The low-pass filter can be directly obtained from the high-pass filter using the quadrature mirror relationship; see Percival and Walden (2000, p. 75).
- 10.
For two integer a and b, a modulus b is basically the remainder after dividing a by b, i.e., a mod b = a–c ċ b with c = ⌊a/b⌋.
- 11.
See Crowley (2007) for more details about the properties of MODWT.
- 12.
Our presentation of the multiresolution analysis is restricted to the case of the MODWT. Nevertheless, a very similar procedure exists for the DWT; see Percival and Walden (2000).
- 13.
One may notice that the variance of the scaling coefficient at scale J is 0 as v J is a scalar (the sample mean of x).
- 14.
Los Angeles (LA), San Francisco (SF), Denver (De), Washington (Wa), Miami (Mi), Chicago (Chi), Boston (Bos), Las Vegas (LV), New York (NY), Portland (Po), Charlotte (Cha), and Cleveland (Cl).
- 15.
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Masset, P. (2015). Analysis of Financial Time Series Using Wavelet Methods. In: Lee, CF., Lee, J. (eds) Handbook of Financial Econometrics and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7750-1_19
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