An introduction to wavelet analysis methods

This introduction to wavelet analysis starts with a discussion on spectral analysis. More technical description of wavelets are provided in Appendix A. Spectral analysis techniques are used to decompose a given series into the sum of sinusoids of different frequencies (a process called ‘complex demodulation’). When comparing the amplitudes or power of these sinusoids, the fact that one frequency has more associated power than others, suggests that much of the variance in a given series can be explained in terms of a well defined cycle at this or an equivalent frequency. However, these elementary sinusoids themselves do not change over time, either in their frequency or their amplitude. This is one of the limitations of spectral analysis, although from time to time suggestions have been made as to developing time varying spectra (e.g. Priestley, 1965). However, even in that approach, the changes had to be very slow over an extended period of time.

Limitations of this kind were effectively removed by the recent development in wavelet theory and practice. Wavelets analysis now decomposes data into different components along both time and scale (or frequency). A wavelet is rather like a sinusoid localised at a particular point in time, so that its power drops off rapidly on either side of that time point (see Appendix A). Wavelets come with a variety of scales. Thus one might have a scale representing a 6-12 month fluctuation (loosely, one cycle), another for a 1-2 year fluctuation, a third for 3-5 year fluctuation and others. Moving through time, one fits a succession of wavelets for each scale. Each

time point contains contributions from wavelets of the same ‘scale’ (quasi-frequency) but centred at neighbouring points. This feature enables one to model cycles that do not remain constant in amplitude, so that, in this respect, wavelet analysis overcomes the limitations of ordinary spectral analysis.

Discrete wavelet transform

The wavelet transform can be either continuous or discrete. Figure 5.1 is a schematic description of a discrete wavelet transform (DWT) outcome. Level 1 is the smallest scale or highest quasi-frequency, so D1 represents the cycle at this highest level of detail. The given series is then split into D1 and A1, where A1 is the series once the highest frequency fluctuations have been removed. Levels 2, 3…. contain successively less high frequency complexity. When more fluctuations have been extracted, the residual series becomes broader time-frame approximations, which reveal longer run cycles and ultimately the trend. An ‘average period’ constructed for a given level of detail D can be derived by finding the sinusoid where the period most closely matches that of the wavelet fitted at any point in time, suitably adjusted for its scale (see Appendix A).

S ign al S A 1 D 1 A 2 D 2 A 3 A 4 A 5 A 6 A 7 D 3 D 4 D 5 D 6 D 7

Figure 5.1 Decomposition into successive details and approximations

The overall effect can be viewed as an operational generalisation of the schematic time series decomposition that is familiar within economic literature:

The successive details in the above would be irregular = D1, seasonal = D2, cycle = D3, while the trend would correspond to the subsequent remaining low frequency

variations, namely approximations. However, the decomposition can be more refined than this. There are further levels of cyclical detail, and the approximation itself can take the form of a sum. The highest approximations, for example, A7 in figure 5.1, can be considered the trend, as it shows no residual cyclical character. The wavelet decomposition is a dyadic data analysis process and thus the scale of sample determines how many levels of decomposition can be made. For a data set with 128 (=27) observations, the wavelet decomposition can be achieved up to level 7.

The technique has become popular because the methodology relies on few assumptions yet is powerful in its ability to analyse patterns in data. In applications of wavelet analysis, exposures can be linear or non-linear, stationary or non-stationary. Those assumptions usually required for a normative model are not necessary for utilising wavelet analysis. Short and long term exposures are decomposed and examined at different scales. The overall effect is rather like adjusting more and more exactly the focus of a microscope. One of the more celebrated images in wavelet exposition is that of Madame Daubechies’ eye as viewed from successively closer up. At long range one sees only the general features, while the rest as blurred. These appear like D6 or D7 plus the A7, although in this case two dimensional. Moving closer, one sees higher level details of the iris, ultimately up to D1. Instead of a limited one dimension, wavelets allow users to examine multi-dimensions of data.

Wavelet functions

A large number of wavelet functions have been developed for the wavelet transform. Collectively across different scales, the existing wavelet functions are flexible enough to allow for asymmetric local cycles of rather arbitrary form, no longer requiring regular sinusoidal patterns. Although all are chosen from the same generic family, wavelets are normalised to refer either to the cycles (‘mother wavelets’) or long term trend or quasi trends (‘father wavelets’). The mother wavelets integrate to zero while father wavelets are normalised to integrate to 1. The results of fitting mother (cyclical) wavelets of different scales are the details (D1, D2, …) and they are additive in their effect. The progressive sums, by adding more details, are the approximations (A1, A2, …). A more detailed discussion on the wavelet function is given in Appendix A,

which also depicts the wavelet family used in the present study, namely the ‘coif5’ wavelets.

Wavelet variation metrics

By decomposing the time series into orthogonal components, the variance of components at different scales can be derived using wavelet methods. Though this is a local concept that differs over periods, one can compute the average variance over the given time horizon. For time series Xt , t=1, 2, ……N, the decomposed wavelet details at scale j are indicated asD(_j,X_t), which should have a mean of zero as the integral of mother wavelet is 0. The wavelet variance at scale j can then be calculated with the

following equations:

∑

With a discrete wavelet transform up to level J, the total volatility of the variable can

be expressed as the sum of all the detail volatilities and the remaining volatility at level J approximation.

The variance decomposition enables users to quantify how time series differ in terms of multi-scale variation behaviour. For time series with a range of persistence characteristics, the wavelet variance analysis may disclose the main contributors to the overall volatility (Percival & Walden, 2000). For instance, a random walk process should exhibit the concentration of variance at the highest approximations and details. On the other hand, the variance of a white noise series would mainly come from the smallest level detail, such as D1. For a long memory process that has longer persistence of shocks than ARMA (autoregressive moving average) class of time series but shorter than a random walk, Percival and Walden (2000) show that a plot of log of wavelet variance versus log of scale exhibits approximately linear variation.

In addition to the wavelet variance, the wavelet correlation can also be constructed on a scale basis. For example, if the original signal S is composed of X, Y,

the covariance and correlation indicators at a specific scale can be expressed as follows: . ) ( ) ( ) ( ) ( 1 ) ( , , ) ( , ) ( , , j Var j Var j Cov j Corr D D N j Cov Y X Y X Y X t Y t j X t j Y X = ∑ =

This decomposed correlation can be particularly useful in economics and finance as the component of the exposures might have different correlations for low frequency and high frequency variations. For instance, the components might move in different ways over a short term period, as the high frequency variations usually reflect market transients that are hard to predict and vary from market to market. However, as most of financial variables are affected by macroeconomics fundamentals, they may still show a close relationship with each other in the long run.

The above wavelet variance and related indicators are mainly constructed through the discrete wavelet transform. However, the wavelet variance can be estimated more effectively with the maximal-overlap discrete wavelet transform (MODWT) (Percival, 1995; Gencay, Selcuk & Whitcher, 2002). Unlike DWT, the MODWT is not orthonormal and not subsampling the filtered output (details see Appendix A). The MODWT is invariant to circularly shifting the original time series. The size of sample that the MODWT can handle is thus not limited to a multiple of 2n. The MODWT extends the sample by assigning the observable values to those unseen as if it were periodic, e.g. the unobserved samples X-1, X-2, …, X-N are assumed the

same as the observed values XN, XN-1, …, X1.

As Percival and Walden (2000) argue, the wavelet variance, covariance and correlation in terms of MODWT coefficients result in better statistical interpretations than DWT. However, the MODWT can be biased as it introduces artificial circularity into the sample data. The coefficients involving the beginning and end data thus need to be excluded from the above formula to get the unbiased variance. In the current context, excluding the boundary coefficients, such as variation over a five year to ten year frequency will rule out the outcomes of interest. Therefore, the following empirical section focuses on the DWT based variance, in which the variance over different scales is the sum of squared details. The biased MODWT type wavelet variance will be constructed only for verifying the results.