Filtering With Wavelets May Be Worse Than You Think

One of most promising applications of wavelets is in the field of nonparametric statistical estimation, in which one wants to estimate an unknown signal from some noisy data. Donoho and Johnstone (1994) have developed a simple and yet powerful methodology for nonparametric regression and smoothing based on the principle of wavelet shrinkage (removing noising by shrinking wavelets coefficients towards zero) referred to as the Waveshrink algorithm. Shrinkage essentially rests on three simple principles: signal features can be represented by just a few wavelet coefficients, noise affects all wavelets coefficients, and by shrinking wavelet coefficients towards zero, the noise can be removed while preserving features. The algorithm can be described as a three step procedure: 1) data are transformed into a set of wavelets coefficients applying the discrete wavelet function; 2) a shrinkage of the coefficients is performed; 3) the shrunken wavelet coefficients are transformed back in the domain of the original data. In order to select the best values for the parameters of the waveshrink algorithm several approaches have been proposed. Each of the proposed methods has its pros and cons, depending on the particular domain of application. Nevertheless a basic question remains: which of the methods dominates under a forecasting criterion?The purpose of this paper is twofold, first we analyze the potential advantages of wavelet shrinkage methods for financial time series prediction; then we identificate the best combination of parameters. Our approch consists in applying wavelet shrinkage methods to obtain a "clean'' signal, which is then used to estimate several linear models. If too little noise is removed from the data, the linear models estimated over the clean signal should perform approximately as well as a linear model on the corresponding noisiy time series. Also, if the function is "oversmoothed'', the patterns present in the data are also removed, and forecasts should be poor. This suggest that there may be a balance between too much filtering and too little filtering, so that forecasts may be improved.The approach adopted is conceptually simple but computationally intensive: we consider all the posible combinations of parameters and use each combination to filter the noise from a set of financial time series. Then, we construct a variety of linear models for each of the filtered series and perform a dynamic forecasting exercise. Finally we conduct statistical analyses to test the relative superiority of each of the filtering methods.The main results show that, in the particular application chosen, filtering with wavelets does not provide more accurate forecasts and, in fact, it can damage them. Also shrinkage does not seem to be sensitive to some parameters (e.g., the type of wavelet) while, for others (e.g., hard or soft shrinkage), the differences are clear.