Time series that are used for forecasting asset returns can carry information on trends of different persistence. Therefore, frequency decomposition of standard signals based on wavelets can improve and expand potential predictors. Similarly, asset returns can be decomposed into parts of different persistence. These can be forecast separately and summed up eventually. This “sum-of-parts” method seems to improve forecast accuracy because its aligns predictors and return trends and helps separating signal from noise.

*The post ties in with the subject of “best practice for tracking macro trends” in the lecture on macroeconomic trend indicators.*

*The below are excerpts from the paper. Headings and the introductory cursive text have been added for context and convenience of reading.*

### What is frequency decomposition?

*Frequency** describes the number of cycles that pass in a given amount of time. For cyclical processes frequency is defined as a number of cycles per unit time. The reciprocal of frequency is period.*

*In frequency domain signals are represented as a function of frequencies, rather than time. Frequency-domain analysis is a tool of utmost importance in signal processing applications…While time-domain analysis shows how a signal changes over time, frequency-domain analysis shows how the signal’s energy is distributed over a range of frequencies… The frequency-domain representation of a signal carries information about the signal’s magnitude and phase at each frequency.*

*A signal can be converted between the time and frequency domains with a pair of mathematical operators called a transform. An example is the Fourier transform, which decomposes a function into the sum of a (potentially infinite) number of sine wave frequency components. The ‘spectrum’ of frequency components is the frequency domain representation of the signal.*

### What are wavelets?

“Wavelets are a signal processing technique developed to overcome some of the limitations of traditional frequency domain methods, such as spectral analysis and Fourier transforms…__A wavelet is a function of finite length that oscillates around a time axis and loses power as it moves away from the center__…There are two kinds of wavelets: father wavelets that capture the smooth and low-frequency part of the series, and mother wavelets that capture the detail and high-frequency components of the series.”

“Unlike Fourier analysis, wavelets are defined over a finite support/window in the time domain, with the size of the window resized automatically according to the frequency of interest. In essence, using a short window allows to isolate the high frequency features of the time series, while looking at the same signal with a large window reveals the low frequency features. Hence, __by varying the size of the time window, it is possible to capture simultaneously both time-varying and frequency-varying features of the time series__. Wavelets are thus extremely useful when the time series have structural breaks or jumps.”

“Wavelets tools allow for a decomposition of a given time series into different time series, each associated with a different time scale or frequency band. This decomposition process is known as **multiresolution analysis**…Frequency decomposition…through a discrete wavelet transform multiresolution analysis…__consists in decomposing a time series into n orthogonal time series components, each capturing the oscillations of the original variable within a specific frequency interval__. Lower frequencies represent the long-term dynamics of the original time series, while the higher frequencies capture the short-term dynamics.”

“Frequency decomposition of the predictors is not only a methodological contribution per se, but it also __represents an expansion of the set of possible predictors__, as each frequency of each predictor can be understood and potentially used as a new predictor.”

### What is the sum-of-parts/wavelet method?

“Our methodology to forecast stock market returns builds on two blocks: the **sum of parts** method… and the **discrete wavelet transform** decomposition of the different predictors.

- The sum-of-the-parts (SOP) method for forecasting stock market returns…consists in decomposing the stock market return into three components, which are
__first forecasted separately and then added in order to obtain the forecast__of the stock market return. - We directly use (some of) the
__frequency-decomposed time series of a set of popular predictors__from the literature. The frequency decomposition is implemented through a discrete wavelet transform multiresolution analysis, which is gaining popularity as a tool for econometric analysis.”

### How does the method improve forecasts?

“The sum-of-parts method improves the forecast accuracy (as compared to the historical mean benchmark) because it __exploits the different time series persistence __of the stock market returns components.”

“The…method, by explicitly decomposing the different predictors of stock returns into their frequency time-series components, allows to identify the best predictors and to exclude the noisy parts. In other words, it __only retains the components that have the greatest predictive power__. This leads to expressive statistically and economically forecasting improvements.”

“The strong performance of this method comes from its ability to isolate the frequencies of the predictors with the highest predictive power from the noisy parts, and from the fact that the __frequency-decomposed predictors carry complementary information that captures both the long-term trend and the higher frequency__ movements of stock market returns.”

### What is the empirical evidence?

“We __focus on the out-of-sample predictability of monthly stock returns__, proxied by the S&P500 index total return. We use monthly data from January 1927 to December 2015 for a set of potential predictors… Specifically, we use the dividend-price ratio, the growth rate of dividends, the book-to-market ratio, the default return spread, the default yield spread, the dividend-payout ratio, the earnings-price ratio, the inflation rate, the long-term government bond return, the long-term government bond yield, the net equity expansion, the return on equity, the stock variance, the term spread and the treasury bill rate.”

“One-step-ahead out-of-sample forecasts of stock market returns are generated __using a sequence of expanding windows__. We use an initial sample (January 1927 to December 1949) to make the first one-step-ahead out-of-sample forecast.”

“Using the historic mean as a benchmark, the monthly out-of-sample R2 for the sum-of-parts/wavelet method is 3.27% for the full out-of-sample period (January 1950 to December 2015). When examining the economic significance of the sum-of-parts/wavelet predictive performance through an asset allocation analysis, we find that a mean-variance investor who allocates her wealth between equities and risk-free bills enjoys significant utility gains from using a sum-of-parts/wavelet-based trading strategy…The __annualized Sharpe ratio of the strategy based on the sum-of-parts/wavelet method is 0.65, which is about 1.8 times the Sharpe ratio generated by the historic mean-based strategy__. These results are robust to the introduction of transaction costs, different sets of portfolio constraints and different out-of-sample forecasting periods.”