A Case for iPCA in Financial Forecasting

Principle Component Analysis (PCA) is used to reduce dimensionality and noise, while still preserving the majority of the variance in the data. It however gives little guarantee on the predictive value of the remaining data. This paper proposes an inverted Principle Component Analysis (iPCA), to achieve dimensionality and noise reduction, by removal of the largest principle components. In addition a three-part decomposition of the financial markets is proposed and its validity is tested. By applying iPCA we were able remove the market component that is less predictable, while at the same time keeping the components that carry most of the predictive information. Our method is applied to various regression models including recurrent neural networks and an improvement of 57% is observed. 1 Time Series Prediction in Finance A time series is a sequence of observations at successive points in time. Time series prediction consists of predicting future values of a time series, given past history of the same series. In finance, the return r = p (t)−p(t−1) p(t−1) is typical to viewed as the variable, where p is the stock price at time t. The objective is to predict r (τ > t), given r, .., r. Frequently one has multiple stocks, in which case each stock is viewed as a variable. Time series prediction in finance is a constantly evolving field. The proliferation of electronic trading means that an individual can invest in more instruments and markets than ever before. It is not uncommon for an investor to maintain a portfolio of derivatives, ETFs, and equities in markets all over the world. This behavior has made individual stocks and markets as a whole significantly more correlated. The additional advancement of algorithmic trading has removed many of the inefficiencies in the markets, making stock prices look increasingly noisy. Whereas simple trend following strategies worked well in the 20th century, increasingly sophisticated algorithms and trading systems are required to maintain an edge over the market. To deal with the large number of financial instruments and the high degree of what can be classified as noise, both dimensionality reduction and filtering are necessary preprocessing steps. Principle Component Analysis (PCA) has a long history of being used for both these purposes in many fields of statistics including time series prediction. PCA projects the data into the d-dimensional subspace, that accounts for the most variability in the data, among all spaces of dimension d. Doing so, implicitly assumes that any noise in the data either has low variance or is uncorrelated among the different variables. In finance, neither of these assumptions can be made. Data can be extremely noisy and unpredictable events can have highly correlated effects on the stock market. We hypothesize that the directions of largest variation in the data in fact correspond to directions of most noise and greatest unpredictability. If this is true, then one would be better off removing the largest principle components and keeping the smaller principle components. This is opposite to the way PCA is typically employed, accordingly we ∗e-mail:[email protected] †e-mail:[email protected] ‡e-mail:[email protected] refer to this procedure as inverted PCA (iPCA). We seek to verify this hypothesis through application to high frequency data of the 100 largest stocks by market capitalization, listed on the New York Stock Exchange (NYSE). 2 Model of Financial Markets We first introduce some notation. Let X ∈ Rm×n be the data matrix of returns (r), where each column represents one stock, and each row is the return of each individual stock at a specific time t, where the row number corresponds to t: X =  − r T − − r T − .. − r ) T −  Next, we propose the following decomposition of the return of a stock i: