An Information-Theoretic Approach to Dimension Reduction of Financial Data

Abstract

factors or components is often a first step 3 . While principal component analysis has been a standard dimension reduction tool for many decades, a theoretically sound measure of the number of components that should be retained has been lacking 4 . Here we show that the effective rank 5 offers a potential model-independent solution to the problem. We demonstrate that the explanatory power of the number of components indicated by the effective rank is remarkably stable for a wide range of global financial market data while the effective rank itself can vary dramatically over time, offering a potential indicator of systemic risk. The results suggest a certain universality to the measure and we provide some theoretical results supporting this view, derive lower bounds for its explanatory power and highlight links to measures of diversification in areas ranging from ecology to quantum mechanics. Our results demonstrate that the timevarying drivers of financial markets do exhibit some persistent structure. We anticipate our results will prompt further investigation of the effective rank in principal component analysis given the latters’ wide appeal in diverse fields of research ranging from psychology to atmospheric science 4 . We also hope our results provide some direction to solving related dimensional problems such as in cluster analysis where the longstanding question of how many clusters should be used remains unanswered 6,7 . Accounting for the correlations between asset prices is fundamental to constructing and measuring the risk of investment portfolios 8 . In the years following the onset of the financial crisis in 2007 asset prices have exhibited persistently high correlations as the usual idiosyncratic considerations of investors give way to macro concerns about the financial system. The dominance of such concerns has resulted in a significant proportion of the variation of asset prices within large investment universes being explained by only a small number of drivers 9,10 . While this effective reduction in the dimension of the investment universe is attractive from the perspective of permitting a simpler model of the underlying risks, it directly inhibits the ability of investors to diversify their portfolio due to the reduced degrees of freedom. Tools for exploring dimension reduction are thus of great interest and principal component analysis (PCA) is possibly the most commonly applied. The application of PCA is primarily an exercise in projecting data from an m dimensional space down onto a smaller set of k principal components that are constructed to capture the maximum amount of variation in the original data as measured by the proportion of total variance explained 4 (henceforth variance explained ). In spite of the prevalence of this technique there is however no widespread agreement on the appropriate value of k to use; while various ad hoc rules of thumb have been