Hidden Noise Structure and Random Matrix Models of Stock Correlations

📝 Abstract

We find a novel correlation structure in the residual noise of stock market returns that is remarkably linked to the composition and stability of the top few significant factors driving the returns, and moreover indicates that the noise band is composed of multiple subbands that do not fully mix. Our findings allow us to construct effective generalized random matrix theory market models that are closely related to correlation and eigenvector clustering. We show how to use these models in a simulation that incorporates heavy tails. Finally, we demonstrate how a subtle purely stationary risk estimation bias can arise in the conventional cleaning prescription.

💡 Analysis

We find a novel correlation structure in the residual noise of stock market returns that is remarkably linked to the composition and stability of the top few significant factors driving the returns, and moreover indicates that the noise band is composed of multiple subbands that do not fully mix. Our findings allow us to construct effective generalized random matrix theory market models that are closely related to correlation and eigenvector clustering. We show how to use these models in a simulation that incorporates heavy tails. Finally, we demonstrate how a subtle purely stationary risk estimation bias can arise in the conventional cleaning prescription.

📄 Content

arXiv:0909.1383v3 [q-fin.RM] 15 Dec 2009 Hidden Noise Structure and Random Matrix Models of Stock Correlations Ivailo I. Dimov,1, ∗Petter N. Kolm,1, † Lee Maclin,1, ‡ and Dan Y. C. Shiber1, § 1Courant Institute of Mathematical Sciences, New York University Corresponding author: Dan Y. C. Shiber We find a novel correlation structure in the residual noise of stock market returns that is remarkably linked to the composition and stability of the top few significant factors driving the returns, and moreover indicates that the noise band is composed of multiple subbands that do not fully mix. Our findings allow us to construct effective generalized random matrix theory market models [3, 4] that are closely related to correlation and eigenvector clustering [6, 12]. We show how to use these models in a simulation that incorporates heavy tails. Finally, we demonstrate how a subtle purely stationary risk estimation bias can arise in the conventional cleaning prescription [3]. Introduction: Originally started in the context of nuclear physics [1], random matrix theory (RMT) has thereafter found numerous applications in a variety of fields such as number theory, disordered systems, neural networks, and signal pro- cessing [1, 2]. Recently the pioneering work of Laloux et al [3], as well as much subsequent research [4, 5], have shown that RMT can also be a valuable tool for analyzing stock mar- ket correlations, where noise can account for more than 2/3 of the eigenvalue spectrum, and a typical large portfolio has size comparable to the measurement time frame. Thus, much of the empirical eigenvalues are spurious and represent measure- ment noise and biases. The remarkable insight provided by Laloux et al was to show that a suitable fit to RMT can clean these spurious contributions, and moreover identify the statis- tically significant signal, or common market risk factors that drive the individual stock returns. The most prominent such non-idiosyncratic factor is the nearly equal-weight top eigen- vector, whose eigenvalue is more than 20 times bigger than the average spectrum. Secondary factors, are long-short port- folios of certain liquidity [3] and industry structure [6, 7], but their contribution is typically an order of magnitude smaller. Most of the rest of the eigenvectors are unstable in time, ap- pear random, and their spectral contribution can be fitted to the Marcenko-Pasteur (MP) distribution [8] derived in the context of Gaussian RMT (GRMT). The noisy eigenvalue correlations [4] also agree with theory [1]. These results have been veri- fied over many stock selections, as well as return frequencies [3, 4, 5]. Despite the apparent success of the theory, subsequent re- search suggests several empirical aspects that the original RMT cleaning may not account for properly. (1) Tails and their correlations have non-trivial effects, and are known to both broaden the spectrum above the upper noise-band edge, as well sharpen it near the lower edge [4, 9, 10], thus mak- ing the fit to the MP distribution problematic. The above re- distribution of spectral weight appears in conjunction with an enhancement of the inverse participation ratios around both ends of the noise spectrum, the so-called localization effect [4], unlike GRMT where the participations are flat [1]. (2) In addition to being partially localized, the band itself may be split due to the same separation of correlation scales [11] that is thought to give rise to clustering of stocks between indus- tries [12]. So far this effect has not been observed, however, due to the large amount of mixing that depletes the stability of all the noisy eigenvectors. It is important to empirically distinguish between the single and multiple band cases. (3) Non-stationarity effects are insufficiently understood. They are suggested [3, 4] to be the source of a residual bias in the risk estimates obtained after RMT cleaning. However, in light of the abovementioned considerations, it is not clear that the original cleaning procedures are unbiased to begin with. In this work, we consider both N = 484 2 minute S&P500 TAQ midquote returns between June 20 - Sep 20, 2007, as well as N = 451 daily S&P500 returns between Jan 2001- Dec 2007 [13]. (1) We reveal a novel correlation structure of the residuals that is linked to the structure and stability of the top few empirical factors. Mainly, we find that the inverse par- ticipations of the localized edge-eigenmodes of the band are dominated by the outlier stocks in the composition of the top few factors, thus indicating that most of the noise there is due to these stocks. The upper edge fluctuations are mainly due to weakly correlated stocks with the smallest relative weight in the market portfolio while lower edge fluctuations are due to strongly correlated stocks identified as the outliers in the secondary factors. The groups in the lower edge belong to major industrial sectors [4, 12], while the upper edge con- tains a large diversified portfo

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