[Paper Review] Periodicity and scaling of eigenmodes in an emerging market

Diane Wilcox, Tim Gebbie|arXiv (Cornell University)|Apr 18, 2004

Complex Systems and Time Series Analysis4 citations

TL;DR

This study analyzes eigenmodes of correlation matrices from the Johannesburg Stock Exchange (1993–2002) using spectral analysis, rescaled-range methods, and detrended fluctuation analysis (DFA). It finds that only eigenmodes linked to eigenvalues outside the Wishart random matrix range exhibit periodic or aperiodic behavior, while those within the Wishart range are dominated by noise, and DFA reveals no long-term memory in these noise-dominated modes.

ABSTRACT

We investigate periodic, aperiodic and scaling behaviour of eigenmodes, i.e. daily price fluctuation time-series derived from eigenvectors, of correlation matrices of shares listed on the Johannesburg Stock Exchange (JSE) from January 1993 to December 2002. Periodic, or calendar, components are investigated by spectral analysis. We demonstrate that calendar effects are limited to eigenmodes which correspond to eigenvalues outside the Wishart range. Aperiodic and scaling behaviour of the eigenmodes are investigated by using rescaled-range methods and detrended fluctuation analysis (DFA). We find that the eigenmodes which correspond to eigenvalues within the Wishart range are dominated by noise effects. In particular, we find that interpolating missing data or illiquid trading days with a zero-order hold introduces high frequency noise and leads to the overestimation of uncorrected (for serial correlation) Hurst exponents. DFA exponents of the eigenmodes suggest an absence of long-term memory.

Motivation & Objective

• To investigate periodic, aperiodic, and scaling behaviors in eigenmodes derived from stock return correlation matrices on the Johannesburg Stock Exchange (JSE).
• To determine whether calendar effects (periodic components) are present in eigenmodes and whether they correlate with eigenvalues outside the Wishart random matrix range.
• To assess the impact of data interpolation (zero-order hold) on the estimation of Hurst exponents in eigenmodes.
• To evaluate the presence of long-term memory in eigenmodes using detrended fluctuation analysis (DFA).

Proposed method

• Construct correlation matrices from daily price fluctuations of JSE-listed shares from January 1993 to December 2002.
• Apply spectral analysis to detect periodic (calendar) components in eigenmodes associated with eigenvalues outside the Wishart random matrix distribution range.
• Use rescaled-range analysis and detrended fluctuation analysis (DFA) to examine aperiodic and scaling behavior of eigenmodes.
• Compare Hurst exponents estimated from raw and interpolated data (using zero-order hold for missing or illiquid trading days).
• Classify eigenmodes based on their eigenvalue position: within or outside the Wishart range, to isolate noise-dominated from signal-dominated modes.
• Evaluate the robustness of Hurst exponent estimates by correcting for serial correlation in the time series.

Experimental results

Research questions

• RQ1 Which eigenmodes of the JSE correlation matrix exhibit periodic (calendar) effects, and are these effects linked to eigenvalues outside the Wishart range?
• RQ2 How does data interpolation using zero-order hold affect the estimation of Hurst exponents in eigenmodes?
• RQ3 Do eigenmodes within the Wishart range display long-term memory, as indicated by DFA and rescaled-range analysis?
• RQ4 What is the relative contribution of noise versus genuine market dynamics in eigenmodes with eigenvalues inside the Wishart range?
• RQ5 How do spectral, DFA, and rescaled-range methods collectively characterize the scaling and persistence properties of eigenmodes?

Key findings

• Calendar effects are present only in eigenmodes corresponding to eigenvalues outside the Wishart range, indicating that these modes reflect non-random, structured market behavior.
• Eigenmodes with eigenvalues within the Wishart range are dominated by noise, as confirmed by their lack of persistent scaling behavior.
• Interpolating missing data or illiquid trading days with a zero-order hold introduces high-frequency noise, leading to overestimation of uncorrected Hurst exponents.
• Detrended fluctuation analysis (DFA) of eigenmodes shows no evidence of long-term memory, suggesting that persistent trends are not a feature of the noise-dominated modes.
• The combination of spectral analysis and DFA indicates that only a small subset of eigenmodes (those outside the Wishart range) carry meaningful market information.
• The study demonstrates that standard Hurst exponent estimation methods are sensitive to data preprocessing, particularly interpolation, when applied to eigenmodes.