Journal of Network Theory in Finance

Fractional differencing: (in)stability of spectral structure and risk measures of financial networks

Arnab Chakrabarti and Anindya S. Chakrabarti

  • Standard filtering techniques on financial networks preprocess asset price data by first-differencing, while the empirical roots d often exhibit magnitudes less or more than one.
  • We apply fractional differencing to rectify over- and under-differencing, which we term as d-corrections.
  • Resulting filtered networks in the form of minimum spanning trees and triangulated maximally filtered graphs, show non-robustness with respect to d-corrections.
  • Spectral structures of the networks show moderate stability with respect to d-corrections, while centrality-based measures of risk show nonmonotonic behavior.

The computation of spectral structures and risk measures from networks of multivariate financial time series data has been at the forefront of the statistical finance literature for a long time. A standard mode of analysis is to consider log returns from the equity price data, which is akin to taking the first difference (d = 1) of the log of the price data. In this paper we study how correcting for the order of differencing leads to altered filtering and risk computation for inferred networks. We show that filtering methods with extreme information loss, such as the minimum spanning tree, as well as those with moderate information loss, such as triangulated maximally filtered graph, are very susceptible to d-corrections; the spectral structure of the correlation matrix is quite stable although the d-corrected market mode almost always dominates the uncorrected (d = 1) market mode, indicating underestimation in the standard analysis; and a PageRank-based risk measure constructed from Granger-causal networks shows an inverted-U-shaped evolution in the relationship between d-corrected and uncorrected return data for historical Nasdaq data for the period 1972–2018.

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