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

Arnab Chakrabarti and Anindya S. Chakrabarti

Abstract

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.