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

Computation of spectral structure 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 first difference ($d = 1$) of the log of the price data. Sometimes authors have considered simple growth rates as well. Either way, the idea is to get rid of the nonstationarity induced by the {\it unit root} of the data generating process. However, it has also been noted in the literature that often the individual time series might have a root which is more or less than unity in magnitude. Thus first differencing leads to under-differencing in many cases and over differencing in others. In this paper, we study how correcting for the order of differencing leads to altered filtering and risk computation on inferred networks. In summary, our results are: (a) the filtering method with extreme information loss like minimum spanning tree as well as filtering with moderate information loss like triangulated maximally filtered graph are very susceptible to such d-corrections, (b) 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 under-estimation in the standard analysis, and (c) the PageRank-based risk measure constructed from Granger-causal networks shows an inverted U-shape evolution in the relationship between d-corrected and uncorrected return data over the period of analysis 1972-2018 for historical data of NASDAQ.