Multiscale correlation networks analysis of the US stock market: a wavelet analysis

Note that scaling and multiscaling behaviors of financial time series have become “stylized facts” in the literature (see, e.g., Mantegna and Stanley 1995; Xu and Gençay 2003; Di Matteo et al. 2005; Di Matteo 2007; Segnon and Lux 2013; Buonocore et al. 2016). According to Di Matteo et al. (2005) and Di Matteo (2007), the scaling behavior in finance can be classified into two types: one is the scaling behavior of price fluctuations as a function of time intervals, which is usually characterized by the Hurst exponent; and the other is the scaling behavior in tails of the distribution of returns, which is often measured by a tail index of the distribution. In the first type, the scale invariance is called as self-similarity that is a crucial feature of fractals. Multiscaling (or multifractality) is an extension of self-similarity and is a form of generalized scaling (Xu and Gençay 2003). Unlike the above research on scaling and multiscaling analyses, our study uses the wavelet multiscaling (or multiresolution) approach to decompose financial data (i.e., stock returns) into multiple time scales associated with multiple time horizons. That is to say, our work focuses on the interaction behavior across stocks over various time scales using MST and PMFG networks.

The GICS denotes the Global Industry Classification Standard, which is an industry taxonomy developed by the Morgan Stanley Capital International (MSCI) and S&P. For the GICS sector definitions, see the link https://​www.​msci.​com/​resources/​pdfs/​GICSSectorDefini​tions.​pdf

Note that tests using (Pearson’s) correlation coefficients may be misleading because they ignore the fact that financial data are characterized by a high degree of heterogeneity. Besides, Forbes and Rigobon (2001, (2002), who investigated stock market comovements during financial crises and market crashes (e.g., the 1997 Asia financial crisis, the 1994 Mexican peso collapse, and the 1987 US stock market crash), pointed out that tests for contagion based on cross-market correlation coefficients can be biased in the presence of heteroscedasticity, endogeneity, and omitted variables. Using an adjusted correlation coefficient, they found that there is no contagion and only interdependence across stock markets during those crises. Therefore, the heteroscedasticity, endogeneity, and omitted variable bias in (Pearson’s) correlation coefficients may affect our empirical results. We leave this interesting topic of how to overcome the bias of (Pearson’s) correlation coefficients for future study.

The Pearson’s correlation coefficient (PCC) between two returns of stocks X and Y is defined as \(\rho _{X,Y} = \frac{\left\langle {r_X r_Y} \right\rangle - \left\langle {r_X } \right\rangle \left\langle {r_Y } \right\rangle }{\sqrt{\left( {\left\langle {r_X^2 } \right\rangle - \left\langle {r_X } \right\rangle ^2} \right) \left( {\left\langle {r_Y^2 } \right\rangle - \left\langle {r_Y } \right\rangle ^2} \right) } }\), where \(\left\langle \cdots \right\rangle \) indicates the time average of the period studied.

MST and PMFG graphs in this paper are drawn by the tool of Pajek. For detailed information of the tool of Pajek, see the link http://​mrvar.​fdv.​uni-lj.​si/​pajek/​.

The remaining MSTs for other wavelet time scales are not presented in this study due to space limitations, but they can be obtained from the authors upon request.

Note that in Figs. 14 and 15, we present the composition of communities detected by the Louvain method in terms of GICS sectors.

We thank a reviewer for pointing out the DBHT literature. For details of the DBHT method and the corresponding code, see Song et al. (2012).

The reason for choosing the GICS sector classification of stocks as a benchmark clustering partition is the evident sectoral clustering effect in the PMFGs for PCC and most time scales. But we should note that for stocks there is no benchmark community partition that can distinguish which clustering approach is better than others.

For details on ARI and AMI and their difference, see Vinh et al. (2010).

For a detailed introduction and the code for the MLE+KS approach, see Clauset et al. (2009) and the link http://​tuvalu.​santafe.​edu/​~aaronc/​powerlaws/​.

But we should keep in mind that algorithms like MST have a natural tendency to create scale-free graphs based on financial data.

Due to the limitation of the number of wavelet time scales, the multistep survival ratio proposed by Onnela et al. (2003), which is used to examine the long-run evolution of networks, is not suitable for discussion in our study.