Abstract
We construct a Pearson correlation-based network and a partial correlation-based network, i.e., two minimum spanning trees (MST-Pearson and MST-Partial), to analyze the correlation structure and evolution of world stock markets. We propose a new method for constructing the MST-Partial. We use daily price indices of 57 stock markets from 2005 to 2014 and find (i) that the distributions of the Pearson correlation coefficient and the partial correlation coefficient differ completely, which implies that the correlation between pairs of stock markets is greatly affected by other markets, and (ii) that both MSTs are scale-free networks and that the MST-Pearson network is more compact than the MST-Partial. Depending on the geographical locations of the stock markets, two large clusters (i.e., European and Asia-Pacific) are formed in the MST-Pearson, but in the MST-Partial the European cluster splits into two subgroups bridged by the American cluster with the USA at its center. We also find (iii) that the centrality structure indicates that outcomes obtained from the MST-Partial are more reasonable and useful than those from the MST-Pearson, e.g., in the MST-Partial, markets of the USA, Germany, and Japan clearly serve as hubs or connectors in world stock markets, (iv) that during the 2008 financial crisis the time-varying topological measures of the two MSTs formed a valley, implying that during a crisis stock markets are tightly correlated and information (e.g., about price fluctuations) is transmitted quickly, and (v) that the presence of multi-step survival ratios indicates that network stability decreases as step length increases. From these findings we conclude that the MST-Partial is an effective new tool for use by international investors and hedge-fund operators.
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Notes
To verify that the partial correlation-based distance metric \(d_{ij}^{*}\) satisfies the third axiom (i.e., the triangle inequality, TI), we follow Wang et al. (2012) and conduct the TI verification based on the empirical data. Given a triplet (i, j, k), if the TI is fulfilled, \(d_{ik}^{*}+d_{kj}^{*}\ge d_{ij}^{*}\) and thus \(H_{ikj}=d_{ik}^{*} +d_{kj}^{*} - d_{ij}^{*}\ge 0\). For 57 stock market indices, we have a total number of \(C_{57}^{3} \times 3\) = 87,780 triplets for testing \(H_{ikj}\). Figure 14 shows the histogram of {\(H_{ikj}\)} and its descriptive statistics for 57 stock market indices during the entire period from 2005 to 2014. The minimum of {\(H_{ikj}\)} is 0.6821, meaning all \(H_{ikj}\ge 0\) and that \(d_{ij}^{*}\) fulfills the third axiom. Figure 15 shows that the minimums of {\(H_{ikj}^{t}\)} for dynamic MST-Partial networks using rolling windows are all larger than 0.3, which further confirms that \(d_{ij}^{*}\) is a Euclidean distance based on our empirical data.
An extreme example of the difference between these two networks is that in the MST-Pearson network DEU and POL are indirectly linked by FRA, and POL links to CZE and HUN together, but in the MST-Partial network DEU–FRA and POL–CZE–HUN are detached and mediated by several remote markets. We find that the Pearson and partial correlation coefficients between FRA and POL are 0.75 and \(-0.04\) and those between FRA and DEU are 0.95 and 0.42. This suggests (i) that the “net” correlation between FRA and DEU is far stronger than that between FRA and POL and this is why FRA–DEU stays in the MST-Partial network, and (ii) that the original Pearson correlation between FRA and POL is an “apparent” correlation resulting primarily from correlations with other markets. We thank a reviewer for pointing out this example.
Our estimated power-law exponent of the MST-Pearson network is similar to the result obtained by Górski et al. (2008) who investigate cumulative distribution furcations (CDFs) of the node degree of MST-Pearson networks for a comparable amount of world-currencies exchange rates based on different numeraires, but note that there is a difference in unity for the estimated exponents between the CDF and the probability distribution function (PDF), see, e.g., Clauset et al. (2009) and Wang et al. (2013a).
For example, Rizvi et al. (2015) use wavelet analysis to investigate co-movements and the contagion behavior in Islamic and conventional equity markets across the USA and Asia-Pacific. They find that the recent US subprime mortgage crisis is a fundamental-based contagion and that there is an increasing co-movement (i.e., a wavelet coherency) during most crises.
For example, when computing the correlation between DEU and USA (a case of European-African and American), we do not need to lag the time series of DEU but only that of the USA because the closing price of the USA market on day \(t-1\) can affect the closing price of the DEU market on day t. In contrast, when computing the correlation between DEU and CHN (a case of European-African and Asian-Pacific), we do not need to lag the time series of CHN but only that of DEU. These lag operations for the time series of DEU are contradictory and inconsistent.
All original time series are lagged by 1 day.
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Acknowledgments
We are grateful to the Editor (Hans M. Amman) and four anonymous referees for their insightful comments and suggestions. The work was supported by the National Natural Science Foundation of China (Grant Nos. 71501066 and 71373072), the China Scholarship Council (Grant No. 201506135022), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20130161110031), and the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 71521061).
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Appendix
Appendix
See Figs. 14 and 15 for the triangle inequality verification of the partial correlation-based distance metric \(d_{ij}^{*}\).
Histogram of \(\{H_{ikj}\}\) and its descriptive statics for 57 stock market indices during the period from 2005 to 2014
Minimums of \(\{H_{ikj}^{t}\}\) for dynamic MST-Partial networks of 57 stock market indices using rolling windows
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Wang, GJ., Xie, C. & Stanley, H.E. Correlation Structure and Evolution of World Stock Markets: Evidence from Pearson and Partial Correlation-Based Networks. Comput Econ 51, 607–635 (2018). https://doi.org/10.1007/s10614-016-9627-7
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DOI: https://doi.org/10.1007/s10614-016-9627-7