Abstract
We consider a networkbased framework for studying causal relationships in financial markets and demonstrate this approach by applying it to the entire U.S. stock market. Directed networks (referred to as “causal market graphs”) are constructed based on publicly available stock prices time series data during 2001–2020, using Granger causality as a measure of pairwise causal relationships between all stocks. We consider the dynamics of structural properties of the constructed network snapshots, group stocks into networkbased clusters, as well as identify the most “influential” market sectors via the PageRank algorithm. Interestingly, we observed drastic changes in the considered network characteristics in the years that corresponded to significant globalscale events, most notably, the financial crisis of 2008 and the COVID19 pandemic of 2020.
1 Introduction
Stock markets are complex interconnected systems, where various “local” factors can cause “global” changes in the behavior of the entire market. For instance, favorable or unfavorable economic conditions in certain market segments, or in certain countries, may affect other countries and industries and potentially cause positive or negative fluctuations that span the entire U.S. and international markets. The idea of describing causal relationships between different components of the market system has been addressed in several recent studies. For instance, the survey [24] discussed the concept of contagion in financial markets, which essentially implies the propagation of impact (such as risk) between different components of the market. A networkbased model is a natural way to mathematically represent these “contagion” processes; however, the principles for constructing the networks that reflect certain types of processes may vary depending on the respective goals and assumptions of a study.
A simple and intuitive technique for constructing a networkbased (or, graphical) model of the market is to represent its elements (e.g., stocks) as nodes and connect the nodes by links (arcs) based on pairwise correlations between the corresponding entities (i.e., the correlations between stock price fluctuations over a certain period of time). Such an approach was studied in [5,6,23] in the context of identifying large correlated clusters and diversified portfolios in the U.S. stock market. Although pairwise correlationbased similarity measures have merit in certain applications, a substantial drawback of such measures is in the inability to produce directed links between entities, that is, to establish the direction of “contagion” (i.e., the propagation from node
In this work, we construct and analyze a directed network model, which describes causal relationships between all pairs of stocks in the U.S. stock market using the concept of Granger causality [15,16]. It should be noted that Granger causality (which will be formally defined later in the article) can be used to determine whether the time series describing stock
There are several previous studies constructing networks based on Granger causality, such as [12,26]; however, there has been no thorough analysis of the resulting networks. Also, the current literature contains little discussion about the influence of market sectors. Further, little attention has been paid to approaches widely used in network science, such as PageRank and
As it will be discussed in the next sections, networks constructed using Granger causality appear to capture certain structural properties of the stock market that reflect overall tendencies in its behavior. In particular, we investigate various aspects of connectivity patterns and the evolution of structural properties of the constructed network snapshots. In addition, the considered network representation is used to group stocks into networkbased clusters and to identify the most “influential” market entities (sectors/industries).
2 Basic concepts, data description, network construction
2.1 Relevant graphtheoretic concepts
Let
Given a node
A directed graph
In some situations, clusters based on strongly connected components can be extremely large and comparable with the size of the whole graph (which is in fact the case for the considered graphs, as it will be shown later). Therefore, the clustering approach based on connected components may not be necessarily appropriate for drawing meaningful conclusions regarding specific groups of nodes within a graph. There is a variety of definitions for “tighter” structures that may be interpreted as clusters that have specific cohesive properties of their connectivity patterns. In this study, we utilize the concepts of kdegenerate graph and kcores for undirected graphs, introduced by [22], and modify them for the case of directed graphs. A simple undirected graph is called
An extension of the notion of a
For a given
2.2 Granger causality
In the aforementioned previous studies of the networkbased model of the U.S. stock market [5,6], the market graph was constructed in such a way that a given pair of nodes is connected by an undirected edge if the corresponding stocks exhibit a similar behavior over a certain period of time. The similarity was measured by Pearson’s correlation between the time series representing the returns of corresponding stocks. In this study, we propose a different technique for constructing the set of arcs: the similarity between stocks is measured by Granger causality [15,16], which is extensively used across many application areas because of its simplicity, robustness, and flexibility [9,13]. The details of the network construction will be presented in Section 2.3, whereas here we introduce the definition of causality and the procedure for conducting Granger causality test between two time series.
Consider two scalarvalued, stationary time series
where
To test this hypothesis, one can apply the
It should be noted that the Granger causality test is valid only if the time series are covariance (or weak) stationary. In this article, we used the Augmented DickeyFuller test [20] to check the stationarity of time series. Further, we assume homoscedasticity, i.e., constant variance of
2.3 Network construction
In the constructed directed unweighted network, the nodes are stocks represented as “ticker” symbols. We used all the stocks listed at NYSE, NASDAQ, and AMEX as of December 31, 2020: There were 7,240 stock symbols in total. The list of stock symbols was obtained from EODdata.^{[1]} We obtained historical stock prices data from Yahoo Finance using yfinance ^{[2]} Python library.
The adjusted close prices data were transformed into the time series of daily returns, since returns possess scalability property (i.e., the values in time series representing each stock returns have the same order of magnitude) and thus are easily comparable. Furthermore, the logarithms of returns were calculated, due to the fact that logreturns have more attractive statistical properties [11], including weak stationarity, which was verified for all considered time series. If
where
A directed network (referred to as a causal market graph) was constructed for each time period (calendar year) to reflect the causal relationships between stocks. It should be noted that a network constructed for each time period contains only those stocks that were present in the market during that entire time period; therefore, the cardinality and composition of the sets of nodes change from period to period. Every stock is represented by a node, and the existence of an arc
Mean  Std  min  25%  50%  75%  max  



0.001923 


0.000108  0.000621  0.046510 
The Granger causality test can be performed with different numbers of lags. In our preliminary computations, we found that in many cases the Bayesian information criterion (BIC) [21] produced the optimal quantity of one or two lags. Moreover, the corresponding
3 Dynamics of structural properties of causal market graph
To reveal the longterm evolution of causal market graph characteristics over time, we consider 20 nonoverlapping 1year periods spanning the most recent two decades. We consider the dynamics of characteristics of the causal market graph, including the number of nodes, arc density, node degrees, connectivity, and degree distribution. In addition, we compute strongly connected components,
3.1 Basic characteristics
The set of stocks traded on NASDAQ, NYSE, and AMEX has undergone significant changes during 2001–2020. As it is shown in Table 2, the number of nodes (stocks) increased from 2087 in period 1 to 7240 in period 20. The number of publicly traded stocks increased by 246% despite the fact that many companies present in the market in earlier periods ceased to exist in later periods.
Year  #Nodes  #Arcs  Max. o.d.  Max i.d.  Arc density (%)  GCC size (%)  Inin assort.  Outout assort. 

2001  2,087  30,082  176  456  0.69  95.35  −0.028  0.111 
2002  2,253  33,625  292  641  0.66  94.85  −0.048  0.115 
2003  2,352  15,230  84  153  0.28  89.71  −0.020  0.092 
2004  2,482  27,947  154  242  0.45  93.03  0.160  0.244 
2005  2,656  22,133  103  257  0.31  93.34  −0.084  0.100 
2006  2,841  33,701  128  878  0.42  94.65  −0.067  0.177 
2007  3,084  134,188  791  855  1.41  98.51  −0.028  0.053 
2008  3,418  651,729  1,649  2,591  5.58  99.44  −0.214  0.060 
2009  3,558  110,200  997  2,212  0.87  98.37  −0.069  −0.009 
2010  3,700  95,185  1,432  1,820  0.70  95.62  0.009  0.023 
2011  3,944  185,331  2,004  2,495  1.19  98.07  −0.091  −0.006 
2012  4,130  82,420  450  1,598  0.48  97.34  −0.011  0.168 
2013  4,410  93,113  598  978  0.48  96.67  −0.063  0.175 
2014  4,697  121,270  557  1,630  0.55  98.59  −0.025  0.163 
2015  5,061  224,480  999  2,120  0.88  99.19  −0.017  0.144 
2016  5,403  205,611  558  2,106  0.70  99.44  −0.034  0.173 
2017  5,720  101,240  335  1,742  0.31  99.28  −0.069  0.058 
2018  6,147  407,644  1,198  3,081  1.08  99.74  −0.017  0.172 
2019  6,701  273,246  1,101  2,417  0.61  99.69  −0.003  0.184 
2020  7,240  3,416,051  4,720  5,685  6.52  99.90  −0.118  −0.091 
(max. o.d. and max i.d. are maximum outdegree and maximum indegree, respectively; GCC size is the size of the giant connected component as the percentage of the total number of nodes; the last two columns show the respective in and outdegree assortativity).
The threshold value used to identify whether two nodes are connected controls the total number of arcs in the graph. Although the threshold specified in the previous section was chosen to be rather conservative, one can see that the number of arcs can still be large; however, it varies greatly: from 15,230 arcs in 2003 to over 3.4 million arcs in 2020. Due to the difference in the number of nodes in the networks corresponding to different time periods, it makes sense to calculate the arc density (i.e., the ratio of the number of arcs to the maximum possible number of arcs), which is a unitless measure; thus, it can be used to compare graphs with different numbers of nodes. Table 2 summarizes basic characteristics of the networks corresponding to all considered time periods.
In the case of correlationbased (undirected) graph instances constructed over a shorter timeframe, the arc density steadily increased over time [5]. However, the causal market graph does not have this property: Table 2 presents the nonmonotonic dynamics of the number of arcs and the arc density, the latter being also visualized in Figure 1. One can interpret the arc density of the causal market graph as a proportion of ordered pairs of stocks, such that the data corresponding to returns of one stock can be potentially used in order to forecast the future return values of the other. Table 2 presents two other fields related to the network structure: maximum out and indegrees. Based on the model of causality, the stocks with high outdegrees are the most “informative” in the sense that their statistics could be used for investigating the behavior of a large number of adjacent stocks (successor nodes in the causal market graph). The indegree of a node can be treated as the property reflecting the number of stocks containing unique information about this stock. Although this characteristic may be meaningful in certain contexts, in this part of the study, we concentrate mainly on outdegrees of nodes due to the aforementioned considerations.
The evolution of the density in the causal market graph is shown in Figure 1. One can observe that it has relatively small values during 2001–2006, but it starts to increase in 2007. Further, the arc density attains its highest values in 2008 and 2020. Many economists associate 2007 with the beginning of the worst financial downfall since the Great Depression (started with the U.S. subprime mortgage crisis). The most significant economic event of 2008 is the collapse of the stock market when Dow Jones and S&P500 endured their worst year since 1930. In 2009, although the US economy was still weak, the stock market started to slowly recover after hitting the bottom in March 2009. As one can see, the values of arc density fell drastically compared to 2008, and they stayed relatively stable until 2020, when COVID19 pandemic started. It can also be observed that inin assortativity has its lowest values in 2008 and 2020.
Although the analysis of these basic properties of the constructed networks may not by itself be sufficient to draw comprehensive conclusions, it can be seen that extreme values of arc density and maximum outdegree of the causal market graph correspond to extreme events in the stock market, and the trends can be noted for the transition periods as well. The different nature of events that impacted the market between 2008 and 2020 may explain the difference in the magnitude of these metrics. In particular, it can be observed that two drastic “spikes” of arc density of the causal market graph (in 2008 and 2020) appear to be inherently different: the 2008 spike was preceded by a smaller yet still significant increase of arc density in 2007 (in fact, the arc density of the 2007 graph is the third largest among the considered time periods), whereas the 2020 spike was not preceded by such an increase. The difference between the respective underlying events that affected the market in 2008 and 2020 is that the 2008 crisis was anticipated by experts based on market trends that started during 2007, but the crisis associated with the 2020 COVID19 pandemic was not anticipated during 2019.
In addition, we consider the specific nodes (stocks/companies) that are most “influential” in the sense that their time series data contain useful information about a large number of other stocks. Figure 2 presents the aggregate distribution of highest outdegree stocks by sector for all considered periods. As one may intuitively expect, the top sectors in this diagram are Funds (that corresponds to Funds, Trusts, and Tracking Stocks) and Financial Services, followed by several other important sectors of the market.
3.2 Degree distribution
As mentioned in Section 2.1, many previous studies have shown that the powerlaw distribution of out and indegrees appears to be a common property for many realworld networks. The degree distribution of most of the constructed causal market graphs also appears to follow a power law, although the quality of powerlaw fit varies between different network snapshots. Table 3 summarizes the evolution of the powerlaw parameter
Year 





2001  1.3114  0.8517  1.4595  0.8856 
2002  1.2130  0.8418  1.4457  0.8497 
2003  1.5744  0.8433  1.9149  0.8822 
2004  1.4431  0.8929  1.5107  0.8469 
2005  1.4516  0.8614  1.7908  0.8578 
2006  1.1817  0.7709  1.6346  0.8683 
2007  1.0375  0.8090  1.1608  0.8565 
2008  0.7791  0.7800  0.7257  0.6368 
2009  1.0251  0.7584  1.2947  0.8085 
2010  1.1015  0.8113  1.1790  0.7953 
2011  1.0251  0.7867  1.1766  0.8145 
2012  1.0573  0.7613  1.4694  0.8593 
2013  1.0987  0.8123  1.5229  0.8362 
2014  1.0875  0.7789  1.3902  0.8391 
2015  1.0006  0.7807  1.2824  0.8642 
2016  0.9873  0.7449  1.3971  0.7651 
2017  1.3792  0.8172  1.6743  0.7321 
2018  0.8402  0.6907  1.1444  0.7974 
2019  1.1093  0.7716  1.3782  0.8196 
2020  0.7239  0.7070  0.7314  0.7109 
Although the value of the parameter
3.3 Strongly connected components
Another interesting question concerning the causal market graph is whether it is strongly connected. If the answer is “yes,” then it would mean that each stock
Returning to Table 3, it can be seen that the parameter
3.4 Identifying cohesive clusters based on
k
outcores
Due to the presence of a giant strongly connected component discussed in the previous subsection, strongly connected components cannot be used for clustering (i.e., partitioning a graph into subgraphs according to some similarity criterion), since one cluster would contain virtually all nodes in the graph. Therefore, in this section, we focus our attention on
Recall from Section 2.1 that a
Year  Degeneracy 

Proportion (%) 

2001  6  519  24.87 
2002  7  656  29.12 
2003  2  1,754  74.57 
2004  5  172  6.93 
2005  3  1,735  65.32 
2006  5  48  1.69 
2007  17  952  30.87 
2008  92  923  27.00 
2009  9  493  13.86 
2010  9  402  10.86 
2011  17  365  9.25 
2012  7  164  3.97 
2013  9  394  8.93 
2014  9  1,415  30.13 
2015  29  173  3.42 
2016  11  2,713  50.21 
2017  7  3,774  65.98 
2018  17  2,475  40.26 
2019  19  524  7.82 
2020  256  1,797  24.82 
Taking a closer look at the
4 Identifying influential market sectors using pagerank
While a stock’s outdegree appears to be a reasonable quantitative measure of the stock’s importance, it treats all links as equal and does not take into account the difference in importance of outneighbors. The PageRank method, which was proposed in [7] for ranking webpages in Google’s search engine, is a simple yet very effective technique that overcomes this drawback. It can be applied to rank nodes in a directed network according to their importance or “centrality” expressed by a certain score. [10] describes the PageRank method as a “democracy,” with links interpreted as votes in favor of the webpages they are directed to. Each webpage can vote for other webpages, and its score is divided evenly over the set of webpages it is voting for. In the realm of a causal market graph, webpages are replaced with stocks and hyperlinks – with causality relations. In addition, we reverse the directions of arcs in the causal market graph to reflect the idea that stock
or, in the matrix form,
Hence, the problem of finding the scores reduces to computing the eigenvector of the columnstochastic matrix
In our experiments, we use PageRank to identify market sectors and industries within a given sector that are most important with respect to aggregated causal relationships. To rank the market sectors over a certain time period, we apply PageRank to the newly introduced causal market sector graph
where
where
Figure 5 shows the breakdown of most influential market sectors for each time period according to their PageRank scores. One can observe that Funds, Trusts, and Tracking Stocks is the topranked sector in all time periods except 2002, when Financial Services sector had the same PageRank score. The fact that Funds, Trusts, and Tracking Stocks is the most influential market sector is not surprising, since many stocks in this sector are by definition reflective of the behavior of the entire market. The fact that Financial Services is the secondmost influential sector in most of the considered time periods is also somewhat expected; however, it is interesting to observe that the PageRank scores of Financial Services, Industrial, and Technology sectors have decreased in the most recent years. Although the PageRankbased approach has limitations since it takes into account only the respective network topology, these observations may be worth investigating further from more traditional economics, and financebased perspectives.
5 Conclusion
In this article, we constructed a networkbased map of causal relationships in the entire U.S. stock market. The considered networkbased model of the stock market is based on publicly available stock prices data and a quantitative causality measure, which makes the model easily interpretable and reproducible. The proposed approach enables one to apply the rich arsenal of network analysis tools toward revealing market trends and investigating the properties of individual nodes and market clusters that may not be apparent otherwise. We focused on studying the basic structural properties of the causal market graph and detecting its most influential entities. The considered networkbased metrics are nonmonotonic, with an interesting observation that significant changes over time appear to coincide with globalscale events, such as COVID19 pandemic and the 2008 financial crisis. In addition, the proposed PageRankbased technique for identifying “influential” market sectors revealed interesting observations that may be worth investigating further.
In terms of other possible methods for constructing the respective networks, another potential direction of further research would be to analyze networks constructed using other connectedness computation methods such as [17,27]. It would also be of interest to consider heteroscedasticity in Granger causality and see its effect on the resulting networks. Future research may also include the investigation of the possibility of constructing a market index solely based on Granger causality metrics. The implication of the presence of powerlaw degree distribution in many of the networks is that a relatively small number of stocks have a large number of strong causal links to a large remaining portion of the market. Further, this observation suggests that the set of stocks comprising the
A limitation of this study, which may be addressed in future research, is the problem of multiple comparisons. In order to construct the edges, we do pairwise Granger causality tests between each pair of nodes. For each pairwise comparison, the employed statistical tests may result in incorrect rejection of the null hypothesis and adding a wrong edge with 0.1% chance. Even though the probability of adding a “wrong” edge is low, the networks analyzed in this article contain thousands of nodes, and considering independent tests, these networks may contain a few “wrong” edges. Despite the fact that these potential effects cannot be completely ruled out, the results presented in the article networks still contain interesting properties, such as the presence of powerlaw degree distributions, patterns of arc density changes corresponding to financial crises, and other observations, which are unlikely to appear solely due to statistical anomalies.
The considered approaches can potentially be applied in a wider variety of settings. One interesting future research direction would be to consider networks of causal relationships that span stock markets of multiple countries. Another potential area of interest would be applying these techniques to shorter time periods, possibly with smaller time increments between data points (e.g., one could consider hourly, or minutebyminute stock prices data over a time period of several days or weeks). In particular, although this article focused mainly on a descriptive rather than predictive/prescriptive analysis of stock market data, it would be interesting to see if the considered networkbased approaches (perhaps with some modifications) could be used in the context of predictive models of market trends.
Acknowledgments
Preliminary results related to this research were presented at the 7th International Conference on Computational Data and Social Networks (CSoNet 2018, Shanghai, China, December 2018) and at the 1st International Conference on Econometrics and Business Analytics (iCEBA 2021, Saint Petersburg, Russia, July 2021).

Conflict of interest: The authors declare no conflict of interest.
References
[1] Aiello, W., Chung, F., & Lu, L. (2000). A random graph model for power law graphs. Experimental Mathematics, 10, 53–66. 10.1080/10586458.2001.10504428Search in Google Scholar
[2] Aiello, W., Chung, F., & Lu, L. (2001). Random evolution in massive graphs. Annual Symposium on Foundations of Computer Science, 42, 510–521. 10.1109/SFCS.2001.959927Search in Google Scholar
[3] Barabasi, A., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286, 509–512. 10.1515/9781400841356.349Search in Google Scholar
[4] Barabasi, A., Albert, R., & Jeong, H. (1999). Scalefree characteristics of random networks: The topology of the world wide web. Physica, A 272, 173–187. 10.1016/S03784371(00)000182Search in Google Scholar
[5] Boginski, V., Butenko, S., & Pardalos, P. (2006). Mining market data: A network approach. Computers and Operations Research, 33, 3171–3184. 10.1016/j.cor.2005.01.027Search in Google Scholar
[6] Boginski, V., Butenko, S., & Pardalos, P. M. (2005). Statistical analysis of financial networks. Computational Statistics and Data Analysis, 48, 431–443. 10.1016/j.csda.2004.02.004Search in Google Scholar
[7] Brin, S., & Page, L. (1998). The anatomy of a largescale hypertextual web search engine. Computer Networks, 30, 107–117. 10.1016/S01697552(98)00110XSearch in Google Scholar
[8] Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., … Wiener, J. (2000). Graph structure in the web. Computer Networks, 33, 309–321. 10.1515/9781400841356.183Search in Google Scholar
[9] Brovelli, A., Ding, M., Ledberg, A., Chen, Y., Nakamura, R., & Bressler, S. L. (2004). Beta oscillations in a largescale sensorimotor cortical network: Directional influences revealed by granger causality. Proceedings of the National Academy of Sciences of the United States of America, 101, 9849–9854. 10.1073/pnas.0308538101Search in Google Scholar PubMed PubMed Central
[10] Bryan, K., & Leise, T. (2006). The $25,000,000,000 eigenvector: The linear algebra behind Google. SIAM Review, 48, 569–581. 10.1137/050623280Search in Google Scholar
[11] Campbell, J., Lo, A., & MacKinlay, C. (1997). The econometrics of financial markets. Princeton, NJ: Princeton University Press. 10.1515/9781400830213Search in Google Scholar
[12] Corsi, F., Lillo, F., Pirino, D., & Trapin, L. (2018). Measuring the propagation of financial distress with grangercausality tail risk networks. Journal of Financial Stability, 38, 18–36. 10.1016/j.jfs.2018.06.003Search in Google Scholar
[13] Diks, C., & Panchenko, V. (2004, August). Modified HiemstraJones test for Granger noncausality. Computing in Economics and Finance, 192. Society for Computational Economics. 10.2202/15583708.1234Search in Google Scholar
[14] Giatsidis, C., Thilikos, D., & Vazirgiannis, M. (2011). Dcores: Measuring collaboration of directed graphs based on degeneracy. In IEEE 11th International Conference on Data Mining (ICDM) (pp. 201–210). 10.1109/ICDM.2011.46Search in Google Scholar
[15] Granger, C. W. J. (1969). Investigating causal relations by econometric models and crossspectral methods. Econometrica, 37, 424–438. 10.2307/1912791Search in Google Scholar
[16] Granger, C. W. J. (1980). Testing for causality: A personal viewpoint. Journal of Economic Dynamic and Control, 2, 329–352. 10.1016/01651889(80)90069XSearch in Google Scholar
[17] Hecq, A., Margaritella, L., & Smeekes, S. (2021, Nov). Granger causality testing in highdimensional VARs: A postdoubleselection procedure. Journal of Financial Econometrics, 11, nbab023. 10.1093/jjfinec/nbab023Search in Google Scholar
[18] Hull, J. (2008). Options, futures, and other derivatives (7th ed.). Lebanon, Indiana, USA: Prentice Hall. Search in Google Scholar
[19] Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Review, 45, 167–256. 10.1137/S003614450342480Search in Google Scholar
[20] Said, S. E., & Dickey, D. A. (1984). Testing for unit roots in autoregressivemoving average models of unknown order. Biometrika, 71, 599–607. 10.1093/biomet/71.3.599Search in Google Scholar
[21] Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6(2), 461–464. 10.1214/aos/1176344136Search in Google Scholar
[22] Seidman, S. B. (1983). Network structure and minimum degree. Social Networks, 5, 269–287. 10.1016/03788733(83)90028XSearch in Google Scholar
[23] Shirokikh, O., Pastukhov, G., Boginski, V., & Butenko, S. (2013). Computational study of the us stock market evolution: A rank correlationbased network model. Computational Management Science, 10(2–3), 81–103. 10.1007/s1028701201604Search in Google Scholar
[24] Sowers, R., & Giesecke, K. (2011, October 18). Contagion. SIAM News. Search in Google Scholar
[25] Tarjan, R. (1972). Depthfirst search and linear graph algorithms. SIAM Journal on Computing, 2, 146–160. 10.1109/SWAT.1971.10Search in Google Scholar
[26] Výrost, T., Lyócsa, Š., & Baumöhl, E. (2015). Granger causality stock market networks: Temporal proximity and preferential attachment. Physica A: Statistical Mechanics and its Applications, 427, 262–276. 10.1016/j.physa.2015.02.017Search in Google Scholar
[27] Wu, F., Zhang, D., & Zhang, Z. (2019). Connectedness and risk spillovers in china’s stock market: A sectoral analysis. Economic Systems, 43(3), 100718. 10.1016/j.ecosys.2019.100718Search in Google Scholar
© 2022 Oleg Shirokikh et al., published by De Gruyter
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