We analyze the system of the daily returns time series of the 300 largest capitalized stocks traded at NYSE in the time period 2001–2003. Each stock is classified according to its sector and sub-sector of economic activity. There are 12 different economic sectors of activity, within the classification of stocks of Yahoo Finance (2004) we use. The sectors are: basic material (BM, 24 stocks), consumer cyclical (CC, 22 stocks), consumer non cyclical (CN, 25 stocks), capital goods (CG, 12 stocks), conglomerates (CO, 8 stocks), energy (EN, 17 stocks), financial (FI, 53 stocks), healthcare (HE, 19 stocks), services (SE, 69 stocks), technology (TE, 34), transportation (TR, 5 stocks), and utilities (UT, 12 stocks). The sub-sectors of activity are 80. In Table S1, we provide the list of the 300 stocks, together with the associated sector and sub-sector of activity. The system is investigated in two different ways. The first analysis is performed by considering the whole period of three years under investigation. This analysis gives an overall description of the system, and takes advantage from the length of time series ( daily records), in order to keep small the statistical uncertainty associated with the partial correlation estimator given in Eq.(1). The second analysis describes the dynamics of influence over time. This is achieved by performing the PCTN and PCPG analysis for shorter time periods in a case with a sliding window approach and in another by considering non overlapping windows. For each time window we compute the partial correlation network and we study the dynamics of influence of individual stocks, as well as economic sectors and sub-sectors of activity, over time.
Stationary network analysis
The first question we shall answer is about the most influential stocks. As a proxy of influence of a stock we use the outdegree of the stock in the PCPG, i.e. the total number of directed links outgoing from in the network. As a proxy of influence of stock in the PCTN we instead use the weighted outdegree, i.e. the sum of the weights of directed links outgoing from in the network. The rationale behind the choice of using different measures of influence for the two networks lies on the distinct nature of the two networks. In the PCPG, which is a sparse network with only links, information about interactions among the elements of the system is kept in the topology of the network. Therefore using the outdegree to measure the influence of a stock is a good choice, because the outdegree only depends on the topology of the network.
On the other hand, for low values of the threshold the PCTN is a quite dense network. For example, when the total number of directed links in the PCTN of the system is 40924, which is of order . For such a dense network information about relevant interactions in the system is largely kept by link weights and therefore weights need to be taken into account for an appropriate description of network characteristics. In Fig. 2 we report indegree and outdegree of the 10 most influential stocks for both the networks, together with their economic sector of activity. Most of the top 10 influential stocks belong to the financial sector. In order to better understand the mutual influence of economic sectors, in Table 1 we list all the 12 economic sectors of activity, together with some information about their overall influence in the system. The order of economic sectors in Table 1 is according to the outdegree in the PCPG and to the weighted outdegree in the PCTN. The outdegree of a sector is defined as the total number of links in the PCPG outgoing from stocks belonging to sector and pointing to stocks belonging to other sectors of activity. We indicate this quantity as . Similarly the quantity is the indegree of sector s, i.e. the total number of links from stocks not belonging to the sector that are directed to stocks belonging to the sector . The weighted outdegree and the weighted indegree of a sector , which are used in the PCTN, are defined in a similar way by summing up weights over all links selected as indicated above. Large values of and indicate that sector is very influential in the system, while large values of and indicate that sector is strongly influenced by other economic sectors of activity. In Table 1 we also report a measure of relative influence of economic sectors based on these indicators. For a given sector , these relative influence measures are defined as: (5)
where the unweighted relative influence of a sector is used in the PCPG, and the weighted relative influence in the PCTN. The relative influence is a quantity ranging in the interval . Positive (negative) value of the relative influence of a sector indicates that the sector influences other sectors more (less) than the amount it is influenced by other sectors. Although the financial sector has the highest outdegree in the PCPG and the highest weighted outdegree in the PCTN, its relative influence is quite different in the two cases. Table 1 shows that for most sectors the sign of relative influence or is the same in both networks although the observed value can be quite different. Furthermore, the ranking of sectors according to the outdegree is different for the two networks, with only the financial sector (top) and transportation sector (bottom) ranked the same for both networks. The differences between the rankings of outdegree and between the relative influence of sectors in the networks are probably due to the different information contained in the two networks. The PCPG focuses on the influence of the stock averaged over the entire market whereas a similar constraint is not present in the PCTN. In three cases the sign of the relative influence is opposite in the two networks. This three sectors are energy, utilities and transportation suggesting that the influence of stocks of these sectors might be quite localized. However, in spite of these differences, the correlation between the relative influence values in the two networks is . This number is quite high, indicating a similar overall description of the relative influence of different sectors in the two networks.
By looking at both Fig. 2 and Table 1, it is evident that the financial sector plays a key role in the system. However such relevance could be due to some specific economic sub-sector of activity. In other words, there could be heterogeneity of behavior also inside the sector. In order to better understand the role of economic sub-sectors we analyze the partial correlation networks by merging together all the stocks belonging to the same economic sub-sector of activity in a single vertex of a new sub-sector PCPG. The result is a weighted directed network in which each vertex correspond to a specific economic sub-sector of activity, and the weight of a directed link from sub-sector to sub-sector is given by the total amount of directed links outgoing from stocks belonging to sub-sector and incoming into stocks of the sub-sector in the PCPG of stocks. In Fig. 3 we show the PCPG of economic sub-sectors. We note from the figure that there are three central sub-sectors of the financial sector in the network. They are (i) Investment services, (ii) Insurance Life and (iii) Regional Banks sub-sectors. These three sub-sectors influence many of the other sub-sectors in the network, and play a major role in the topology of the sub-sector network. It is to notice that such a prominent role of the financial sector and of some of its sub-sectors does not emerge in standard correlation analysis of stock returns at NYSE. A major difference between the economic information carried by standard correlations and the one carried by partial correlations is observed by comparing the role of economic sectors in the corresponding planar networks. The PMFG associated with standard correlations is an undirected network with links, i.e. with the same number of links observed in the directed PCPG. The total number of links bridging stocks belonging to different economic sectors is 283 in the PMFG, while this number reaches 476 in the PCPG. This fact indicates that the mutual influence of stocks according to partial correlations is not localized within economic sectors, as it is mostly for standard correlations, but it is spread over the whole partial correlation network. An even more striking difference between standard correlation and partial correlation can be observed by looking at the specific relevance of each economic sector in the planar networks. In Table 1 we report the indegree and outdegree of each economic sector in the PCPG. We note that the outdegree of the financial sector is 304, while its indegree is 4 in the PCPG. On the other hand, the degree of the financial sector is just 119 in the standard correlation PMFG. This finding shows that the influence of the financial sector in PCPG is about 3 times larger than its influence in the standard correlation PMFG. A rather opposite behavior is observed for the services sector of activity. The degree of the services sector is just second to the financial sector in both the planar networks. Its degree is 85 in the standard correlation PMFG, whereas it is 152 in the PCPG. The degree 152 of the services sector can be disaggregated in terms of indegree and outdegree in the PCPG (see Table 1). Its indegree is equal to 136, while its outdegree is just 16. This result shows that the services sector is strongly influenced by other sectors, while it is poorly influential for the whole system. This behavior is exactly the opposite than what has been observed for the financial sector, and this crucial difference between Financial and Services sectors cannot be inferred by looking at networks obtained by using standard correlation as a similarity measure. For the sake of comparison, in the next subsection we show the sub-sector network associated with the standard correlation PMFG, and we list the sector degree in the PMFG.
PCPG analysis of the 300 stocks, grouped by their corresponding sub-sector.
In this network we present how each sub-sector is affecting the other sub-sectors. The color of vertices is according to the economic sector each sub-sector belongs to. Specifically, basic materials (violet), capital goods (light green), conglomerates (orange), consumer cyclical (tan), consumer non cyclical (yellow), energy (blue), financial (green), healthcare (gray), services (cyan), technology (red), transportation (brown), and utilities (magenta). Sub-sectors with a positive relative influence according to Eq.(5) are labeled in the figure. Sub-sectors labeled with numbers are listed in Table S2. We find two main hubs in the network – the investment services and the insurance life sub-sectors. The thickness and gray level of links is proportional to the logarithm of the weight of the link.