4.2. Network Construction
We built a standard vector autoregression model (VAR model) on 30 return series
. Based on the VAR model, according to final prediction error and Akaike information criterion, the lag length for Granger causality test was set to be 2. Next, we conducted pair-wise Granger causality test in software Eviews to obtain F statistics for each pair of
and
, where
i and
j each denotes a stock index from our sample. Granger causality test results are shown in
Table S2. Then we were able to build a Granger causality square matrix
. In
Table S2, the first column is a list of
and the first row is a list of
; numbers in
Table S2 represent the F statistics results of the pair-wise Granger causality test under the null hypothesis that
did not Granger cause
. Higher F statistics indicated smaller
p value of the test, which led to stronger Granger causality relation between
and
.
In Panel A of
Table S2, vertically, 17 stock indices have fairly large values of F statistics relating to AXJO, HSI, N225 and TWII, showing that AXJO, HSI, N225 and TWII were strongly Granger caused by these 17 stock indices; 21 stock indices, including AEX, BFX, BVSP, DE30, DJI, ES35, FCHI, GSPTSE, JTOPI, MXX, NDX, OMXS30, OMXC20, PSI20, S&P500, STOXX50, SWI20, TA35, TASI, WIG20 and XU100, had notably small values of F statistics, indicating that the daily lognormal returns of these 21 stock indices were not significantly Granger caused by returns in other markets. We will use “influence” and “impact” interchangeably with “Granger cause” in the following study. It should be noted that these deduced “influence” and “impact” reflect Granger causality relations and do not necessarily prove causal relations per se. The majority values of F statistics were small, indicating that there were no significant causal relations among these stock indices. Next, we analyzed Panel A horizontally. DJI, GSPTSE, NDX and S&P500 had notably large values of F statistics relating to at least 10 stock indices, implying that DJI, GSPTSE, NDX and S&P500 probably had strong influence on the daily returns in a significant number of global markets; 13 stock indices, including AXJO, HSI, JTOPI, N225, NSE20, NSEI, OMXC20, OMXS30, SSEC, TWII, VNI30, WIG20 and XU100, had significantly small values of F statistics associated with other indices, implying that their daily returns cannot greatly impact returns in other markets.
Next, we repeated the above procedures to obtain Granger causality square matrices
and
for VaR series and ES series and show the results in Panel B and Panel C of
Table S2, respectively. With regards to
in panel B, vertically, BSESN and NSEI had significantly large values of F statistics relating to other stock indices, implying that daily returns of these 2 indices were probably strongly influenced by other markets. It is worth noting that the large values of F statistics were clearly smaller than the large values of F statistics of
. Out of 30 stock indices, 18, including AEX, AXJO, BVSP, DJI, HSI, JTOPI, NDX, OMXC20, OMXS30, PSI20, S&P500, SSEC, SWI20, TA35, TASI, VNI30, WIG20 and XU100, had relatively small values for F statistics. In many cases the numbers were less than 1, indicating that there were essentially no causal relations between them. In regard to
, F statistics notably had the smallest values. In Panel C, vertically, AXJO and BSESN are 2 indices that had relatively large F statistics associating with other indices.
Lastly, we inputted matrices
and
into software Ucinet to obtain 3 networks
,
and
. Results are shown in
Figure 4.
4.3. Network Analysis
(1) Centrality Analysis
We illustrate 6 major network measures of
,
and
in
Figure 5. Specifically, we are interested in analyzing out-degree and in-degree of each index, because out-degree, out-to-local-degree and out-beta centrality give us the same order of nodes, and it is the same for the 3 types of in degrees.
In regard to network , firstly we analyzed out-degree. S&P 500 was placed at the top of the ranking of out-degree with a value of 64.31. The top 2–5 stock indices were DJI (57.242), NDX (56.25), GSPTSE (42.445) and AEX (33.14). Several European stock indices also have above average out-degrees: STOXX50 (33.04), DE30 (32.92), FCHI (30.943), BFX (28.152) and OMXS30 (25.094). The stock index that has the smallest out-degree is VNI (0.974), which implies that the Vietnamese stock market has less than 1 edge associated with other nodes in the network. Other nodes with notable small out-degrees were NSE20 (2.605), TWII (2.887), N225 (2.912), AXJO (3.624), HSI (3.934), NSEI (5.142) and XU100 (6.568), most of which are Asian markets. It is worth noting that OMXC20 had an out-degree of 12.283, which was considerably less than the average out-degree of the network and was in sharp contrast to OMXS30 (25.094). In terms of in-degree, the top ones in the ranking were Asian markets: N225 (120.666), AXJO (102.665), TWII (83.148), HSI (56.492), NSEI (32.089) and BSESN (30.696). Markets with smaller in-degrees were mostly from emerging markets such as MXX (2.272), BVSP (3.402), XU100(3.873), ES35(4.932), TA35(5.682) and TASI (5.724). European markets generally had below average or average level in-degrees. Based on these observations, we can conclude that regarding stock index return, the three major indices in the United States have the most powerful contagion capabilities on other stock markets in the world, and western European markets also have considerable influence around the globe. In contrast, Asian markets generally have the least influence on world markets but are the most easily influenced by other nodes in the network.
Nodes in network with higher out-degrees were DJI (13.944), JTOPI (13.037), S&P500 (11.004), AEX (10.988) and WIG20 (7.736), while nodes with smaller out-degrees were VNI30 (1.204), ES35 (1.224), DE30 (1.51), N225 (1.391) and OMXS30 (1.824). European markets mostly had out-degrees of 2–4. In contrast to the Dow Jones Industrial Average and Standard & Poor 500 that have high out-degrees, NASDAQ had a less than average out-degree of 3.962. On the other hand, BSESN (20.954) and NSEI (18.195) had the largest in-degrees, followed by DE30 (9.552), GSPTSE (8.938), MXX (7.63) and BFX (7.509). The top 5 stock indices with the smallest in-degrees were PSI20 (0.383), HIS (0.506), TASI (1.082), NDX (1.356) and AEX (1.809), followed closely by XU100 (1.841). These results were distinctly different from results drawn from network . We cannot simply classify our stock indices based on geographical regions or traditional reputations. Dow Jones Industrial Average and Standard & Poor still have the greatest impact on other stocks and have more levels of resistance to being affected by others; meanwhile, out-degree and in-degree of NASDAQ do not align with them, and this contradicts the previous results.
In regard to network , BSESN had high values in all 6 network measures. AEX, JTOPI, SWI20 and WIG20 had large out-degrees, out-to-local degrees and out-beta centrality measures. AXJO is the only index that had significantly large values of in-degree. In comparison, XU100 was the opposite of BSESN and had small numbers across all 6 measures. Other stock indices that were significantly small in centrality measures are HSI, MXX, N225, NDX, NSE20, NSEI, OMXC20, OMXS30, SSEC, TA35, TASI, TWII and VNI30, among which HSI, MXX, N225, NSE20, NSEI, OMXS30, SSEC, TA35, TWII and VNI30 had significantly small out-degrees, out-to-local degrees and out-beta centrality measures, while NDX, OMXC20 and TASI had significantly small in-degrees, in-to-local degrees and in beta centrality measures.
(2) Hierarchical Clustering
For all 3 networks we found that the number of hierarchical clusters was 29. Tree diagrams are presented in
Figure 6 and measures of cluster adequacy are presented in
Figure 7. The measures of cluster adequacies capture the goodness of fit of the partitions. The routine takes a proximity matrix and a partition matrix where the clusters are defined in the matrix columns. There were 4 measures that we took particular interest in: ETA, Newman and Girvan’s modularity Q, Q-Prime and Krackhardt and Stern’s E-I. In terms of ETA, at first the curves for all 3 networks gradually dropped to −0.2 from 0. After the first 20 stock indices, 3 curves of ETA dropped to −0.3 through very different routes. Since ETA denotes the correlation between the input matrix and an ideal structure matrix, it seems that all nodes were within 30% deviation from their “perfect” position or structure. In terms of indicator Q, the curve of
started from −0.017 and dropped to −0.088 at some point, but eventually bounced back to −0.039; the curve of
started from −0.03 and went through a similar passage as
, lowered to −0.058 but eventually adjusted back to −0.024; the curve of
started from −0.04, the most negative value of all, and ended up at −0.15. Since Q represents the percentage of edges that fell within the partition less than the expected fraction when the edges were randomly distributed, we may conclude that for all 3 networks, based on Q, at least 2/3 of nodes extended edges just as “expected”. Q-Prime, a normalized version of Q, led us to the same conclusion as before, except that in the graph of Q-Prime,
did not have any bounce-backs and plunged directly to −0.3. Lastly, E-I index is the difference between external and internal edges, divided by the total number of edges. In terms of E-I,
,
and
, all started from 1 and stayed close to 1 for the first half of nodes, but later quickly dived to 0.502, −0.258 and 0.367, respectively.
(3) Roles and Structures
We base our analysis of roles and structures on the Concor method. Regarding network
, we discovered that the maximum depth of splits (not blocks) was 7, convergence criteria was set to be 0.2 and maximum iteration was set to be 25. The resulting structure had a R
2 of 0.961. We called the top governing layer that encompasses all 30 nodes layer#1, and the most basic layer of units layer#7. On layer#2, we had 4 large groups: the first group consisted of AEX, SWI20, BFX, FCHI, STOXX50, DE30, ES35, PSI20, TA35, MXX, BVSP, DJI, S&P500, NDX, GSPTSE and TASI, almost all of which were from Europe and North America; the second consisted of JTOPI, OMXC20, OMXS30, XU100 and WIG20; the third consisted of N225, AXJO, TWII, HIS, SSEC and VNI30, all of which were from Asia; lastly, the group that had the smallest number of nodes included NSEI, BSESN and NSE20. Regarding network
, we found that the maximum depth of splits was 6. Convergence criteria and maximum iteration were set to be 0.2 and 25, respectively. For
, layer#2 had 4 groups: the first group was composed of AEX, SWI20, PSI20, DJI, AXJO, XU100, BVSP, NSE20, JTOPI, WIG20 and TA35; the second group included HSI, SSEC, NDX, TASI and OMXC20; the third group consisted of BFX, MXX, GSPTSE, BSESN, NSEI and VNI30; and the last group had DE30, OMXS30, ES35, FCHI, STOXX50, TWII and N225. Geographical and economical groups and allies were seemingly irrelevant in the structure partitioning for
, nor were other political characteristics of these stock markets. Lastly, for network
, the maximum depth of splits was 7 and other parameters were set the same as
and
. As a result, R
2 = 0.721. We still had 4 groups in layer#2 of
. The first group had the greatest number of nodes: AEX, SWI20, PSI20, BVSP, JTOPI, GSPTSE, WIG20, S&P500, DJI, NDX, BFX, STOXX50, FCHI and TA35; the second group consisted of OMXC20, TASI, SSEC and XU100; the third group consisted of AXJO, BSESN, MXX, NSE20, TWII, HIS, VNI30 and the last group consisted of OMXS30, DE30, ES35, NSEI and N225. In this case, we can see that geographical factors influenced structure partitioning in the most basic layers, especially layer#6 and layer#7, such as the following units: S&P500 and DJI, STOXX50 and FCHI, TWII and HSI, and OMXS30 and DE30. We conducted structure analysis again but using the Euclidean method as a comparison to results based on the Concor method. Evidently, results based on Euclidean method had many irrelevant and inaccurate details, because using correlation to partition the stock indices is more adequate practice compared to using distance to partition. Results are shown in
Figure 8.
(4) Core and Periphery
When conducting core and peripheral analysis, number of iterations was set to be 50 and population size was set to be 100. For each network, nodes were divided into core class and peripheral. Blocked adjacency matrices were calculated and the resulting
density matrices are shown in
Table 2. Final fitness values for matrices
,
,
were 0.438, 0.435 and 0.557, respectively. In theory, when the value of
density is higher, the class is “heavier” or “meatier”. Core-to-core
density of
was 56.627, the highest among all results. Core-to-core densities of
and
were 12.794 and 20.498, respectively, indicating that the core class of
was moderately dense while the core class of
can be considered sparse. Peripheral-to-peripheral
density of
was 8.094, which was not too much smaller than the core-to-core
density of
. The peripheral blocks in both
and
were remarkably sparse, since their peripheral-to-peripheral
density values were 2.964 and 2.657, respectively. In terms of connections between core and peripheral classes,
had comparatively connected core class and peripheral class, in that the peripheral-to-core and core-to-peripheral densities were 29.120 and 15.683 each. The obvious difference in values of peripheral-to-core and core-to-peripheral densities implies that nodes in peripheral class had more directed connections to core class than core class to peripheral class. This pattern was also true for
and
, since peripheral-to-core densities for these matrices were 4.862 and 7.498, while core-to-peripheral densities were 3.128 and 4.288.
(5) Correlation Analysis
We present the correlation matrix of network centrality measures in
Figure 9. Out-degree and in-degree of nodes in
are denoted as
and
. Similarly,
,
and
represent out-degree and in-degree of nodes in
and
, respectively. We conducted correlation analysis on 3 networks. Correlation between
and
was 0.49, the largest among all indicators. The second largest correlation was between
and
—0.44. Correlations between
and
,
and
, and
and
were 0.25, 0.17 and 0.15, respectively, which also indicated some perceptible correlations. Correlations less than 0.1 were considered too small in our study. The most negative correlation was −0.31, which was between
and
. The second most negative correlation was −0.16, which was between
and
.
(6) QAP Correlation Analysis
We conducted Quadratic Assignment Procedure (QAP) correlation analysis in this section. We used QAP correlation to further analyze similarities and differences among matrices
,
and
. We carried out the calculation process in Ucinet to evaluate whether we can predict the structure of one matrix based on the other matrix. Results are shown in
Table 3. The QAP correlation analysis showed a significant correlation between
and
(
r = 0.597,
p = 0.000), which was certainly expected because VaR and ES are inherently correlated; correlation between
and
is also worth highlighting (
r = 0.151,
p = 0.047); correlation between
and
had the smallest r value and largest
p value (
r = 0.105,
p = 0.107), which could be considered an insignificant result because the
p value was greater than 10%. Our results seem to suggest that, based on a Pearson correlation value of nearly 0.6 and a
p value of 0, there was a significant intercorrelation between the VaR Granger causality network and ES Granger causality network for the selected 30 stock indices during the last decade. The correlation between the return Granger causality network and ES Granger causality network was statistically significant since
p value was less than 5%, but
r value was only 0.151 and not relatively high. The statistical significance was even smaller for the VaR Granger causality network since the
p value exceeded 10%.
4.4. Impulse Response Analysis
Next, we will examine more in-depth financial risk contagion patterns among key players in the network. Based on centrality measurements of
, we found that PSI20 had the lowest in-degree, in-to-local degree and in-beta (0.383, 1648.713, 0.29, respectively) and BSESN had the highest in-measures (20.954, 78,923.57, 16.652, respectively). Examining Granger causality matrix
we detected that JTOPI, GSPTSE, DJI and AXJO had the strongest Granger causal relations with BSESN; on the other hand, DE30, ES35, TASI and WIG20 had zero Granger causal relations with PSI20. Based on these observations, we performed impulse response analysis on them because they were the most likely to give us deeper insights into risk transmission patterns between highly risk-sensitive stock indices, as well as highly risk-proof stock indices. To perform the analysis, we built VAR models on the variables first, lag length was found to be 2, and then calculated response standard error using the Monte Carlo method, with repetitions set as 100. We based our impulse response function on Cholesky decomposition. Results are shown in
Figure 10.
First, we observed the responses of VaR of BSESN and PSI20 for a period of 1000 trading days. We denote VaR of stock indices as and trading day as . It is clear that the response of to exceeded 0.04% at around and continued to grow till around , and then steadily declined to around 0.03% on . The impulse response curve eventually touched 0.00% at about , indicating a fairly long-lasting effect of the initial shock. The response of to was developed in a similar manner, except that after the initial impulse the curve instantly elevated to about 0.015% and then peaked at around 0.03% at about before falling to almost 0.00% at the end of the observation period. Response curves of to and to both started from about 0.005% and exceeded well above 0.03%, but the former decreased to 0.00% at around , whereas the latter steadily lowered down but never touched 0.00% by the end of the observation period. Although responses varied, we can still summarize that 1 unit of impulse from the selected 4 stock indices typically led to a 0.03–0.04% increase at peak level in . The elevation in was non-negligible and long-lasting, validating research result from network analysis.
As a comparison, we conducted impulse response analysis on
. We generated impulses from
,
,
and
in turn and demonstrate results in the lower half of
Figure 10. It is evident that
did not respond almost at all to
and
. The response of
to
was the strongest as the curve instantly jumped to 0.02% after the initial shock, and then steadily descended to 0.00%. The response of
to
was a little bit less than 0.01% originally, before almost uniformly declining to 0.00% by the end of the period. So far, we have detected the striking differences between a highly risk-sensitive stock index (BSESN) and a highly risk-insensitive or risk-proof one (PSI20). For the former, the shock curve gradually ascended to 0.03–0.04% before smoothly returning to 0.00%, but for the latter, either the curve started and peaked at around 0.01% and then fell through a linear route to 0.00%, or the curve showed little to no response to the original trigger. We have also discovered that even if Granger causality test indicates that there is no causal relation between 2 variables, such as
and
or
and
, VaR modelling and impulse response analysis reveal that there could still be some response in the target stock index, though the response was considerably mild.
We carried out impulse response analysis on selected ES series. Based on centrality measurements of
, we found that XU100 had the lowest in-degree, in-to-local degree and in-beta (0.601, 1932.666, 0.307, respectively) and BSESN had the highest in-measures (22.714, 69,889.42, 21.769, respectively). Examining Granger causality matrix
we detected that JTOPI, GSPTSE, BVSP and WIG20 had the strongest Granger causal relations with BSESN; on the other hand, DJI, GSPTSE and NDX had zero Granger causal relations with PSI20, and AEX, FCHI and BFX had F statistics of 0.01. Based on these observations, we performed impulse response analysis on them in a similar way as before and present the results in
Figure 11.
First, we observed the responses of and for a period of 1000 trading days. It is clear that the response of to and that of to were comparatively similar; both overpassed 0.08% at around and both fell back to 0.00% at around , except that the former went beyond 0.00%. The response of to rose to as high as 0.1% just before , before sliding back to 0.00% at around . The curve dipped in the negative realm, but eventually adjusted back to zero. The smallest peak value was about 0.07% and was from to . The curve slowly fell back to 0.00% by the end of the period.
Responses of to 6 chosen stock indices were developed in almost identical style. After the initial instigation of 1 unit of impulse, was instantly lifted by 0.022–0.025%, and then retreated to 0.00% steadily in an almost linear passage. Moreover, both bounds of standard deviation in 6 subplots had nearly identical shapes. The distance between upper and lower bounds became wider consistently.
We arrived at the same conclusion regarding the difference in curve patterns of highly risk-sensitive stock index and highly risk-proof index. Moreover, compared to
Figure 10, response rates shown in
Figure 11 were markedly higher: 0.07–0.08% compared to 0.03–0.04%, and 0.022–0.025% compared to 0.01–0.02%. This finding also confirms our core-peripheral analysis that showed that overall
was much more densely formed, indicating that connections between nodes in
were stronger. Furthermore, we have demonstrated that when we base our calculation on different risk indicators, we can still arrive at the same conclusion about which index can cause the greatest response in another index, such as the impact of JTOPI on BSESN. However, when we want to follow the ranking and select a few more influential stock indices, we may find that the picks are different according to different risk indicators.