1. Introduction
Many countries have equity markets. The overall performance of these markets is typically summarized by stock market indices. Economic globalization has interconnected financial markets of different countries. Market movements, and economic and financial news generated or associated with a specific market are almost immediately transmitted to the other markets by professional information providers, media, and social media, making the global financial system highly interconnected. The influence of foreign investment on emerging countries has been investigated thoroughly in [
1], and it has been shown that emerging and mature markets are much more integrated today than in the past. The influence among Pacific Rim countries has been explored in [
2]. Moreover, it is well known that China and Japan hold huge investments in Brazil. As a result of these interconnections, the log-return dynamics of market indices are correlated among them and fluxes of information are present between market indices. Examples of quantitative estimation of correlations observed between market indices can be found in [
3,
4,
5,
6,
7,
8,
9].
A few years after the first studies on correlation of market indices [
3,
4,
5], other studies investigated information flow between these indices by using transfer entropy as a quantitative indicator of the flux. In [
10], a causality analysis has shown that higher foreign investment flows can lead to the development of the domestic banking system. Transfer entropy [
11] was introduced by Schreiber in 2000, and the first application to finance this information-theoretical measure distinguishing driving and responding elements in subsystems was proposed in [
12]. Other investigations of transfer entropy between pairs of market indices or sector indices have been published in [
13,
14,
15,
16,
17,
18,
19,
20,
21]. The investigation of information flows in complex systems has also considered the role of global variables in the process of information flow. In a system with several agents, the so-called global transfer entropy (GTE) averages the information flow to a given agent from all other agents [
20,
22].
Transfer entropy [
22] has deep relations with Granger causality [
23]. In fact, the two quantities are strictly proportional when the underlying stochastic processes are Gaussian [
24]. Granger causality has also been used to detect directed networks of elements starting from the analysis of their time series. A first influential paper extracting a directed network from the statistical analysis of a financial multivariate time series on the basis of Granger causality tests was reference [
25]. Since then, Granger causality has also been used to investigate Granger causation between pairs of stock market indices [
26,
27,
28].
Pairwise interdependencies may not fully describe the complexity of information flows. Higher-order multivariate effects may play a role in the global financial systems. Information theory, which is commonly used to quantify interactions between two variables, has recently been extended to provide multivariate information measures that have been increasingly used to investigate interactions between groups of three or more variables, often with an emphasis on the so-called synergistic and redundant interactions. While bivariate information measures are commonly agreed upon, the multivariate information measures are still the topic of active research, see, e.g., in [
22]. In this context, redundancy and synergy are intuitive yet elusive concepts, with definitions that range from the purely operative to the most conceptual ones, see in [
29,
30] and the references therein. When analyzing interactions in multivariate time series, redundancy may arise if some channels are all influenced by another signal which is not included in the regression; another source of redundancy may be the emergence of synchronization in a subgroup of variables, without the need of an external influence. Redundancy manifests itself through a high degree of correlation in a group of variables, both for instantaneous and lagged influences. A complementary concept to redundancy is synergy, corresponding to variables which provide more information when treated jointly [
31]. Redundancy and synergy have been further connected to information transfer in [
32], where an expansion of the information flow has been proposed, and in the partial information decomposition (PID) introduced in [
33], where a multiscale approach has been proposed. Recently, it has been demonstrated that the physical quantity that reveals an upcoming transition, at least in the two dimensional Ising model, is actually the synergy, which peaks in the disordered phase, while the redundancy reaches its maximum at the critical temperature [
34]. It is remarkable that while the evaluation of GTE requires the knowledge of all the variables, PID can be applied to just three variables: a triplet of interacting variables can thus be seen as a proxy of the critical behavior of the whole system. The PID framework has been applied to the cardiovascular and respiratory systems in [
35], to fMRI data in [
36], and to the analysis of the muscular system in [
37].
The purpose of this work is the application of PID to a set of financial market indices in order to characterize the market dynamics in terms of the level of predictability of each given market exploiting three-body interactions and the corresponding synergy, so as to go beyond the description of the causality pattern based only on pairwise influences. Specifically, we analyze the information flow among market indices of financial markets located in Europe, America, and Asia. We focus on the information flow originating in the European and American markets and impacting on Asian financial markets.
2. Data
We consider seventeen stock market indices that belong to three groups: 4 indices of American stock markets (labeled as AM), 7 indices of European stock markets (labeled as EU), and 6 indices of Asian stock markets (labeled as AS). In particular, the indices of American stock markets are S&P500, Russell 2000, Ibovespa index (Brazil), and TSX Composite Index (Toronto). The indices of European stock markets are German DAX (Frankfurt), FTSE 100 (London), CAC40 (Paris), BFX (Brussels), Italian FTSE MIB, Spanish IBEX3,5 and Swiss Market Index SMI. The third group consists of stock market indices Nikkei 225 (Tokyo), Hang Seng HSI (Hong Kong), SSE Composite Index (Shanghai), KOSPI 200 Composite Stock Price Index (Korea), BSE Sensex Index (Bombay), and FTSE Straits Times Index (Singapore). Daily openings and closing prices from 1 January 2000 to 31 December 2019 have been collected from Quandl [
38] and Yahoo Finance [
39]. During the investigated years, several financial crises occurred. It is worth mentioning (i) the crash of the dotcom bubble, whose bubble burst lasted from March 2000 to October 2002, with effects until the beginning of 2003; (ii) the Global Financial Crisis of 2007–2009, which had such a global impact as it spread over most of the countries like an unstoppable domino; (iii) the European sovereign debt crisis, started in correspondence of the August 2011 stock markets fall, when the European stock markets suffered heavy losses due to fears about the world economic outlook; and (iv) the Chinese stock markets turbulence in 2015–2016. As so many events occurred, each with its own peculiarities, we decide to adopt a window approach, selecting non-overlapping windows. Varying the width of the time windows, we realize that the synergistic information flow appears to be localized in time rather than being a continuous exchange of information. However, application of PID requires a suitable number of samples, therefore a proper localization of the events (when synergistic dependencies occur) is unfeasible; indeed, in order to have statistical reliability of results, the window cannot be too small. In this paper, we show the results for windows corresponding to one calendar year, a conventional and easily interpretable duration, and leave to further research the development of methods to deal locally in time the issue of synergistic information flow.
Denoting
the closing price of the
i-th stock market index on day
t, daily logarithmic returns are calculated for every market index as
The same procedure is applied to the opening price for the i-th stock market index
to obtain the overnight change
We verify that both x and y variables can be treated as stationary variables by performing an Augmented Dickey-Fuller test. The property of stationarity is a necessary condition for the information theoretical analyses that we apply in this work.
In this type of study, it is very important to properly take into account the time zone effect [
26]. The selected stock markets operate in different time zones and the opening and closing times of markets differ accordingly. In order to avoid the bias due to the time zone effect, in this paper we analyze only the information flowing in circuits made of three markets, where the target belongs to the AS group and two drivers belong to AM and/or EU groups. Moreover, we concentrate on the prediction of the overnight change of asiatic markets based on the knowledge of European and American markets closing prices at the day before. This choice ensures that the target variable cannot receive information from the driving variables in the same day. Consequently, we label stock market indices of the AS group as the
time series, while markets in AM and EU groups are associated with the
time series. In other words, we study the predictive information flow in pairwise directed interactions
and triplet circuits
, where
and
are in AM or EU groups, and
is in AS group. It is worth mentioning that due to the timing of markets openings, the same analysis would not be possible for circuits with drivers in Asia and Europe and the target being American, indeed the European markets close when American markets are already open, therefore the informational character of such triplets would not be comparable with those of circuits America-Europe → Asia. Particular care has been spent to cope with the problem of missing records arising, e.g., when stock markets are closed in some countries due to national holidays. To cope with this, for each triplet of stock market indices, the samples for the estimation of causalities have been constructed taking just the days where data of all the three indexes were available as well as records of the following day.
5. Discussion and Conclusions
Use of causality analysis of stock market index returns in the description of the information flow occurring in the global financial system has received growing attention during the last years. In the present work, we provide the first study of the information flow detected among groups of three stock market indices over a period of twenty years. Our analysis is performed by investigating the so-called Synergy, an information theoretical measure that has been recently introduced to account for multivariate interaction effects in causality analysis.
The global financial system is operating worldwide in all continents. For this reason, the activity of different markets is scheduled at different time intervals due to the presence of different time zones. To investigate information flows compatible with the sequence of market activities occurring worldwide in a trading day, we consider information flow that has targets in Asian markets and driving signals in previous European and American markets.
Moreover, in the regression models we choose to focus on a specific form of information flow. We consider the driving signals as originated by the closing returns (close to close daily return) of European and American stock market indices, and we consider as target signal the subsequent overnight change of Asian stock market return (open to close daily return). To our knowledge, this is the first time this choice is adopted in a causality analysis of stock market indices. Our results show that predicting the open to index return leads to higher causality metrics with respect to those that one would obtain predicting the close to close returns, see
Figure 4. We interpret this result as an evidence that markets digest quite quickly the information flow originated in stock markets of other countries.
In addition to the Synergy investigation, we also estimated bivariate GC and GTE between driving and target indices. Concerning bivariate GC analysis, we find that the most important sources of information are the US indices SP500 and Russell 2000, whereas the most influenced Asian stock market indices are KOSPI 200, NIKKEI 225, HSI, and STI (especially from American stock market indices). For these indices, the information flow is detected for all years. The information flow of European stock market indices is less pronounced and more localized in time especially during the years of the financial crisis originated in 2007–2008 and turned out into sovereign debt crisis into 2011–2012. This years of crisis are also the years when the information flow is observed for SSE Composite Index and BSE Sensex Index. A similar temporal pattern is observed for the GTE with highest values of this metrics observed during the years 2007–2012.
Coming back to Synergy results, it is worth noting that the highest values of Synergy are observed when the two stock market index drivers involve an European and an American stock market index (see
Figure 2).
Moreover, Synergy seems more relevant when a middle size American market is involved. In fact, the highest values of the Synergy are observed when driving indices include IBOVESPA or TSX, although their influence is rather low with respect to SP500 and Russell 2000 in the bivariate GC analysis. It is well known that both China and Japan hold huge investments in Brazil, and our analysis suggests that information about the Brazil main stock market index is informative for HSI and NIKKEI 225, jointly with information from other European stock market indices.
Our results thus show that the Synergy, i.e., a proxy of higher order information flow rooted in information theory, provides details that are complementary to those obtained from the bivariate and global GC analysis, and can thus be used to get a better characterization of the global financial system.
In order to better characterize higher order dependencies of global financial market, further research will be devoted to develop methodologies capable to estimate locally in time the synergistic information flow, indeed the synergistic information flow appears to have a localized nature rather than resembling a nearly continuous exchange of information.