# Synergistic Information Transfer in the Global System of Financial Markets

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## Abstract

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## 1. Introduction

## 2. Data

## 3. Methods

#### 3.1. Bivariate Granger Causality

#### 3.2. Global Granger Causality

#### 3.3. Partial Information Decomposition

## 4. Results

#### 4.1. Pairwise and Global Granger Causality

#### 4.2. Synergy

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Granger Causality at different calendar years. Each panel refers to a different Asian stock market index target. In each panel, the Granger-causing stock market indices (each one associated with a row) are divided into 2 continental groups (separated by an horizontal thick black line). The top group comprises American stock market indices and the bottom group European stock market indices. Within each group, the Granger-causing stock market indices are ordered by the GC averaged on all years.

**Figure 2.**The Synergy S for each calendar year. Each panel represents an Asian stock market index target. The 55 triplets of stock market indices in each panel are divided into 3 groups (from top to bottom separated by a black thick line): the first group includes both driving stock market indices from the American continent (6 triplets originating from 4 American stock market indices and the Asian target), the second group includes one driving stock market index from the American continent and the other from the European continent (28 triplets originating from 4 American stock market indices, 7 European stock market indices, and the Asian target), and the third group includes both driving stock market indices from the European continent (21 triplets originating from 7 European stock market indices, and the Asian target). Within each group triplets are ordered according to the average value of Synergy averaged on all years. In each panel, for the sake of simplicity the driving stock market indices are all labeled with a three letters code. The American stock market indices are labeled as

**SP5**for S&P500,

**Rus**for Russel 2000,

**Ibo**for IBOVESPA and

**Tsx**for TSX. The European stock market indices are labeled as

**Dax**for DAX,

**Bfx**for BFX,

**Mib**for FTSE MIB,

**F10**for FTSE 100,

**Cac**for CAC 40,

**Ibe**for IBEX 35, and

**Smi**for SMI.

**Figure 3.**Scatter plot of the average Synergy associated with each triplet of stock market indices averaged over all 20 time windows as a function of the number of validated windows. The color of dots is chosen according to the target stock market index as indicated in the legend box.

**Figure 4.**Average Synergy for opening or closing returns of the targets. The bar plot compares the average Synergy estimated by using overnight returns (blue bars) and closing logarithmic price returns (red bars) for the target stock market index. The average Synergy is shown for all 330 triplets as a function of the rank of each triplet. The rank is determined by considering the value of the average Synergy for the overnight returns. The same rank is used also when showing the average Synergy of the closing return.

**Table 1.**Global Granger causality on Asian stock market indices. Table shows ${G}_{i}$ (as defined in (4)) for each calendar year. For each Asian stock market index target, the GGC is computed by using the 11 American and European stock market indices investigated in this paper. The values in parenthesis represent the 5 and the 95 percentile of the GGC computed for the IAAFT surrogates. Values labeled with an asterisk are compatible with the values obtained for surrogate data. When this occurs we say that the estimation of the variable is not statistically validated in the considered time window.

SSE | KOSPI 200 | BSE | HSI | Nikkei 225 | STI | |
---|---|---|---|---|---|---|

2000 | $0.17{\phantom{\rule{0.166667em}{0ex}}}^{*}\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ | $0.83\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.21}{0.05}\right)$ | $0.76\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.17}{0.05}\right)$ | $1.50\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.21}{0.04}\right)$ | $0.86\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.05}\right)$ | $1.37\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.04}\right)$ |

2001 | $0.11{\phantom{\rule{0.166667em}{0ex}}}^{*}\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.23}{0.03}\right)$ | $1.60\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.04}\right)$ | $0.44\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.17}{0.04}\right)$ | $1.58\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.21}{0.05}\right)$ | $1.08\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.04}\right)$ | $1.75\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.04}\right)$ |

2002 | $0.14{\phantom{\rule{0.166667em}{0ex}}}^{*}\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.03}\right)$ | $1.76\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ | $0.61\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.04}\right)$ | $1.84\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.15}{0.04}\right)$ | $1.43\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ | $1.81\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.16}{0.04}\right)$ |

2003 | $0.08{\phantom{\rule{0.166667em}{0ex}}}^{*}\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.16}{0.03}\right)$ | $1.25\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ | $0.58\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.16}{0.03}\right)$ | $1.21\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.17}{0.04}\right)$ | $1.13\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.16}{0.03}\right)$ | $1.39\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.17}{0.04}\right)$ |

2004 | $0.09{\phantom{\rule{0.166667em}{0ex}}}^{*}\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ | $0.85\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ | $0.33\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.04}\right)$ | $1.12\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ | $1.31\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.16}{0.05}\right)$ | $0.85\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.03}\right)$ |

2005 | $0.08{\phantom{\rule{0.166667em}{0ex}}}^{*}\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.15}{0.04}\right)$ | $1.67\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.04}\right)$ | $0.53\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.16}{0.05}\right)$ | $1.53\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.16}{0.04}\right)$ | $1.56\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.16}{0.03}\right)$ | $1.22\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.04}\right)$ |

2006 | $0.23\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.05}\right)$ | $2.12\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.21}{0.04}\right)$ | $0.86\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.05}\right)$ | $1.76\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.22}{0.05}\right)$ | $1.95\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.05}\right)$ | $1.46\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.04}\right)$ |

2007 | $0.30\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.04}\right)$ | $1.77\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ | $0.96\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ | $1.68\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ | $1.87\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ | $2.12\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.05}\right)$ |

2008 | $0.80\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.21}{0.04}\right)$ | $2.00\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.05}\right)$ | $1.60\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.04}\right)$ | $1.52\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.04}\right)$ | $1.35\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.05}\right)$ | $1.82\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.05}\right)$ |

2009 | $1.56\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.03}\right)$ | $1.53\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.04}\right)$ | $1.00\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.16}{0.03}\right)$ | $1.18\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.04}\right)$ | $1.48\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ | $1.25\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.16}{0.05}\right)$ |

2010 | $1.11\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.17}{0.03}\right)$ | $1.53\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.15}{0.04}\right)$ | $0.53\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.17}{0.04}\right)$ | $1.11\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.15}{0.04}\right)$ | $1.50\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.16}{0.04}\right)$ | $1.56\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.17}{0.04}\right)$ |

2011 | $2.05\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.17}{0.04}\right)$ | $1.94\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.05}\right)$ | $1.36\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.04}\right)$ | $1.94\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.21}{0.04}\right)$ | $1.54\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.22}{0.06}\right)$ | $1.76\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.05}\right)$ |

2012 | $0.79\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.05}\right)$ | $1.63\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.23}{0.05}\right)$ | $0.64\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.05}\right)$ | $1.45\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.05}\right)$ | $1.49\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.23}{0.04}\right)$ | $1.00\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.05}\right)$ |

2013 | $0.60\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.04}\right)$ | $1.16\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.21}{0.06}\right)$ | $0.49\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.05}\right)$ | $1.26\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.22}{0.05}\right)$ | $0.98\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.23}{0.05}\right)$ | $1.10\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.05}\right)$ |

2014 | $0.27\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ | $1.75\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.17}{0.04}\right)$ | $1.16\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.05}\right)$ | $0.98\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.05}\right)$ | $2.06\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.22}{0.05}\right)$ | $0.82\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.05}\right)$ |

2015 | $0.44\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ | $1.20\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.17}{0.04}\right)$ | $0.66\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.03}\right)$ | $1.19\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.16}{0.04}\right)$ | $1.64\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.17}{0.05}\right)$ | $1.31\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ |

2016 | $0.46\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.22}{0.04}\right)$ | $1.32\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.04}\right)$ | $0.17{\phantom{\rule{0.166667em}{0ex}}}^{*}\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.21}{0.04}\right)$ | $0.76\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.05}\right)$ | $0.84\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.05}\right)$ | $0.83\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.04}\right)$ |

2017 | $0.50\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.06}\right)$ | $1.02\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.05}\right)$ | $0.28\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.05}\right)$ | $1.18\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.22}{0.05}\right)$ | $0.87\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.05}\right)$ | $0.98\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.05}\right)$ |

2018 | $1.77\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.05}\right)$ | $1.62\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.05}\right)$ | $0.92\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.21}{0.05}\right)$ | $1.41\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.05}\right)$ | $1.27\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ | $1.30\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.19}{0.04}\right)$ |

2019 | $0.67\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.16}{0.04}\right)$ | $0.98\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.18}{0.05}\right)$ | $0.31\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.21}{0.04}\right)$ | $0.91\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.20}{0.05}\right)$ | $1.37\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.21}{0.04}\right)$ | $0.85\phantom{\rule{0.277778em}{0ex}}\left(\genfrac{}{}{0pt}{}{0.22}{0.06}\right)$ |

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## Share and Cite

**MDPI and ACS Style**

Scagliarini, T.; Faes, L.; Marinazzo, D.; Stramaglia, S.; Mantegna, R.N. Synergistic Information Transfer in the Global System of Financial Markets. *Entropy* **2020**, *22*, 1000.
https://doi.org/10.3390/e22091000

**AMA Style**

Scagliarini T, Faes L, Marinazzo D, Stramaglia S, Mantegna RN. Synergistic Information Transfer in the Global System of Financial Markets. *Entropy*. 2020; 22(9):1000.
https://doi.org/10.3390/e22091000

**Chicago/Turabian Style**

Scagliarini, Tomas, Luca Faes, Daniele Marinazzo, Sebastiano Stramaglia, and Rosario N. Mantegna. 2020. "Synergistic Information Transfer in the Global System of Financial Markets" *Entropy* 22, no. 9: 1000.
https://doi.org/10.3390/e22091000