# Information Flow between Bitcoin and Other Investment Assets

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

#### 2.2. Granger Causality

#### 2.3. Transfer Entropy

## 3. Results

#### 3.1. Granger Causality Test

#### 3.2. Normality Test

#### 3.3. Transfer Entropy

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Transfer entropy by histogram analysis; and (

**b**) Transfer entropy by STSA ($S=5$). We display the information flow between the assets. The arrow represents the causal link, and the number indicates the estimated value of transfer entropy (the value of effective transfer entropy is also denoted in parentheses). The arrows are colored differently for each value and statistical significance. The significance level was evaluated by bootstrapping the underlying Markov process [31,34]. * and *** indicate the significance at the 10% and 1% levels, respectively.

Min. | Max. | Mean | Std. | Skewness | Kurtosis | |
---|---|---|---|---|---|---|

Bitcoin | $-0.27$ | $0.25$ | $8.96\times {10}^{-4}$ | $3.85\times {10}^{-2}$ | $-0.24$ | $6.25$ |

S&P 500 | $-0.04$ | $0.05$ | $2.50\times {10}^{-4}$ | $8.35\times {10}^{-3}$ | $-0.50$ | $3.79$ |

Gold | $-0.03$ | $0.04$ | $5.66\times {10}^{-5}$ | $8.75\times {10}^{-3}$ | $0.15$ | $1.83$ |

USD/EUR | $-0.03$ | $0.03$ | $-1.35\times {10}^{-4}$ | $5.33\times {10}^{-3}$ | $0.11$ | $2.52$ |

Null Hypothesis (H_{0}) | F-Statistics (p = 1) |
---|---|

Gold ↛ Bitcoin Bitcoin ↛ Gold | 0.04 2.83 * |

S&P 500 ↛ Bitcoin Bitcoin ↛ S&P 500 | 2.88 * 3.24 * |

USD/EUR ↛ Bitcoin Bitcoin ↛ USD/EUR | 0.84 0.08 |

**Table 3.**Jarque–Bera, skewness, and kurtosis tests on the residuals of the bivariate VAR($p$) model.

${\mathit{H}}_{0}:\mathbf{Residuals}\mathbf{Are}\mathbf{Normally}\mathbf{Distributed}$ | |||
---|---|---|---|

Jarque–Bera | Skewness | Kurtosis | |

${M}_{B,G}$ | $2.49\times {10}^{3}$ *** | $7.43$ * | $2.48\times {10}^{3}$ *** |

${M}_{B,S}$ | $2.81\times {10}^{3}$ *** | $44.19$ *** | $2.77\times {10}^{3}$ *** |

${M}_{B,U}$ | $2.64\times {10}^{3}$ *** | $2.96$ | $2.64\times {10}^{3}$ *** |

^{2}statistics. * and *** indicate significance at the 10% and 1% levels, respectively. ${M}_{X,Y}$ is the residuals of the bivariate VAR($p$) model between the two asset returns, such as $X$ and $Y$, where $B$, $G$, $S$, and $U$ represent Bitcoin, gold, S&P 500, and USD/EUR, respectively.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Jang, S.M.; Yi, E.; Kim, W.C.; Ahn, K. Information Flow between Bitcoin and Other Investment Assets. *Entropy* **2019**, *21*, 1116.
https://doi.org/10.3390/e21111116

**AMA Style**

Jang SM, Yi E, Kim WC, Ahn K. Information Flow between Bitcoin and Other Investment Assets. *Entropy*. 2019; 21(11):1116.
https://doi.org/10.3390/e21111116

**Chicago/Turabian Style**

Jang, Sung Min, Eojin Yi, Woo Chang Kim, and Kwangwon Ahn. 2019. "Information Flow between Bitcoin and Other Investment Assets" *Entropy* 21, no. 11: 1116.
https://doi.org/10.3390/e21111116