# What Coins Lead in the Cryptocurrency Market: Using Copula and Neural Networks Models

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Neural Network Autoregression Model

`nnetar`function in

`R`package forecast by (Hyndman et al. 2019).

#### 2.2. Copula and Directional Dependence

- For all $(u,v)\in {[0,1]}^{2}$, $C(u,0)=C(0,v)=0$ if at least one coordinate of $(u,v)$ is 0;
- $C(u,1)=u$, $C(1,v)=v$, for $u,v\in [0,1];$ and
- C is 2-increasing (see Nelsen 2006).

#### 2.3. Gaussian Copula Marginal Beta Regression

`R`package gcmr (Masarotto and Varin 2017) and choose beta marginal distribution to find the estimates of $({\beta}_{0},{\beta}_{1})$ from Gaussian marginal regression. With these estimates of $({\beta}_{0},{\beta}_{1})$ and the covariate ${v}_{t}$, we compute $\mathrm{E}\left({U}_{t}\right|{v}_{t})$ and then calculate $\mathrm{Var}\left(\mathrm{E}\left({U}_{t}\right|{v}_{t})\right)$ and $\mathrm{Var}\left({U}_{t}\right)$. With these computed values, we compute the estimation of ${\rho}_{{V}_{t}\to {U}_{t}}^{2}$ in Equation (4). In a similar way, the directional dependence ${\rho}_{{U}_{t}\to {V}_{t}}^{2}$ is obtained.

## 3. Data Analysis

#### 3.1. Data and Summary Statistics

#### 3.2. Results

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Azoff, E. Michael. 1994. Neural Network Time Series Forecasting of Financial Markets. New York: John Wiley and Sons. [Google Scholar]
- Bação, Pedro, António Duarte, Helder Sebastião, and Srdjan Redzepagic. 2018. Information Transmission Between Cryptocurrencies: Does Bitcoin Rule the Cryptocurrency World? Scientific Annals of Economics and Business 65: 97–117. [Google Scholar] [CrossRef]
- Bouri, Elie, Syed Jawad Hussain Shahzad, and David Roubaud. 2019a. Co-explosivity in the cryptocurrency market. Finance Research Letters 29: 178–83. [Google Scholar] [CrossRef]
- Bouri, Elie, Rangan Gupta, and David Roubaud. 2019b. Herding behaviour in cryptocurrencies. Finance Research Letters 29: 216–21. [Google Scholar] [CrossRef]
- Bouri, Elie, Brian Lucey, and David Roubaud. 2019c. The volatility surprise of leading cryptocurrencies: Transitory and permanent linkages. Finance Research Letters. [Google Scholar] [CrossRef]
- Breitung, Jörg, and Bertrand Candelon. 2006. Testing for short-and long-run causality: A frequency-domain approach. Journal of Econometrics 132: 363–78. [Google Scholar] [CrossRef]
- Chang, Eric C., Joseph W. Cheng, and Ajay Khorana. 2000. An examination of herd behaviour in equity markets: An international perspective. Journal of Banking & Finance 24: 1651–99. [Google Scholar]
- Cherubini, Umberto, Fabio Gobbi, Sabrina Mulinacci, and Silvia Romagnoli. 2011. Dynamic Copula Methods in Finance, 1st ed. New York: John Wiley and Sons. [Google Scholar]
- Corbet, Shaen, Andrew Meegan, Charles Larkin, Brian Lucey, and Larisa Yarovaya. 2018. Exploring the dynamic relationships between cryptocurrencies and other financial assets. Economics Letters 165: 28–34. [Google Scholar] [CrossRef]
- Diebold, Francis X., and Kamil Yilmaz. 2012. Better to give than to receive: Predictive directional measurement of volatility spillovers. International Journal of Forecasting 28: 57–66. [Google Scholar] [CrossRef]
- Diebold, Francis X., and Kamil Yilmaz. 2016. Trans-Atlantic equity volatility connectedness: U.S. and European financial institutions, 2004–2014. Journal of Financial Econometrics 14: 81–127. [Google Scholar] [CrossRef]
- Dyhrberg, Anne Haubo. 2016. Bitcoin, gold and the dollar-A GARCH volatility analysis. Financial Research Letters 16: 85–92. [Google Scholar] [CrossRef]
- Faraway, Julian, and Chris Chatfield. 1998. Time series forecasting with neural networks: A comparative study using the airline data. Applied Statistics 47: 231–50. [Google Scholar] [CrossRef]
- Ferrari, Silvia, and Francisco Cribari-Neto. 2004. Beta regression for modelling rates and proportions. Journal of Applied Statistics 31: 799–815. [Google Scholar] [CrossRef]
- Gkillas, Konstantinos, and Paraskevi Katsiampa. 2018. An Application of Extreme Value Theory to Cryptocurrencies. Economics Letters 164: 109–11. [Google Scholar] [CrossRef]
- Masarotto, Guido, and Cristiano Varin. 2017. Gaussian Copula Regression in R. Journal of Statistical Software 77: 1–26. [Google Scholar] [CrossRef]
- Guolo, Annamaria, and Cristiano Varin. 2014. Beta regression for time series analysis of bounded data, with application to Canada Google Flu Trends. The Annals of Applied Statistics 8: 74–88. [Google Scholar] [CrossRef]
- Hencic, Andrew, and Christian Gouriéroux. 2015. Noncausal Autoregressive Model in Application to Bitcoin/USD Exchange Rate. In Econometrics of Risk. Studies in computational intelligence. Cham: Springer, vol. 583, pp. 17–40. [Google Scholar]
- Hertz, John, Anders Krogh, and Richard Palmer. 1991. Introduction to the Theory of Neural Computation. Redwood City: Addison-Wesley. [Google Scholar]
- Hyndman, Rob, George Athanasopoulos, Chrisoph Bergmeir, Gabriel Caceres, Leanne Chhay, Mitchell O’Hara-Wild, Fotios Petropoulos, Slava Razbash, Earo Wang, and Farah Yasmeen. 2019. Forecast: Forecasting Functions for Time Series and Linear Models, R Package Version 8.8; Available online: http://pkg.robjhyndman.com/forecast (accessed on 5 August 2019).
- Ji, Qiang, Elie Bouri, Chi Keung Lau, and David Roubaud. 2019. Dynamic connectedness and integration in cryptocurrency markets. International Review of Financial Analysis 63: 257–72. [Google Scholar] [CrossRef]
- Jondeau, Eric, and Michael Rockinger. 2006. The copula-garch model of conditional dependencies: An international stock market application. Journal of International Money and Finance 25: 827–53. [Google Scholar] [CrossRef]
- Katsiampa, Paraskevi. 2017. Volatility estimation for Bitcoin: A comparison of GARCH models. Economics Letters 158: 3–6. [Google Scholar] [CrossRef][Green Version]
- Katsiampa, Paraskevi. 2019. An empirical investigation of volatility dynamics in the cryptocurrency market. Research in International Business and Finance 50: 322–35. [Google Scholar] [CrossRef]
- Kim, Jong-Min, and Sun-Young Hwang. 2017. Directional Dependence via Gaussian Copula Beta Regression Model with Asymmetric GARCH Marginals. Communications in Statistics: Simulation and Computation 46: 7639–53. [Google Scholar] [CrossRef]
- Kojadinovic, Ivan, and Jun Yan. 2010. Modeling multivariate distributions with continuous margins using the copula R package. Journal of Statistical Software 34: 1–20. [Google Scholar] [CrossRef]
- Kristoufek, Ladislav. 2013. Bitcoin meets Google Trends and Wikipedia: Quantifying the relationship between phenomena of the Internet era. Scientific Reports 3: 3415. [Google Scholar] [CrossRef]
- Kuan, Chung-Ming, and Halbert White. 1994. Artificial neural networks: An econometric perspective (with discussion). Econometric Reviews 13: 1–143. [Google Scholar] [CrossRef]
- Masarotto, Guido, and Cristiano Varin. 2012. Gaussian copula marginal regression. Electronic Journal of Statistics 6: 1517–49. [Google Scholar] [CrossRef]
- Nelsen, Roger B. 2006. An Introduction to Copulas, 2nd ed. New York: Springer. [Google Scholar]
- Phillips, Peter, Shuping Shi, and Jun Yu. 2015. Testing for multiple bubbles: Historical episodes of exuberance and collapse in the S&P 500. International Economic Review 56: 1043–78. [Google Scholar]
- Ripley, Brian D. 1996. Pattern Recognition and Neural Networks. Cambridge: Cambridge University Press. [Google Scholar]
- Sklar, Abe. 1973. Random variables, joint distribution functions, and copulas. Kybernetika 9: 449–60. [Google Scholar]
- Stavroyiannis, Stavros, and Vassilios Babalos. 2017. Herding, faith-based investments and the global financial crisis: Empirical evidence from static and dynamic models. Journal of Behavioral Finance 18: 478–489. [Google Scholar] [CrossRef]
- Sungur, Engin A. 2005. A note on directional dependence in regression setting. Communications in Statistics: Theory and Methods 34: 1957–65. [Google Scholar] [CrossRef]
- Ziȩba, Damian, Ryszard Kokoszczyński, and Katarzyyna Śledziewska. 2019. Shock transmission in the cryptocurrency market. Is Bitcoin the most influential? International Review of Financial Analysis 64: 102–25. [Google Scholar] [CrossRef]

**Figure 2.**Residuals from neural network autoregression NNAR models to the daily log-returns in percentage of the five cryptocurrencies.

**Figure 3.**Copula directional dependence among the five cryptocurrencies. The thickness of the arrows reflects the strength of copula directional dependence, with thicker arrows indicating stronger pairwise directional dependence.

**Figure 5.**Linear and LOESS regression models between prices and volumes of the five cryptocurrencies: linear fit (dashed line) and LOESS fit (solid line).

**Table 1.**Descriptive statistics of the five cryptocurrencies. The sample consists of 1165 daily log-returns of the five cryptocurrencies from 8 August 2015 to 15 October 2018.

LBTC | LETH | LLTC | LXLM | LXRP | |
---|---|---|---|---|---|

Minimum | −20.7530 | −31.5469 | −39.5151 | −36.6358 | −61.6273 |

Q1 | −0.9719 | −2.6806 | −1.6928 | −3.2394 | −2.1367 |

Q2 | 0.2967 | −0.0894 | 0.0000 | −0.4066 | −0.3564 |

Mean | 0.2775 | 0.4836 | 0.2283 | 0.3886 | 0.3407 |

Q3 | 1.8199 | 3.2674 | 1.7673 | 3.1298 | 1.8960 |

Maximum | 22.5119 | 41.2337 | 51.0348 | 72.3102 | 102.7356 |

Skewness | −0.2519 | 0.5072 | 1.3240 | 2.0427 | 2.9787 |

Kurtosis | 7.9613 | 7.6037 | 16.1616 | 18.0902 | 40.6386 |

**Table 2.**Correlation coefficients, r and the coefficients of determination, ${r}^{2}$. This table presents the Pearson’s correlation coefficients and the coefficients of determination of log-returns of the five cryptocurrencies.

r | ${\mathit{r}}^{2}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|

LETH | LLTC | LXLM | LXRP | LETH | LLTC | LXLM | LXRP | ||

LBTC | 0.369 | 0.591 | 0.342 | 0.285 | LBTC | 0.136 | 0.350 | 0.117 | 0.081 |

LETH | 0.359 | 0.257 | 0.239 | LETH | 0.129 | 0.066 | 0.057 | ||

LLTC | 0.366 | 0.345 | LLTC | 0.134 | 0.119 | ||||

LXLM | 0.540 | LXLM | 0.292 |

**Table 3.**The Optimal model structure using neural networks autoregression models, $\mathrm{NNAR}(p,k)$. This table shows the optimal model structure of estimating nonlinear autocorrelations in log-returns of five cryptocurrencies. The p and k represent the lags and hidden nodes, respectively, used in the model.

LBTC | LETH | LLTC | LXLM | LXRP |
---|---|---|---|---|

NNAR(1,1) | NNAR(4,2) | NNAR(8,4) | NNAR(9,5) | NNAR(18,10) |

**Table 4.**The results of copula directional dependence. The higher value of copula directional dependence between each direction is highlighted in bold.

(U,V) | V → U | U→ V | Diff = (V→ U - U→ V) | ||||
---|---|---|---|---|---|---|---|

Estimate (Diff) | Bias (Diff) | Std. Error (Diff) | MSE (Diff) | Boot 95% CI of Diff | |||

(LBTC, LETH) | 0.1325 | 0.1028 | 0.0295 | 0.000002 | 0.000175 | 0.00000003 | (0.0291, 0.0298) |

(LBTC, LLTC) | 0.3852 | 0.3890 | −0.0037 | 0.000003 | 0.000360 | 0.00000013 | (−0.0044, −0.0030) |

(LBTC, LXLM) | 0.1299 | 0.1159 | 0.0143 | 0.000001 | 0.000180 | 0.00000003 | (0.0139, 0.0146) |

(LBTC, LXRP) | 0.1141 | 0.1006 | 0.0138 | 0.000004 | 0.000186 | 0.00000003 | (0.0134, 0.0142) |

(LETH, LLTC) | 0.1240 | 0.1578 | −0.0333 | 0.000007 | 0.000197 | 0.00000004 | (−0.0337, −0.0329) |

(LETH, LXLM) | 0.1016 | 0.1090 | −0.0075 | −0.000003 | 0.000161 | 0.00000003 | (−0.0078, −0.0072) |

(LETH, LXRP) | 0.0893 | 0.0974 | −0.0082 | −0.000002 | 0.000160 | 0.00000003 | (−0.0085, −0.0079) |

(LLTC, LXLM) | 0.1863 | 0.1611 | 0.0250 | −0.000001 | 0.000219 | 0.00000005 | (0.0246, 0.0255) |

(LLTC, LXRP) | 0.1587 | 0.1450 | 0.0134 | −0.000006 | 0.000224 | 0.00000005 | (0.0129, 0.0138) |

(LXLM, LXRP) | 0.2111 | 0.2208 | −0.0093 | 0.000004 | 0.000208 | 0.00000004 | (−0.0097, −0.0089) |

**Table 5.**Summary statistics of the volumes of the five cryptocurrencies. Daily trading volumes (in $) of Bitcoin (BTC), Ethereum (ETH), Litecoin (LTC), Stella (XLM), and Ripple (XRP) are shown in the table.

BTC | ETH | LTC | XLM | XRP | |
---|---|---|---|---|---|

Minimum | 12,712,600 | 102,128 | 507,480 | 491 | 24,819 |

Q1 | 68,338,000 | 8,933,050 | 2,374,230 | 31,416 | 722,260 |

Q2 | 277,084,992 | 69,245,600 | 12,755,200 | 543,934 | 5,013,190 |

Mean | 2,347,742,132 | 821,315,142 | 224,843,171 | 37,125,273 | 307,798,172 |

Q3 | 3,961,080,064 | 1,475,939,968 | 302,471,008 | 40,041,100 | 249,264,000 |

Maximum | 23,840,899,072 | 9,214,950,400 | 6,961,679,872 | 1,513,270,016 | 9,110,439,936 |

**Table 6.**Pearson’s correlation coefficient r, determination of coefficients ${r}^{2}$ of linear regression models between prices and volumes, and pseudo ${R}^{2}$ values from LOESS fit are provided in the table.

BTC | ETH | LTC | XLM | XRP | |
---|---|---|---|---|---|

r | 0.937 | 0.893 | 0.763 | 0.706 | 0.778 |

${r}^{2}$ | 0.879 | 0.797 | 0.582 | 0.499 | 0.606 |

${R}^{2}$ | 0.950 | 0.862 | 0.757 | 0.774 | 0.778 |

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**MDPI and ACS Style**

Hyun, S.; Lee, J.; Kim, J.-M.; Jun, C. What Coins Lead in the Cryptocurrency Market: Using Copula and Neural Networks Models. *J. Risk Financial Manag.* **2019**, *12*, 132.
https://doi.org/10.3390/jrfm12030132

**AMA Style**

Hyun S, Lee J, Kim J-M, Jun C. What Coins Lead in the Cryptocurrency Market: Using Copula and Neural Networks Models. *Journal of Risk and Financial Management*. 2019; 12(3):132.
https://doi.org/10.3390/jrfm12030132

**Chicago/Turabian Style**

Hyun, Steve, Jimin Lee, Jong-Min Kim, and Chulhee Jun. 2019. "What Coins Lead in the Cryptocurrency Market: Using Copula and Neural Networks Models" *Journal of Risk and Financial Management* 12, no. 3: 132.
https://doi.org/10.3390/jrfm12030132