# A Comprehensive Statistical Analysis of the Six Major Crypto-Currencies from August 2015 through June 2020

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Statistical Analyses

## 3. Assessing Crypto-Assets’ Risks

#### 3.1. Unconditional Risk

#### 3.2. Conditional Risk

## 4. A Look at Dependence

#### 4.1. Dependence (by Pair-Copulas)

#### 4.2. Dependence (by Cointegration)

## 5. Discussions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Aas, Kjersti, Claudia Czado, Arnoldo Frigessi, and Henrik Bakken. 2009. Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics 44: 182–98. [Google Scholar] [CrossRef] [Green Version]
- Berg, Daniel, and Kjersti Aas. 2009. Models for construction of multivariate dependence—A comparison study. European Journal of Finance 15: 639–59. [Google Scholar]
- Baillie, Richard T., Tim Bollerslev, and Hans Ole Mikkelsen. 1996. Fractionally integrated generalized autoregressive conditional Heteroskedasticity. Journal of Econometrics 74: 3–30. [Google Scholar] [CrossRef]
- Bariviera, Aurelio F. 2017. The inefficiency of Bitcoin revisited: A dynamic approach. Economic Letters 161: 1–4. [Google Scholar] [CrossRef] [Green Version]
- Bariviera, Aurelio F., María José Basgall, Waldo Hasperué, and Marcelo Naiouf. 2017. Some stylized facts of the Bitcoin market. Physica A: Statistical Mechanics and Its Applications 484: 82–90. [Google Scholar] [CrossRef] [Green Version]
- Baur, Dirk G., Kihoon Hong, and Adrian D. Lee. 2017. Bitcoin: Medium of Exchange or Speculative Assets? Journal of International Financial Markets, Institutions and Money 54: 177–89. [Google Scholar] [CrossRef]
- Bedford, Tim, and Roger M. Cooke. 2001. Probability density decomposition for conditionally dependent random variables modeled by vines. Annals of Mathematics and Artificial Intelligence 32: 245–68. [Google Scholar] [CrossRef]
- Bedford, Tim, and Roger M. Cooke. 2002. Vines—A new graphical model for dependent random variables. Annals of Statistics 30: 1031–68. [Google Scholar] [CrossRef]
- Bollerslev, Tim, and Hans Ole Mikkelsen. 1996. Modeling and pricing long memory in stock market volatility. Journal of Econometrics 73: 151–84. [Google Scholar] [CrossRef]
- Buterin, V. 2013. A Next Generation Smart Contract Decentralized Application Platform. Available online: https://blockchainlab.com/pdf/Ethereum_white_paper-a_next_generation_smart_contract_and_decentralized_application_platform-vitalik-buterin.pdf (accessed on 25 August 2020).
- Chan, Stephen, Jeffrey Chu, Saralees Nadarajah, and Joerg Osterrieder. 2017. A Statistical Analysis of Crypto-currencies. Journal of Risk and Financial Management 10: 12. [Google Scholar] [CrossRef] [Green Version]
- Chaim, Pedro, and Marcio P. Laurini. 2018. Is Bitcoin a Bubble? Physica A: Statistical Mechanics and Its Applications 517: 222–32. [Google Scholar] [CrossRef]
- Cheah, Eng-Tuck, and John Fry. 2015. Speculative bubbles in bitcoin markets? An empirical investigation into the fundamental value of Bitcoin. Economics Letters 130: 32–36. [Google Scholar] [CrossRef] [Green Version]
- Cheah, Eng-Tuck, Tapas Mishra, Mamata Parhi, and Zhuang Zhang. 2018. Long Memory Interdependency and Inefficiency in Bitcoin Markets. Economics Letters 167: 18–25. [Google Scholar] [CrossRef]
- Chu, Jeffrey, Stephen Chan, Saralees Nadarajah, and Joerg Osterrieder. 2017. GARCH modelling of crypto-currencies. Journal of Risk and Financial Management 10: 17. [Google Scholar] [CrossRef]
- Ciaian, Pavel, Miroslava Rajcaniova, and d’Artis Kancs. 2016. The economics of Bitcoin price formation. Applied Economics 48: 1799–815. [Google Scholar] [CrossRef] [Green Version]
- Cont, Rama. 2001. Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues. Quantitative Finance 1: 223–36. [Google Scholar] [CrossRef]
- Corbet, Shaen, Brian Lucey, Andrew Urquhart, and Larisa Yarovaya. 2018. Crypto-currencies as a Financial Asset: A Systematic Analysis. International Review of Financial Analysis 62: 182–99. [Google Scholar] [CrossRef] [Green Version]
- de Haan, Laurens. 1984. Slow variation and characterization of domains of attraction. Statistical Extremes and Applications 131: 31–48. [Google Scholar]
- Deniz, Pinar, and Thanasis Stengos. 2020. Cryptocurrency returns before and after the introduction of Bitcoin futures. J. Risk and Financial Management 13: 116. [Google Scholar] [CrossRef]
- Dickey, David A., and Wayne A. Fuller. 1979. Distribution of the Estimators for Autoregressive Time Series with a Unit Root. Journal of the American Statistical Association 366: 427–31. [Google Scholar]
- Donoho, David L. 1982. Breakdown Properties of Multivariate Location Estimators. Ph.D. thesis, Harvard University, Harvard, UK. [Google Scholar]
- Drożdż, Stanisław, Robert Gȩbarowski, Ludovico Minati, Paweł Óswiȩcimka, and Marcin Wa̧torek. 2018. Bitcoin market route to maturity? Evidence from return fluctuations, temporal correlations and multiscaling effects. Chaos 28: 071101. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Drożdż, Stanisław, Ludovico Minati, Paweł Óswiȩcimka, Marek Stanuszek, and Marcin Wa̧torek. 2019. Signatures of the crypto-currency market decoupling from the Forex. Future Internet 11: 154. [Google Scholar] [CrossRef] [Green Version]
- Drożdż, Stanisław, Ludovico Minati, Pawel Oswiecimka, Marek Stanuszek, and Marcin Watorek. 2020. Competition of noise and collectivity in global cryptocurrency trading: Route to a self-contained market. Chaos 30: 023122. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Dyhrberg, Anne Haubo. 2016. Bitcoin, gold and the dollar—A GARCH volatility analysis. Finance Research Letters 16: 85–92. [Google Scholar] [CrossRef] [Green Version]
- Engle, R. F., and Clive W. J. Granger. 1987. Spurious regression in econometrics. Journal of Econometrics 2: 111–20. [Google Scholar]
- Engle, Robert F., and Andrew J. Patton. 2001. What good is a volatility model? Quantitative Finance 1: 237–45. [Google Scholar] [CrossRef]
- Fry, John, and Eng-Tuck Cheah. 2016. Negative bubbles and shocks in crypto-currency markets. International Review of Financial Analysis 47: 343–52. [Google Scholar] [CrossRef]
- Garnier, Josselin, and Knut Solna. 2018. Chaos and Order in the Bitcoin Market. Available online: https://www.researchgate.net/publication/327858987 (accessed on 31 March 2020).
- Hansen, Bruce E. 1994. Autoregressive Conditional Density Estimation. International Economic Review 35: 3. [Google Scholar] [CrossRef]
- Hencic, Andrew, and Christian Gouriéroux. 2015. Noncausal Autoregressive Model in Application to Bitcoin/USD Exchange Rates. In Econometrics of Risk. Studies in Computational Intelligence. Edited by Van-Nam Huynh, Vladik Kreinovich, Songsak Sriboonchitta and Komsan Suriya. Berlin/Heidelberg: Springer, vol. 583. [Google Scholar]
- Hosking, Jonathan Richard Morley. 1981. Fractional differencing. Biometrika 68: 165–76. [Google Scholar] [CrossRef]
- Hosking, Jonathan Richard Morley, and James R. Wallis. 1987. Parameter and quantile estimation for the generalized pareto distribution. Technometrics 29: 339–49. [Google Scholar] [CrossRef]
- Hosking, Jonathan Richard Morley, and James R. Wallis. 1997. Regional Frequency Analysis. Cambridge: Cambridge University Press. [Google Scholar]
- Joe, Harry. 1997. Multivariate Models and Dependence Concepts. London: Chapman e Hall. [Google Scholar]
- Johansen, Søren. 1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12: 231–54. [Google Scholar] [CrossRef]
- Jones, M. Chris, and M. J. Faddy. 2003. A skew extension of the t-distribution, with applications. Journal of the Royal Statistical Society 65: 159–74. [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. Volatility co-movement between Bitcoin and Ether. Finance Research Letters 30: 221–27. [Google Scholar] [CrossRef] [Green Version]
- Kristoufek, Ladislav. 2015. What are the main drivers of the Bitcoin price? Evidence from wavelet coherence analysis. PLoS ONE 10: e0123923. [Google Scholar] [CrossRef]
- Kwiatkowski, Denis, Peter C. B. Phillips, Peter Schmidt, and Yongcheol Shin. 1992. Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of Econometrics 54: 159–78. [Google Scholar] [CrossRef]
- Lahmiri, Salim, and Stelios Bekiros. 2018. Chaos, randomness and multi-fractality in bitcoin market. Chaos Solitons & Fractals 106: 28–34. [Google Scholar]
- Li, Xin, and Chong Alex Wang. 2017. The Technology and Economic Determinants of Crypto-currency Exchange Rates: The Case of Bitcoin. Decision Support Systems 95: 49–60. [Google Scholar] [CrossRef]
- McLachlan, Geoffrey J., and David Peel. 2000. Finite Mixture Models. New York: Wiley. [Google Scholar]
- McNeil, Alexander J., and Rüdiger Frey. 2000. Estimation of tail-related risk measures for heteroscedastic financial time series: An extreme value approach. Journal of Empirical Finance 7: 271–300. [Google Scholar] [CrossRef]
- McQueen, Grant, and Steven Thorley. 1994. Bubbles, Stock Returns, and Duration Dependence. Journal of Financial and Quantitative Analysis 29: 379–401. [Google Scholar] [CrossRef]
- Mendes, Beatriz V. M., Eduardo F. L. de Melo, and Roger B. Nelsen. 2007. Robust fits for copula models. Communication in Statistics 36: 997–1017. [Google Scholar] [CrossRef]
- Nadarajah, Saralees, and Jeffrey Chu. 2017. On the inefficiency of bitcoin. Economics Letters 150: 6–9. [Google Scholar] [CrossRef] [Green Version]
- Nakamoto, Satoshi. 2009. Bitcoin: A Peer-to-Peer Electronic Cash System. Available online: https://bitcoin.org/bitcoin.pdf (accessed on 22 March 2020).
- Nelsen, Roger B. 2006. An Introduction to Copulas. Berlin/Heidelberg: Springer. [Google Scholar]
- Phillips, Peter C. B., and Pierre Perron. 1988. Testing for a Unit Root in Time Series Regression. Biometrika 75: 335–46. [Google Scholar] [CrossRef]
- Pickands, James, III. 1975. Statistical inference using extreme order statistics. Annals of Statistics 3: 119–31. [Google Scholar]
- Phillip, Andrew, Jennifer S. K. Chan, and Shelton Peiris. 2018. A new look at Crypto-currencies. Economics Letters 163: 6–9. [Google Scholar] [CrossRef]
- Stahel, Werner A. 1981. Robust Estimation: Infinitesimal Optimality and Covariance Matrix Estimators. Zurich: ETH. [Google Scholar]
- Tan, Chia-Yen, You-Beng Koh, and Kok-Haur Ng. 2019. Structural Change Analysis of Active Crypto-currency Market. arXiv arXiv:1909.10679. [Google Scholar]
- Tsay, Ruey S. 2002. Analysis of Financial Time Series. Wiley Series in Probability and Statistics; New York: Wiley-Interscience. [Google Scholar]
- Urquhart, Andrew. 2016. The inefficiency of bitcoin. Economics Letters 148: 80–82. [Google Scholar] [CrossRef]
- Urquhart, Andrew. 2017. Price clustering in Bitcoin. Economics Letters 159: 145–48. [Google Scholar] [CrossRef]
- Zwick, Hélène Syed, and Sarfaraz Ali Shah Syed. 2019. Bitcoin and Gold Prices: A Fledging Long-Term Relationship. Theoretical Economics Letters 9: 7. [Google Scholar] [CrossRef] [Green Version]
- Zhu, Dongming, and John W. Galbraith. 2010. A generalized asymmetric Student-t distribution with application to financial econometrics. Journal of Econometrics 157: 297–305. [Google Scholar] [CrossRef]

**Figure 3.**The first and second rows show the Bitcoin GPD’s graphical diagnosis for the left and right tails. In the third row the return level is shown in days with the corresponding $\alpha $-VaR, for the left tail (blue) and right tail (black) of Bitcoin.

**Figure 4.**Scatter plots of the standardized residuals from the GARCH fits on the upper-left panel (above diagonal). The corresponding Kendall’s $\tau $ coefficients on the bottom-right panel.

Bitcoin | Ethereum | Ripple | Litecoin | Stellar | Monero | Euro | |
---|---|---|---|---|---|---|---|

Mean | 0.21 | 0.34 | 0.20 | 0.17 | 0.19 | 0.29 | 0.00 |

0.99%[LCL, UCL] | [−0.03,0.47] | [−0.06,0.74] | [−0.25,0.66] | [−0.17,0.52] | [−0.90,1.30] | [−0.14,0.71] | [−0.04,0.04] |

Median | 0.23 | −0.06 | −0.30 | −0.05 | −0.36 | −0.07 | −0.01 |

Standard deviation | 3.96 | 6.33 | 7.18 | 5.49 | 17.37 | 6.60 | 0.48 |

Maximum | 25.49 | 39.94 | 101.97 | 47.97 | 269.14 | 56.47 | 2.47 |

Minimum | −23.97 | −33.35 | −63.15 | −40.60 | −244.65 | −29.18 | −2.88 |

Skewness | −0.21 * | 0.43 * | 2.60 * | 1.02 * | 0.06 | 0.93 * | 0.12 |

Kurtosis | 5.55 | 5.22 | 36.75 | 11.41 | 83.51 | 7.46 | 3.43 |

**Table 2.**GPD estimates and long-run risk measures: shape parameter (standard error); percentage of excesses ${p}^{*}$; 1% and 5% VaR and expected loss (EL) estimates.

Estimates | Bitcoin | Ethereum | Ripple | Litecoin | Stellar | Monero | Euro |
---|---|---|---|---|---|---|---|

Left tail estimates | |||||||

Shape(st. er); ${p}^{*}$ | 0.02(0.07); 14% | 0.04(0.08); 13% | 0.25(0.10); 10% | 0.12(0.08); 12% | 0.59(0.10); 15% | 0.01(0.07); 17% | 0.20(0.11); 9% |

VaR: 1% & 5% | −12.08 & −6.29 | −17.56 & −9.50 | −16.91 & −8.29 | −14.29 & −7.64 | −39.13 & −13.40 | −16.67 & −9.63 | −1.18 & −0.71 |

EL: 1% & 5% | 15.24 & 10.03 | 23.04 & 14.75 | 24.48 & 14.15 | 20.70 & 12.14 | 89.38 & 35.66 | 20.41 & 14.10 | 1.75 & 1.00 |

Right tail estimates | |||||||

Shape(st. er); ${p}^{*}$ | 0.15(0.09); 12% | 0.00(0.06); 13% | 0.33(0.12); 9% | 0.21(0.08); 14% | 0.57(0.12); 11% | 0.15(0.07); 16% | 0.24(0.13); 8% |

VaR: 5% & 1% | 6.23 & 11.41 | 11.15 & 20.23 | 10.06 & 24.61 | 8.63 & 18.00 | 15.68 & 46.98 | 10.83 & 20.60 | 0.76 & 1.25 |

EL: 5% & 1% | 9.78 & 14.81 | 16.84 & 27.93 | 19.81 & 39.91 | 15.43 & 27.75 | 38.74 & 86.72 | 16.91 & 30.10 | 1.07 & 1.83 |

**Table 3.**Summary of the results from the ARFIMA$(p,d,q)$-FIGARCH$(m,D,s)$ fits for all series. All parameters estimates are 5% statistically significant. Notations: LEV: leverage parameter; d.f.: degrees of freedom of the F distribution; skew: skewness estimate of the skew-t distribution.

Crypto-Coin | $({\mathit{\varphi}}_{0},{\mathit{\varphi}}_{1})$ | d | ${\mathit{\theta}}_{1}$ | $\mathit{\omega}$ | ${\mathit{\alpha}}_{1}$ | D | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | LEV | d.f. | Skew | AIC | 1%-VaR |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Bitcoin-LM | (0.201,–) | 0.00 | — | 0.144 | 0.258 | 0.689 | 0.780 | — | — | 3.15 | — | 5.0577 | (−6.80,7.20) |

Bitcoin | (0.207,–) | 0.00 | — | 0.212 | 0.225 | — | 0.401 | 0.418 | −0.089 | 3.11 | — | 5.0584 | (−7.23,7.65) |

Ethereum-LM | (0.128,–) | 0.00 | — | 3.470 | 0.138 | 0.646 | 0.515 | — | — | 3.18 | 1.064 | 6.1096 | (−10.59,12.05) |

Ethereum | (0.128,–) | 0.00 | — | 2.726 | 0.261 | — | 0.738 | — | — | 3.08 | 1.063 | 6.1116 | (−11.28,12.82) |

Ripple-LM | (−0.156,–) | 0.00 | −0.154 | 0.623 | 0.678 | 0.451 | 0.744 | — | — | 2.81 | 1.065 | 5.7761 | (−7.58,8.56) |

Ripple | (−0.160,–) | 0.00 | −0.154 | 2.059 | 0.263 | — | 0.736 | — | — | 2.72 | 1.056 | 5.7967 | (−8.99,10.01) |

Litecoin-LM | (–,–) | 0.00 | — | 0.001 | 0.361 | 0.619 | 0.753 | — | — | 3.11 | 1.048 | 5.5577 | (−11.79,12.80) |

Litecoin | (–,–) | 0.00 | — | 0.092 | 0.107 | — | 0.892 | — | — | 3.10 | 1.045 | 5.5676 | (−13.68,14.77) |

Stellar-LM | (–,–) | 0.00 | −0.097 | 0.815 | 0.278 | 0.374 | 0.375 | — | — | 3.47 | 1.143 | 6.2334 | (−9.61,11.65) |

Stellar | (–,–) | 0.00 | −0.093 | 1.801 | 0.212 | — | 0.787 | — | — | 3.17 | 1.133 | 6.2541 | (−9.75,11.86) |

Monero-LM | (–,–) | 0.0 | −0.110 | 3.256 | 0.115 | 0.652 | 0.582 | — | — | 3.30 | 1.056 | 6.2455 | (−12.40,14.33) |

Monero | (–,–) | 0.00 | −0.110 | 2.321 | 0.206 | — | 0.793 | — | — | 3.18 | 1.053 | 6.2472 | (−13.14,15.09) |

Euro-LM | (–,–) | 0.00 | — | 0.002 | 0.043 | 0.503 | 0.274 | 0.484 | — | 4.82 | — | 1.4120 | (−0.78,0.78) |

Euro | (–,–) | 0.00 | — | 0.001 | 0.019 | — | 0.975 | — | — | 6.73 | 1.088 | 1.2403 | (−0.79,0.88) |

**Table 4.**Pair-copulas’ robust fits: copula family and estimates, dependence measures, the losses (gains) BB7-copula-based joint probability associated 1%-VaR in the upper row, and the $\pi $-copula-based risk under independence (bottom row).

Pairs in | Copula Family | Linear Coef $\mathit{\rho}$ | Kendall’s $\mathit{\tau}$ | Lower TDC ${\mathit{\lambda}}_{\mathit{L}}$ | Upper TDC ${\mathit{\lambda}}_{\mathit{U}}$ | Losses: BB7 Risk | Gains: BB7 Risk |
---|---|---|---|---|---|---|---|

Tree 1 | (Parameters) | (Residuals Based) | (Copula Based) | (Copula Based) | (Copula Based) | Losses: ($\mathbf{\Pi}$ Risk) | Gains: ($\mathbf{\Pi}$ Risk) |

(XMR,BTC) | BB7 | 0.523 | 0.601 | 0.746 | 0.566 | 0.745% | 0.556% |

(1.923,2.361) | (0.01%) | (0.01%) | |||||

(BTC,LTC) | BB7 | 0.630 | 0.674 | 0.829 | 0.627 | 0.833% | 0.635% |

(2.186,3.691) | (0.01%) | (0.01%) | |||||

(LTC,ETH) | BB7 | 0.462 | 0.616 | 0.751 | 0.565 | 0.783% | 0.584% |

(1.918,2.632) | (0.01%) | (0.01%) | |||||

(ETH,XRP) | BB7 | 0.311 | 0.588 | 0.750 | 0.486 | 0.741% | 0.492% |

(1.672,2.422) | (0.01%) | (0.01%) | |||||

(XRP,XLM) | BB7 | 0.236 | 0.608 | 0.740 | 0.605 | 0.746% | 0.599% |

(2.083,2.301) | (0.01%) | (0.01%) |

**Table 5.**Estimates of the speed of adjustment parameters $\alpha $ for all pairs. All estimates are 5% statistically significant.

Pairs | (BTC,ETH) | (BTC,RIP) | (BTC,LTC) | (BTC,XLM) | (BTC,XMR) |
---|---|---|---|---|---|

$\alpha $ | (0.00502,0.00063) | (0.00453,0.000002) | (0.00528,0.00029) | (0.00477,0.000004) | (0.00573,0.00033) |

**Table 6.**Estimates of the speed of adjustment parameters $\alpha $ for the six crypto-asset system. All estimates are 1% statistically significant.

BTC | ETH | RIP | LTC | XLM | XMR | |
---|---|---|---|---|---|---|

1st cointegrating vector | −0.13611 | −0.03555 | −0.00010 | −0.00191 | −0.00003 | −0.00891 |

2nd cointegrating vector | −0.00251 | −0.00037 | −0.000002 | −0.00003 | −0.0000004 | −0.00015 |

© 2020 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/).

## Share and Cite

**MDPI and ACS Style**

Vaz de Melo Mendes, B.; Fluminense Carneiro, A.
A Comprehensive Statistical Analysis of the Six Major Crypto-Currencies from August 2015 through June 2020. *J. Risk Financial Manag.* **2020**, *13*, 192.
https://doi.org/10.3390/jrfm13090192

**AMA Style**

Vaz de Melo Mendes B, Fluminense Carneiro A.
A Comprehensive Statistical Analysis of the Six Major Crypto-Currencies from August 2015 through June 2020. *Journal of Risk and Financial Management*. 2020; 13(9):192.
https://doi.org/10.3390/jrfm13090192

**Chicago/Turabian Style**

Vaz de Melo Mendes, Beatriz, and André Fluminense Carneiro.
2020. "A Comprehensive Statistical Analysis of the Six Major Crypto-Currencies from August 2015 through June 2020" *Journal of Risk and Financial Management* 13, no. 9: 192.
https://doi.org/10.3390/jrfm13090192