# Long Memory in the Volatility of Selected Cryptocurrencies: Bitcoin, Ethereum and Ripple

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Data

_{t}represents return at time t, lnP

_{t}is the natural logarithm of the opening price at date t and lnP

_{t}

_{−1}is the natural logarithm of the opening price at date t − 1.

## 4. Methodology

#### 4.1. Long Memory Tests

#### 4.1.1. Rescaled Range (R/S) Statistics

#### 4.1.2. Geweke and Porter-Hudak (GPH) Model

#### 4.1.3. Gaussian Semiparametric (GSP) Method

#### 4.2. Results of Long Memory Tests

#### 4.3. GARCH Models

#### 4.3.1. The Fractional Integrated GARCH (FIGARCH) Model

_{1}and β

_{1}close to one, with α

_{1}small and β

_{1}large. Therefore, the effect of shocks on the conditional variance diminishes very slowly. In these situations, Baillie et al. (1996) suggest the class of Fractionally Integrated GARCH (FIGARCH) models. This model captures slowly decaying volatility as well as recognizing both the long memory and short memory characteristics of conditional variance (Chkili et al. 2014). Fractionally integrated processes are significantly different from both stationary and unit-root processes with their persistence and mean reverting features.

#### 4.3.2. Hyperbolic GARCH (HYGARCH) Model

#### 4.4. The VaR and Backtesting

_{t}= E(|L

_{t}| > |VaR

_{t}|)

_{t}is the expected value of loss if a VaR

_{t}violation occurs. Hendricks (1996) interpreted the ESF1 as the excess value of the losses over the VaR, the ESF2 as expected value of loss exceeding the VaR level, divided by the associated VaR values.

_{0}: α = α

_{0}and where α

_{0}is the pre-specified VaR level.

^{2 (1)}chi squared distributed with one degree of freedom. If the null hypothesis, H

_{0}: α = α

_{0}, cannot be rejected, then the model will be favored for VaR prediction, which exhibits unconditional coverage measured by α = E(N/T) equals the desired coverage level α

_{0}.

## 5. Findings

#### In Sample VaR Estimations

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Time Series Plots of Bitcoin price and return. (

**A**) Presents the plot historical plot of daily closing prices for Bitcoin between January 1, 2014 and February 28, 2018. (

**B**) Presents the daily returns for Bitcoin for the same time period. The data is obtained from Bitfinex.

**Figure 2.**Time Series Plots of Ethereum Price and Return. (

**A**) Presents the plot historical plot of daily closing prices for Ethereum between October 3, 2016 and February 28, 2018; (

**B**) presents the daily returns for Ethereum for the same time period. The data is obtained from Bitfinex.

**Figure 3.**Time Series Plots of Ripple Price and Return. (

**A**) Presents the plot historical plot of daily closing prices for Ripple between May 20, 2017 and February 28, 2018; (

**B**) presents the daily returns for Ripple for the same time period. The data is obtained from Bitfinex.

Statistic | BTC | ETH | XRP |
---|---|---|---|

Mean | 0.158 | 0.605 | 0.351 |

Maximum | 24.348 | 25.859 | 63.137 |

Minimum | −28.703 | −34.48 | −37.713 |

Std. Dev. | 4.09 | 6.613 | 9.859 |

Skewness | −0.442 | −0.076 | 1.54 |

Kurtosis | 9.873 | 6.363 | 11.715 |

Jarque-Bera | 2951.355 | 339.707 | 1014.636 |

ARCH 1-2 | 52.23 *** | 49.4 *** | 6.16 *** |

ARCH 1-5 | 25.45 *** | 22.42 *** | 2.88 *** |

ARCH 1-10 | 14.02 *** | 11.83 *** | 4.33 *** |

Q(20) | 27.75 | 22.62 | 26.94 |

Qsq(20) | 212.32 *** | 159.79 *** | 69.52 *** |

Observations | 1475 | 719 | 285 |

Panel 2A: Bitcoin (BTC) Daily Returns | ||||

Statistic | Hurst–Mandelbrot R/S | Lo R/S | GPH | GSP |

d parameter | - | - | –0.027 (0.025) | –0.01 (0.018) |

Test Statistics | 2.094 | 2.117 | ||

Critical values | Probability | Probability | ||

90% | [0.861, 1.747] | [0.2722] | [0.5635] | |

95% | [0.809, 1.862] | |||

99% | [0.721, 2.098] | |||

Panel 2B: Bitcoin (BTC) Squared Daily Returns | ||||

Statistic | Hurst–Mandelbrot R/S | Lo R/S | GPH | GSP |

d parameter | - | - | 0.175 (0.025) | 0.175 (0.018) |

Test Statistics | 3.252 | 2.900 | ||

Critical values | Probability | Probability | ||

90% | [0.861, 1.747] | [0.0000] | [0.0000] | |

95% | [0.809, 1.862] | |||

99% | [0.721, 2.098] |

Panel 3A: Ethereum (ETH) Daily Returns | ||||

Statistic | Hurst–Mandelbrot R/S | Lo R/S | GPH | GSP |

d parameter | - | - | 0.034 (0.038) | 0.022 (0.026) |

Test Statistics | 1.676 | 1.665 | ||

Critical values | Probability | Probability | ||

90% | [0.861, 1.747] | [0.3796] | [0.3963] | |

95% | [0.809, 1.862] | |||

99% | [0.721, 2.098] | |||

Panel 3B: Ethereum (ETH) Squared Daily Returns | ||||

Statistic | Hurst–Mandelbrot R/S | Lo R/S | GPH | GSP |

d parameter | - | - | 0.264 (0.038) | 0.255 (0.026) |

Test Statistics | 2.482 | 2.139 | ||

Critical values | Probability | Probability | ||

90% | [0.861, 1.747] | [0.0000] | [0.0000] | |

95% | [0.809, 1.862] | |||

99% | [0.721, 2.098] |

Panel 4A: Ripple (XRP) Daily Returns | ||||

Statistic | Hurst–Mandelbrot R/S | Lo R/S | GPH | GSP |

d parameter | - | - | 0.076 (0.063) | 0.038 (0.041) |

Test Statistics | 1.496 | 1.492 | ||

Critical values | Probability | Probability | ||

90% | [0.861, 1.747] | [0.2298] | [0.3613] | |

95% | [0.809, 1.862] | |||

99% | [0.721, 2.098] | |||

Panel 4B: Ripple (XRP) Squared Daily Returns | ||||

Statistic | Hurst–Mandelbrot R/S | Lo R/S | GPH | GSP |

d parameter | - | - | 0.18 (0.063) | 0.122 (0.041) |

Test Statistics | 2.011 | 1.887 | ||

Critical values | Probability | Probability | ||

90% | [0.861, 1.747] | [0.0047] | [0.0035] | |

95% | [0.809, 1.862] | |||

99% | [0.721, 2.098] |

Estimation Method | BTC | ETH | XRP | |||
---|---|---|---|---|---|---|

HYGARCH Student | HYGARCH sk.-t | FIGARCH sk.-t | HYGARCH sk.-t | FIGARCH Student | FIGARCH sk.-t | |

Cst(M) | 0.145 *** | 0.114 ** | 0.368 ** | 0.351 ** | −0.192 | 0.092 |

Cst(V) | 0.128 | 0.146 | 272.83 | 1.127 | 0.722 | 0.377 |

d-Figarch | 0.65 *** | 0.659 *** | 0.68 *** | 0.643 ** | 0.625 ** | 0.60 ** |

ARCH(Alpha1) | 0.207 ** | 0.201 ** | 0.281 ** | 0.326 | 0.594 *** | 0.586 *** |

GARCH(Beta1) | 0.678 *** | 0.68 *** | 0.636 *** | 0.633 *** | 0.896 *** | 0.903 *** |

Student(DF) | 2.737 *** | 3.584 *** | ||||

Asymmetry | -0.023 | 0.103 *** | 0.098 *** | 0.104 | ||

Tail | 2.739 *** | 4.115 *** | 3.73 *** | 3.648 *** | ||

Log Alpha (HY) | 0.241 ** | 0.238 ** | 0.106 | |||

No. Observations | 1475 | 1475 | 719 | 719 | 285 | 285 |

No. Parameters | 7 | 8 | 7 | 8 | 6 | 7 |

Log Likelihood | −3758.126 | −3757.843 | −2256.649 | −2256.91 | −994.076 | −993.21 |

AIC | 5.105 | 5.106 | 6.296 | 6.30 | 7.018 | 7.019 |

SW | 5.130 | 5.134 | 6.341 | 6.351 | 7.094 | 7.108 |

SB | 5.105 | 5.106 | 6.296 | 6.29 | 7.017 | 7.017 |

H-Quinn | 5.114 | 5.116 | 6.313 | 6.319 | 7.048 | 7.054 |

JB | 34298 | 35411 | 177.96 | 168.64 | 210.67 | 281.64 |

Nyblom stability test | 3.964 | 4.152 | 1.989 | 1.679 | 1.070 | 1.171 |

Pearson (50) | 54.322 * | 53.847 * | 81.486 *** | 67.30 *** | 49.912 | 48.50 |

Panel A: VaR Backtesting Results for Bitcoin (BTC) Returns. | ||||||

BTC HYGARCH sk.-t | BTC HYGARCH | |||||

Short positions | Short positions | |||||

Quantile | Success rate | Kupiec LRT | P-value | Success rate | Kupiec LRT | P-value |

0.95 | 0.947 | 0.253 | 0.614 | 0.951 | 0.044 | 0.833 |

0.975 | 0.974 | 0.034 | 0.851 | 0.974 | 0.0348 | 0.851 |

0.99 | 0.989 | 0.104 | 0.746 | 0.989 | 0.1041 | 0.746 |

Long positions | Long positions | |||||

Quantile | Failure rate | Kupiec LRT | P-value | Failure rate | Kupiec LRT | P-value |

0.05 | 0.054 | 0.728 | 0.393 | 0.057 | 1.725 | 0.188 |

0.025 | 0.023 | 0.099 | 0.752 | 0.025 | 0.0004 | 0.983 |

0.01 | 0.010 | 0.004 | 0.947 | 0.010 | 0.004 | 0.947 |

Panel B: VaR Backtesting Results for Ethereum (ETH) Returns. | ||||||

ETH FIGARCH sk.-t | ETH HYGARCH sk.-t | |||||

Short positions | Short positions | |||||

Quantile | Success rate | Kupiec LRT | P-value | Success rate | Kupiec LRT | P-value |

0.95 | 0.933 | 3.864 ** | 0.049 | 0.936 | 2.727 * | 0.098 |

0.975 | 0.973 | 0.058 | 0.808 | 0.979 | 0.534 | 0.464 |

0.99 | 0.988 | 0.088 | 0.765 | 0.991 | 0.210 | 0.646 |

Long positions | Long positions | |||||

Quantile | Failure rate | Kupiec LRT | P-value | Failure rate | Kupiec LRT | P-value |

0.05 | 0.058 | 1.019 | 0.312 | 0.051 | 0.031 | 0.858 |

0.025 | 0.030 | 0.863 | 0.352 | 0.026 | 0.058 | 0.808 |

0.01 | 0.011 | 0.088 | 0.765 | 0.009 | 0.005 | 0.942 |

Panel C: VaR Backtesting Results for Ripple (XRP) Returns. | ||||||

XRP FIGARCH | XRP FIGARCH sk.-t | |||||

Short positions | Short positions | |||||

Quantile | Success rate | Kupiec LRT | P-value | Success rate | Kupiec LRT | P-value |

0.95 | 0.933 | 1.515 | 0.218 | 0.940 | 0.527 | 0.467 |

0.975 | 0.968 | 0.467 | 0.494 | 0.968 | 0.467 | 0.494 |

0.99 | 0.975 | 4.341 ** | 0.037 | 0.982 | 1.337 | 0.247 |

Long positions | Long positions | |||||

Quantile | Failure rate | Kupiec LRT | P-value | Failure rate | Kupiec LRT | P-value |

0.05 | 0.045 | 0.118 | 0.730 | 0.052 | 0.040 | 0.839 |

0.025 | 0.017 | 0.724 | 0.394 | 0.028 | 0.106 | 0.744 |

0.01 | 0.007 | 0.285 | 0.592 | 0.010 | 0.007 | 0.929 |

Panel A: Expected Shortfalls for Bitcoin (BTC). | ||||

BTC | HYGARCH | HYGARCHsk.-t | ||

α quantile | ESF1 | ESF2 | ESF1 | ESF2 |

Short positions | ||||

0.95 | 7.7225 | 1.5586 | 7.5319 | 1.5416 |

0.97 | 9.0107 | 1.4334 | 9.0107 | 1.4624 |

0.99 | 9.6857 | 1.2315 | 9.6857 | 1.26 |

Long positions | ||||

0.05 | –8.72 | 1.6721 | –8.88 | 1.6717 |

0.025 | –11.08 | 1.6698 | –11.38 | 1.6709 |

0.01 | –13.46 | 1.5745 | –13.46 | 1.537 |

Panel B: Expected Shortfalls for Ethereum (ETH). | ||||

ETH | FIGARCH | HYGARCHsk.-t | ||

α quantile | ESF1 | ESF2 | ESF1 | ESF2 |

Short positions | ||||

0.95 | 13.8 | 1.3663 | 13.93 | 1.33 |

0.97 | 15 | 1.3255 | 15.15 | 1.3606 |

0.99 | 15.92 | 1.2213 | 16.59 | 1.2115 |

Long positions | ||||

0.05 | −12.62 | 1.4874 | −13.25 | 1.4821 |

0.025 | −14.19 | 1.3998 | −15.03 | 1.3772 |

0.01 | −19.48 | 1.3946 | −20.11 | 1.3148 |

Panel C: Expected Shortfalls for Ripple (XRP). | ||||

XRP | FIGARCH | FIGARCHsk.-t | ||

α quantile | ESF1 | ESF2 | ESF1 | ESF2 |

Short positions | ||||

0.95 | 22.44 | 1.635 | 23.52 | 1.6362 |

0.97 | 31.35 | 1.6778 | 31.35 | 1.607 |

0.99 | 34.19 | 1.3312 | 42.07 | 1.4082 |

Long positions | ||||

0.05 | −17.24 | 1.3569 | −16.84 | 1.4136 |

0.025 | −23.05 | 1.2638 | −20.02 | 1.2529 |

0.01 | −28.26 | 1.0676 | −27.22 | 1.1236 |

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## Share and Cite

**MDPI and ACS Style**

Kaya Soylu, P.; Okur, M.; Çatıkkaş, Ö.; Altintig, Z.A.
Long Memory in the Volatility of Selected Cryptocurrencies: Bitcoin, Ethereum and Ripple. *J. Risk Financial Manag.* **2020**, *13*, 107.
https://doi.org/10.3390/jrfm13060107

**AMA Style**

Kaya Soylu P, Okur M, Çatıkkaş Ö, Altintig ZA.
Long Memory in the Volatility of Selected Cryptocurrencies: Bitcoin, Ethereum and Ripple. *Journal of Risk and Financial Management*. 2020; 13(6):107.
https://doi.org/10.3390/jrfm13060107

**Chicago/Turabian Style**

Kaya Soylu, Pınar, Mustafa Okur, Özgür Çatıkkaş, and Z. Ayca Altintig.
2020. "Long Memory in the Volatility of Selected Cryptocurrencies: Bitcoin, Ethereum and Ripple" *Journal of Risk and Financial Management* 13, no. 6: 107.
https://doi.org/10.3390/jrfm13060107