# GARCH Modelling of Cryptocurrencies

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## Abstract

**:**

## 1. Introduction

## 2. Data

## 3. Models

#### 3.1. GARCH Models

#### 3.2. Model Selection

- the Akaike information criterion due to Akaike (1974) defined by:$$\begin{array}{c}\hfill {\displaystyle \mathrm{AIC}=2k-2\mathrm{ln}L\left(\widehat{\mathsf{\Theta}}\right),}\end{array}$$
- the corrected Akaike Information Criterion (AICc) due to Hurvich and Tsai (1989) defined by:$$\begin{array}{c}\hfill {\displaystyle \mathrm{AICc}=\mathrm{AIC}+\frac{{\displaystyle 2k(k+1)}}{{\displaystyle n-k-1}};}\end{array}$$
- the Hannan–Quinn criterion due to Hannan and Quinn (1979) defined by:$$\begin{array}{c}\hfill {\displaystyle \mathrm{HQC}=-2\mathrm{ln}L\left(\widehat{\mathsf{\Theta}}\right)+2k\mathrm{ln}\mathrm{ln}n.}\end{array}$$

#### 3.3. Estimation of Value at Risk

## 4. Results

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The histogram of the log returns of the exchange rates of Bitcoin, Dash, Dogecoin, Litecoin, Maidsafecoin, Monero and Ripple from 22 June 2014–17 May 2017.

**Table 1.**Summary statistics of the exchange rates of Bitcoin, Dash, Dogecoin, Litecoin, Maidsafecoin, Monero and Ripple from 22 June 2014–17 May 2017.

Statistic | Bitcoin | Dash | Dogecoin | Litecoin | Maidsafecoin | Monero | Ripple |
---|---|---|---|---|---|---|---|

Minimum | 594.069 | 9.834 | 0.000 | 9.772 | 0.014 | 3.984 | $-0.632$ |

Q1 | 588.454 | 9.551 | 0.000 | 9.741 | 0.014 | 3.224 | $-0.020$ |

Median | 570.611 | 9.045 | 0.000 | 9.241 | 0.014 | 3.299 | $-0.002$ |

Mean | 582.795 | 10.050 | 0.000 | 9.134 | 0.015 | 2.957 | 0.004 |

Q3 | 605.908 | 10.147 | 0.000 | 9.342 | 0.015 | 2.253 | 0.018 |

Maximum | 598.986 | 9.518 | 0.000 | 9.253 | 0.014 | 2.557 | 1.020 |

Skewness | 603.710 | 9.267 | 0.000 | 9.002 | 0.015 | 2.559 | 2.579 |

Kurtosis | 640.815 | 8.958 | 0.000 | 9.008 | 0.017 | 2.517 | 47.042 |

SD | 642.122 | 7.936 | 0.000 | 8.192 | 0.016 | 2.352 | 0.073 |

Variance | 650.489 | 6.635 | 0.000 | 8.185 | 0.017 | 2.309 | 0.005 |

CV | 643.383 | 7.757 | 0.000 | 8.025 | 0.017 | 2.660 | 17.048 |

Range | 630.412 | 7.504 | 0.000 | 7.303 | 0.018 | 2.646 | 1.651 |

IQR | 629.299 | 7.006 | 0.000 | 7.286 | 0.020 | 2.469 | 0.038 |

**Table 2.**p-values of the unconditional (conditional) coverage value at risk exceedance test for the log returns of the exchange rates of Bitcoin from 22 June 2014–17 May 2017.

Exceedance Probability | ||||||
---|---|---|---|---|---|---|

0.05 | 0.025 | 0.01 | 0.005 | 0.001 | 0.0001 | |

SGARCH | 0.078 (0.312) | 0.099 (0.238) | 0.263 (0.173) | 0.284 (0.266) | 0.264 (0.104) | 0.350 (0.138) |

EGARCH | 0.287 (0.125) | 0.238 (0.349) | 0.227 (0.127) | 0.319 (0.385) | 0.231 (0.094) | 0.094 (0.091) |

GJRGARCH | 0.242 (0.120) | 0.356 (0.394) | 0.384 (0.393) | 0.198 (0.297) | 0.299 (0.118) | 0.341 (0.300) |

APARCH | 0.398 (0.334) | 0.280 (0.331) | 0.071 (0.344) | 0.088 (0.193) | 0.267 (0.137) | 0.284 (0.184) |

IGARCH | 0.128 (0.086) | 0.301 (0.356) | 0.144 (0.053) | 0.302 (0.115) | 0.398 (0.249) | 0.145 (0.261) |

CSGARCH | 0.256 (0.243) | 0.179 (0.149) | 0.115 (0.175) | 0.384 (0.140) | 0.137 (0.109) | 0.261 (0.319) |

GARCH | 0.323 (0.336) | 0.387 (0.347) | 0.100 (0.242) | 0.069 (0.396) | 0.334 (0.238) | 0.097 (0.092) |

TGARCH | 0.253 (0.189) | 0.358 (0.165) | 0.259 (0.286) | 0.089 (0.126) | 0.213 (0.102) | 0.143 (0.143) |

AVGARCH | 0.203 (0.347) | 0.348 (0.079) | 0.277 (0.376) | 0.082 (0.082) | 0.371 (0.052) | 0.208 (0.240) |

NGARCH | 0.097 (0.194) | 0.064 (0.290) | 0.199 (0.240) | 0.064 (0.204) | 0.100 (0.127) | 0.266 (0.186) |

NAGARCH | 0.199 (0.069) | 0.072 (0.149) | 0.185 (0.061) | 0.216 (0.167) | 0.285 (0.121) | 0.062 (0.099) |

ALLGARCH | 0.271 (0.327) | 0.201 (0.072) | 0.262 (0.097) | 0.114 (0.320) | 0.162 (0.180) | 0.167 (0.249) |

**Table 3.**p-values of the unconditional (conditional) coverage value at risk exceedance test for the log returns of the exchange rates of Dash from 22 June 2014–17 May 2017.

Exceedance Probability | ||||||
---|---|---|---|---|---|---|

0.05 | 0.025 | 0.01 | 0.005 | 0.001 | 0.0001 | |

SGARCH | 0.358 (0.383) | 0.051 (0.231) | 0.357 (0.129) | 0.122 (0.167) | 0.150 (0.234) | 0.146 (0.162) |

EGARCH | 0.396 (0.340) | 0.333 (0.225) | 0.374 (0.157) | 0.216 (0.170) | 0.220 (0.138) | 0.132 (0.196) |

GJRGARCH | 0.282 (0.254) | 0.115 (0.196) | 0.218 (0.280) | 0.141 (0.262) | 0.069 (0.358) | 0.211 (0.328) |

APARCH | 0.257 (0.304) | 0.290 (0.194) | 0.087 (0.204) | 0.238 (0.212) | 0.304 (0.182) | 0.314 (0.240) |

IGARCH | 0.222 (0.241) | 0.110 (0.262) | 0.305 (0.297) | 0.206 (0.364) | 0.068 (0.374) | 0.390 (0.100) |

CSGARCH | 0.268 (0.055) | 0.378 (0.391) | 0.352 (0.276) | 0.286 (0.302) | 0.100 (0.368) | 0.134 (0.081) |

GARCH | 0.200 (0.145) | 0.165 (0.080) | 0.186 (0.293) | 0.272 (0.077) | 0.323 (0.243) | 0.265 (0.298) |

TGARCH | 0.304 (0.121) | 0.299 (0.155) | 0.231 (0.173) | 0.264 (0.385) | 0.092 (0.337) | 0.376 (0.305) |

AVGARCH | 0.227 (0.335) | 0.261 (0.303) | 0.151 (0.214) | 0.179 (0.101) | 0.359 (0.052) | 0.364 (0.392) |

NGARCH | 0.069 (0.212) | 0.054 (0.246) | 0.216 (0.259) | 0.222 (0.119) | 0.138 (0.145) | 0.162 (0.240) |

NAGARCH | 0.116 (0.180) | 0.129 (0.302) | 0.332 (0.183) | 0.179 (0.354) | 0.344 (0.397) | 0.196 (0.339) |

ALLGARCH | 0.069 (0.195) | 0.380 (0.378) | 0.350 (0.075) | 0.152 (0.263) | 0.243 (0.256) | 0.172 (0.243) |

**Table 4.**p-values of the unconditional (conditional) coverage value at risk exceedance test for the log returns of the exchange rates of Dogecoin from 22 June 2014–17 May 2017.

Exceedance Probability | ||||||
---|---|---|---|---|---|---|

0.05 | 0.025 | 0.01 | 0.005 | 0.001 | 0.0001 | |

SGARCH | 0.063 (0.109) | 0.178 (0.264) | 0.297 (0.230) | 0.101 (0.256) | 0.178 (0.280) | 0.246 (0.393) |

EGARCH | 0.398 (0.069) | 0.077 (0.107) | 0.370 (0.214) | 0.079 (0.339) | 0.347 (0.274) | 0.340 (0.210) |

GJRGARCH | 0.215 (0.095) | 0.178 (0.114) | 0.061 (0.073) | 0.298 (0.266) | 0.116 (0.302) | 0.342 (0.380) |

APARCH | 0.368 (0.178) | 0.264 (0.359) | 0.381 (0.103) | 0.221 (0.326) | 0.225 (0.100) | 0.222 (0.361) |

IGARCH | 0.098 (0.073) | 0.164 (0.070) | 0.172 (0.115) | 0.187 (0.136) | 0.375 (0.227) | 0.382 (0.380) |

CSGARCH | 0.096 (0.267) | 0.063 (0.181) | 0.324 (0.069) | 0.200 (0.354) | 0.223 (0.237) | 0.264 (0.292) |

GARCH | 0.132 (0.377) | 0.342 (0.133) | 0.332 (0.054) | 0.137 (0.388) | 0.137 (0.084) | 0.386 (0.197) |

TGARCH | 0.178 (0.346) | 0.329 (0.211) | 0.250 (0.329) | 0.141 (0.181) | 0.186 (0.205) | 0.137 (0.276) |

AVGARCH | 0.254 (0.218) | 0.181 (0.122) | 0.394 (0.280) | 0.150 (0.385) | 0.118 (0.369) | 0.208 (0.106) |

NGARCH | 0.086 (0.118) | 0.396 (0.314) | 0.144 (0.244) | 0.120 (0.242) | 0.169 (0.318) | 0.341 (0.303) |

NAGARCH | 0.282 (0.284) | 0.106 (0.127) | 0.342 (0.234) | 0.342 (0.321) | 0.309 (0.288) | 0.308 (0.105) |

ALLGARCH | 0.252 (0.267) | 0.223 (0.189) | 0.242 (0.201) | 0.290 (0.249) | 0.319 (0.317) | 0.178 (0.226) |

**Table 5.**p-values of the unconditional (conditional) coverage value at risk exceedance test for the log returns of the exchange rates of Litecoin from 22 June 2014–17 May 2017.

Exceedance Probability | ||||||
---|---|---|---|---|---|---|

0.05 | 0.025 | 0.01 | 0.005 | 0.001 | 0.0001 | |

SGARCH | 0.056 (0.092) | 0.166 (0.267) | 0.383 (0.156) | 0.134 (0.187) | 0.231 (0.099) | 0.352 (0.261) |

EGARCH | 0.134 (0.139) | 0.209 (0.370) | 0.062 (0.390) | 0.297 (0.215) | 0.256 (0.200) | 0.127 (0.074) |

GJRGARCH | 0.052 (0.245) | 0.091 (0.153) | 0.204 (0.275) | 0.081 (0.194) | 0.211 (0.089) | 0.187 (0.055) |

APARCH | 0.362 (0.180) | 0.292 (0.229) | 0.280 (0.294) | 0.322 (0.161) | 0.391 (0.170) | 0.095 (0.138) |

IGARCH | 0.358 (0.242) | 0.106 (0.391) | 0.068 (0.087) | 0.117 (0.051) | 0.298 (0.053) | 0.108 (0.310) |

CSGARCH | 0.092 (0.117) | 0.267 (0.129) | 0.102 (0.318) | 0.379 (0.234) | 0.241 (0.345) | 0.261 (0.371) |

GARCH | 0.158 (0.114) | 0.109 (0.307) | 0.350 (0.265) | 0.399 (0.339) | 0.309 (0.354) | 0.337 (0.361) |

TGARCH | 0.326 (0.268) | 0.207 (0.397) | 0.090 (0.179) | 0.392 (0.223) | 0.148 (0.144) | 0.158 (0.192) |

AVGARCH | 0.108 (0.098) | 0.131 (0.167) | 0.178 (0.069) | 0.054 (0.287) | 0.374 (0.286) | 0.171 (0.104) |

NGARCH | 0.259 (0.072) | 0.297 (0.097) | 0.085 (0.219) | 0.344 (0.085) | 0.227 (0.185) | 0.386 (0.295) |

NAGARCH | 0.313 (0.338) | 0.053 (0.274) | 0.390 (0.148) | 0.251 (0.276) | 0.251 (0.106) | 0.230 (0.094) |

ALLGARCH | 0.080 (0.236) | 0.357 (0.299) | 0.184 (0.202) | 0.115 (0.154) | 0.065 (0.209) | 0.116 (0.199) |

**Table 6.**p-values of the unconditional (conditional) coverage value at risk exceedance test for the log returns of the exchange rates of Maidsafecoin from 22 June 2014–17 May 2017.

Exceedance Probability | ||||||
---|---|---|---|---|---|---|

0.05 | 0.025 | 0.01 | 0.005 | 0.001 | 0.0001 | |

SGARCH | 0.304 (0.218) | 0.099 (0.128) | 0.318 (0.080) | 0.072 (0.287) | 0.271 (0.192) | 0.351 (0.310) |

EGARCH | 0.337 (0.315) | 0.308 (0.128) | 0.229 (0.152) | 0.178 (0.051) | 0.139 (0.324) | 0.382 (0.091) |

GJRGARCH | 0.181 (0.215) | 0.054 (0.353) | 0.375 (0.144) | 0.173 (0.262) | 0.342 (0.378) | 0.090 (0.252) |

APARCH | 0.386 (0.067) | 0.387 (0.343) | 0.116 (0.165) | 0.108 (0.272) | 0.198 (0.197) | 0.374 (0.092) |

IGARCH | 0.385 (0.280) | 0.308 (0.189) | 0.184 (0.177) | 0.096 (0.225) | 0.259 (0.241) | 0.155 (0.284) |

CSGARCH | 0.379 (0.190) | 0.175 (0.158) | 0.315 (0.246) | 0.148 (0.125) | 0.133 (0.054) | 0.379 (0.356) |

GARCH | 0.316 (0.130) | 0.172 (0.083) | 0.388 (0.183) | 0.385 (0.298) | 0.198 (0.104) | 0.239 (0.228) |

TGARCH | 0.397 (0.351) | 0.372 (0.069) | 0.377 (0.305) | 0.243 (0.255) | 0.142 (0.195) | 0.081 (0.150) |

AVGARCH | 0.087 (0.227) | 0.136 (0.278) | 0.397 (0.228) | 0.195 (0.348) | 0.260 (0.308) | 0.124 (0.153) |

NGARCH | 0.077 (0.177) | 0.314 (0.398) | 0.214 (0.247) | 0.384 (0.147) | 0.265 (0.063) | 0.320 (0.135) |

NAGARCH | 0.369 (0.292) | 0.115 (0.205) | 0.058 (0.180) | 0.100 (0.258) | 0.226 (0.144) | 0.330 (0.249) |

ALLGARCH | 0.077 (0.266) | 0.074 (0.207) | 0.244 (0.302) | 0.335 (0.287) | 0.275 (0.352) | 0.091 (0.389) |

**Table 7.**p-values of the unconditional (conditional) coverage value at risk exceedance test for the log returns of the exchange rates of Monero from 22 June 2014–17 May 2017.

Exceedance Probability | ||||||
---|---|---|---|---|---|---|

0.05 | 0.025 | 0.01 | 0.005 | 0.001 | 0.0001 | |

SGARCH | 0.251 (0.095) | 0.105 (0.294) | 0.158 (0.136) | 0.081 (0.388) | 0.386 (0.077) | 0.382 (0.397) |

EGARCH | 0.098 (0.077) | 0.302 (0.319) | 0.364 (0.118) | 0.398 (0.385) | 0.186 (0.198) | 0.326 (0.300) |

GJRGARCH | 0.317 (0.140) | 0.234 (0.124) | 0.088 (0.350) | 0.174 (0.092) | 0.122 (0.240) | 0.072 (0.262) |

APARCH | 0.077 (0.311) | 0.298 (0.260) | 0.149 (0.179) | 0.211 (0.264) | 0.381 (0.090) | 0.232 (0.287) |

IGARCH | 0.275 (0.179) | 0.057 (0.087) | 0.060 (0.398) | 0.379 (0.310) | 0.288 (0.254) | 0.247 (0.133) |

CSGARCH | 0.350 (0.123) | 0.109 (0.330) | 0.331 (0.343) | 0.372 (0.174) | 0.241 (0.051) | 0.238 (0.091) |

GARCH | 0.198 (0.093) | 0.162 (0.380) | 0.316 (0.395) | 0.249 (0.309) | 0.388 (0.097) | 0.224 (0.053) |

TGARCH | 0.228 (0.060) | 0.233 (0.158) | 0.170 (0.156) | 0.148 (0.150) | 0.236 (0.135) | 0.162 (0.333) |

AVGARCH | 0.079 (0.233) | 0.399 (0.293) | 0.376 (0.259) | 0.224 (0.229) | 0.216 (0.240) | 0.371 (0.235) |

NGARCH | 0.233 (0.228) | 0.215 (0.266) | 0.326 (0.385) | 0.231 (0.056) | 0.312 (0.193) | 0.258 (0.370) |

NAGARCH | 0.395 (0.102) | 0.105 (0.130) | 0.292 (0.242) | 0.354 (0.116) | 0.170 (0.207) | 0.121 (0.114) |

ALLGARCH | 0.337 (0.245) | 0.275 (0.131) | 0.221 (0.223) | 0.169 (0.304) | 0.170 (0.197) | 0.086 (0.181) |

**Table 8.**p-values of the unconditional (conditional) coverage value at risk exceedance test for the log returns of the exchange rates of Ripple from 22 June 2014–17 May 2017.

Exceedance Probability | ||||||
---|---|---|---|---|---|---|

0.05 | 0.025 | 0.01 | 0.005 | 0.001 | 0.0001 | |

SGARCH | 0.249 (0.368) | 0.318 (0.256) | 0.353 (0.249) | 0.193 (0.075) | 0.199 (0.243) | 0.063 (0.131) |

EGARCH | 0.259 (0.287) | 0.281 (0.191) | 0.309 (0.157) | 0.167 (0.222) | 0.086 (0.109) | 0.256 (0.084) |

GJRGARCH | 0.143 (0.303) | 0.219 (0.135) | 0.394 (0.085) | 0.095 (0.319) | 0.299 (0.184) | 0.308 (0.224) |

APARCH | 0.312 (0.083) | 0.356 (0.162) | 0.125 (0.097) | 0.216 (0.126) | 0.138 (0.075) | 0.177 (0.103) |

IGARCH | 0.288 (0.331) | 0.071 (0.246) | 0.053 (0.154) | 0.113 (0.063) | 0.367 (0.234) | 0.265 (0.109) |

CSGARCH | 0.096 (0.386) | 0.114 (0.207) | 0.065 (0.312) | 0.117 (0.398) | 0.308 (0.380) | 0.069 (0.070) |

GARCH | 0.305 (0.256) | 0.083 (0.332) | 0.245 (0.070) | 0.333 (0.379) | 0.275 (0.258) | 0.209 (0.277) |

TGARCH | 0.149 (0.293) | 0.327 (0.342) | 0.076 (0.399) | 0.176 (0.236) | 0.369 (0.289) | 0.307 (0.136) |

AVGARCH | 0.093 (0.084) | 0.309 (0.259) | 0.222 (0.210) | 0.071 (0.327) | 0.187 (0.395) | 0.109 (0.300) |

NGARCH | 0.225 (0.143) | 0.246 (0.150) | 0.317 (0.320) | 0.105 (0.099) | 0.134 (0.233) | 0.210 (0.249) |

NAGARCH | 0.141 (0.350) | 0.050 (0.242) | 0.189 (0.207) | 0.259 (0.227) | 0.379 (0.289) | 0.075 (0.205) |

ALLGARCH | 0.283 (0.253) | 0.343 (0.068) | 0.081 (0.236) | 0.309 (0.298) | 0.386 (0.153) | 0.162 (0.304) |

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**MDPI and ACS Style**

Chu, J.; Chan, S.; Nadarajah, S.; Osterrieder, J. GARCH Modelling of Cryptocurrencies. *J. Risk Financial Manag.* **2017**, *10*, 17.
https://doi.org/10.3390/jrfm10040017

**AMA Style**

Chu J, Chan S, Nadarajah S, Osterrieder J. GARCH Modelling of Cryptocurrencies. *Journal of Risk and Financial Management*. 2017; 10(4):17.
https://doi.org/10.3390/jrfm10040017

**Chicago/Turabian Style**

Chu, Jeffrey, Stephen Chan, Saralees Nadarajah, and Joerg Osterrieder. 2017. "GARCH Modelling of Cryptocurrencies" *Journal of Risk and Financial Management* 10, no. 4: 17.
https://doi.org/10.3390/jrfm10040017