# Multivariate Analysis of Cryptocurrencies

## Abstract

**:**

## 1. Introduction

## 2. Methodology

## 3. Empirical Analysis

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DCC | Dynamic conditional correlation model (Engle 2002) |

cDCC | Corrected DCC (Aielli 2013) |

DECO | Dynamic equicorrelation (Engle and Kelly 2012) |

RM | RiskMetrics |

MIDAS | Mixing-Data Sampling |

DAGM | Double Asymmetric GARCH–MIDAS model (Amendola et al. 2019) |

M1 | GARCH for the univariate part and cDCC for the correlations |

M2 | GARCH for the univariate part and DCC-MIDAS for the correlations |

M3 | GARCH for the univariate part and DECO for the correlations |

M4 | DAGM for the univariate part and cDCC for the correlations |

M5 | DAGM for the univariate part and DCC-MIDAS for the correlations |

M6 | DAGM for the univariate part and DECO for the correlations |

M7 | RiskMetrics |

SSM | Set of Superior Models |

MCS | Model Confidence Set (Hansen et al. 2011) |

GMV | Global Minimum Variance |

## Note

1 | In the case of the RiskMetrics model, the covariance matrix is calculated and not estimated. |

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**Figure 2.**Plots of Google trends. Sample period: June 2016 to December 2020. Number of monthly observations: 55.

**Figure 4.**Plots of volatilities (main diagonal) and correlations for the model M4: DAGM for the univariate part and cDCC for the correlations. Sample period: January 2017 to December 2020. Number of daily observations: 1457.

**Figure 5.**Plots of volatilities (main diagonal) and correlations for the model M5: DAGM for the univariate part and DCC-MIDAS for the correlations. Red lines represent the long-run correlation. Sample period: January 2017 to December 2020. Number of daily observations: 1457.

**Figure 6.**Plots of volatilities (main diagonal) and correlations for the model M6: DAGM for the univariate part and DECO for the correlations. Sample period: January 2017 to December 2020. Number of daily observations: 1457.

Model | Functional Form |
---|---|

cDCC (Aielli 2013) | ${H}_{i,t}={D}_{i,t}{R}_{i,t}{D}_{i,t}$ |

${D}_{i,t}=diag({\sigma}_{i,t,1},\cdots ,{\sigma}_{i,t,j},\cdots ,{\sigma}_{i,t,n})$ | |

${R}_{i,t}={(\mathit{diag}({Q}_{i,t}))}^{-1/2}{Q}_{i,t}{(\mathit{diag}({Q}_{i,t}))}^{-1/2}$ | |

${Q}_{i,t}=(1-a-b)\Psi +a({\mathit{\xi}}_{i-1,t}{\mathit{\xi}}_{i-1,t}^{{}^{\prime}})+$ | |

$\hspace{1em}\hspace{1em}b{Q}_{i-1,t}$ | |

${\mathit{\xi}}_{i,t}={D}_{i,t}^{-1}{\mathit{r}}_{i,t}$ | |

$\Psi =E\left({\mathit{\xi}}_{i,t}{\mathit{\xi}}_{i,t}^{{}^{\prime}}\right)$ | |

DCC-MIDAS (Colacito et al. 2011) | ${H}_{i,t}={D}_{i,t}{R}_{i,t}{D}_{i,t}$ |

${D}_{i,t}=diag({\sigma}_{i,t,1},\cdots ,{\sigma}_{i,t,j},\cdots ,{\sigma}_{i,t,n})$ | |

${R}_{i,t}={(\mathit{diag}({Q}_{i,t}))}^{-1/2}{Q}_{i,t}{(\mathit{diag}({Q}_{i,t}))}^{-1/2}$ | |

${Q}_{i,t}=(1-a-b){\overline{R}}_{i,t}(\omega )+a({\mathit{\xi}}_{i-1,t}{\mathit{\xi}}_{i-1,t}^{{}^{\prime}})+$ | |

$\hspace{1em}\hspace{1em}b{Q}_{i-1,t}$ | |

${\mathit{\xi}}_{i,t}={D}_{i,t}^{-1}{\mathit{r}}_{i,t}$ | |

DECO (Engle and Kelly 2012) | ${H}_{i,t}={D}_{i,t}{R}_{i,t}^{DECO}{D}_{i,t}$ |

${D}_{i,t}=diag({\sigma}_{i,t,1},\cdots ,{\sigma}_{i,t,j},\cdots ,{\sigma}_{i,t,n})$ | |

${R}_{i,t}^{DECO}=(1-{\rho}_{i,t}){I}_{n}+{\rho}_{i,t}{J}_{n}$ | |

${\rho}_{i,t}=\frac{1}{n(n-1)}\left({\iota}^{{}^{\prime}}{R}_{i,t}\iota -n\right)$ | |

RiskMetrics | ${H}_{i,t}=\lambda {\mathit{r}}_{i-1,t}{\mathit{r}}_{i-1,t}^{\prime}+(1-\lambda ){H}_{i-1,t}$ |

$\lambda =0.94$ |

**Notes**: The table reports the multivariate model dynamics of the different specifications employed in the empirical section. ${\overline{R}}_{i,t}\left(\omega \right)$ represents the long-run correlation as in Equation (2.12) of Colacito et al. (2011). ${I}_{n}$ denotes the identity matrix, ${J}_{n}$ a matrix of ones, and $\iota $ a $n\times 1$ vector of ones.

Min. | Max. | Mean | SD | Skew. | Kurt. | |
---|---|---|---|---|---|---|

Bitcoin | −0.465 | 0.225 | 0.002 | 0.041 | −0.911 | 13.535 |

Ethereum | −0.551 | 0.290 | 0.002 | 0.056 | −0.538 | 9.586 |

Lithium | −0.449 | 0.511 | 0.002 | 0.058 | 0.717 | 11.599 |

Stellar | −0.410 | 0.723 | 0.003 | 0.077 | 1.968 | 17.156 |

Ripple | −0.616 | 1.027 | 0.002 | 0.071 | 2.407 | 39.494 |

Dashcoin | −0.459 | 0.438 | 0.001 | 0.058 | 0.605 | 9.009 |

Dogecoin | −0.493 | 0.477 | 0.002 | 0.062 | 0.792 | 12.843 |

**Notes**: The table presents the main statistics (the minimum (Min.) and maximum (Max.), the mean, standard deviation (SD), skewness (Skew.) and excess kurtosis (Kurt.)) for the close-to-close log-returns. Sample period: June 2016–December 2020. Number of observations: 1671.

Estimate | Std. Error | t Value | Pr(>|t|) | Sig. | |
---|---|---|---|---|---|

Bitcoin | |||||

$const$ | 0.000 | 0.000 | 2.856 | 0.004 | *** |

$\alpha $ | 0.181 | 0.053 | 3.393 | 0.001 | *** |

$\beta $ | 0.790 | 0.029 | 27.214 | 0.000 | *** |

Ethereum | |||||

$const$ | 0.000 | 0.000 | 2.892 | 0.004 | *** |

$\alpha $ | 0.182 | 0.044 | 4.103 | 0.000 | *** |

$\beta $ | 0.733 | 0.059 | 12.424 | 0.000 | *** |

Lithium | |||||

$const$ | 0.000 | 0.000 | 1.493 | 0.135 | |

$\alpha $ | 0.071 | 0.028 | 2.562 | 0.010 | ** |

$\beta $ | 0.878 | 0.046 | 19.096 | 0.000 | *** |

Stellar | |||||

$const$ | 0.000 | 0.000 | 1.744 | 0.081 | * |

$\alpha $ | 0.219 | 0.070 | 3.135 | 0.002 | *** |

$\beta $ | 0.752 | 0.079 | 9.534 | 0.000 | *** |

Ripple | |||||

$const$ | 0.000 | 0.000 | 2.281 | 0.023 | ** |

$\alpha $ | 0.374 | 0.152 | 2.455 | 0.014 | ** |

$\beta $ | 0.609 | 0.117 | 5.225 | 0.000 | *** |

Dashcoin | |||||

$const$ | 0.000 | 0.000 | 2.663 | 0.008 | *** |

$\alpha $ | 0.236 | 0.070 | 3.394 | 0.001 | *** |

$\beta $ | 0.744 | 0.062 | 11.990 | 0.000 | *** |

Dogecoin | |||||

$const$ | 0.000 | 0.000 | 2.585 | 0.010 | *** |

$\alpha $ | 0.230 | 0.044 | 5.276 | 0.000 | *** |

$\beta $ | 0.769 | 0.039 | 19.623 | 0.000 | *** |

**Notes**: The table reports the estimates of the univariate GARCH models used in the M2, M3 and M4 specifications. Column Std. Error reports the Quasi-Maximum Likelihood standard errors. Sample Period: June 2016 to December 2020. Number of daily observations: 1671. *, ** and *** represent the significance at levels $10\%,5\%,1\%$, respectively.

Estimate | Std. Error | t Value | Pr(>|t|) | Sig. | |
---|---|---|---|---|---|

Bitcoin | |||||

$\alpha $ | 0.192 | 0.048 | 4.037 | 0.000 | *** |

$\beta $ | 0.765 | 0.027 | 27.837 | 0.000 | *** |

m | −6.381 | 0.977 | −6.532 | 0.000 | *** |

${\theta}^{+}$ | 0.364 | 0.169 | 2.156 | 0.031 | ** |

${\omega}_{2}^{+}$ | 1.739 | 0.720 | 2.415 | 0.016 | ** |

${\theta}^{-}$ | 0.183 | 0.129 | 1.421 | 0.155 | |

${\omega}_{2}^{-}$ | 2.698 | 0.976 | 2.763 | 0.006 | *** |

Ethereum | |||||

$\alpha $ | 0.189 | 0.041 | 4.548 | 0.000 | *** |

$\beta $ | 0.698 | 0.066 | 10.622 | 0.000 | *** |

m | −6.029 | 0.314 | −19.201 | 0.000 | *** |

${\theta}^{+}$ | 0.218 | 0.080 | 2.729 | 0.006 | *** |

${\omega}_{2}^{+}$ | 2.323 | 0.638 | 3.639 | 0.000 | *** |

${\theta}^{-}$ | 0.071 | 0.049 | 1.434 | 0.152 | |

${\omega}_{2}^{-}$ | 8.893 | 4.558 | 1.951 | 0.051 | * |

Lithium | |||||

$\alpha $ | 0.072 | 0.034 | 2.156 | 0.031 | ** |

$\beta $ | 0.849 | 0.073 | 11.560 | 0.000 | *** |

m | −5.876 | 0.294 | −19.970 | 0.000 | *** |

${\theta}^{+}$ | 0.043 | 0.022 | 1.944 | 0.052 | * |

${\omega}_{2}^{+}$ | 2.148 | 1.036 | 2.074 | 0.038 | ** |

${\theta}^{-}$ | 0.033 | 0.027 | 1.194 | 0.233 | |

${\omega}_{2}^{-}$ | 3.764 | 4.435 | 0.849 | 0.396 | |

Stellar | |||||

$\alpha $ | 0.274 | 0.091 | 2.991 | 0.003 | *** |

$\beta $ | 0.501 | 0.156 | 3.217 | 0.001 | *** |

m | −5.000 | 0.275 | −18.181 | 0.000 | *** |

${\theta}^{+}$ | 0.059 | 0.020 | 2.988 | 0.003 | *** |

${\omega}_{2}^{+}$ | 1.758 | 0.334 | 5.263 | 0.000 | *** |

${\theta}^{-}$ | 0.092 | 0.020 | 4.677 | 0.000 | *** |

${\omega}_{2}^{-}$ | 1.209 | 0.158 | 7.676 | 0.000 | *** |

Ripple | |||||

$\alpha $ | 0.346 | 0.114 | 3.031 | 0.002 | *** |

$\beta $ | 0.569 | 0.110 | 5.174 | 0.000 | *** |

m | −5.313 | 0.629 | −8.447 | 0.000 | *** |

${\theta}^{+}$ | 0.036 | 0.014 | 2.628 | 0.009 | *** |

${\omega}_{2}^{+}$ | 4.307 | 1.350 | 3.190 | 0.001 | *** |

${\theta}^{-}$ | 0.013 | 0.006 | 2.218 | 0.027 | ** |

${\omega}_{2}^{-}$ | 8.188 | 1.714 | 4.777 | 0.000 | *** |

Dashcoin | |||||

$\alpha $ | 0.347 | 0.114 | 3.039 | 0.002 | *** |

$\beta $ | 0.652 | 0.114 | 5.696 | 0.000 | *** |

m | −0.074 | 0.309 | −0.238 | 0.812 | |

${\theta}^{+}$ | −0.046 | 0.049 | −0.957 | 0.339 | |

${\omega}_{2}^{+}$ | 1.992 | 0.908 | 2.193 | 0.028 | ** |

${\theta}^{-}$ | 0.077 | 0.041 | 1.895 | 0.058 | * |

${\omega}_{2}^{-}$ | 1.986 | 0.827 | 2.402 | 0.016 | ** |

Dogecoin | |||||

$\alpha $ | 0.226 | 0.042 | 5.374 | 0.000 | *** |

$\beta $ | 0.743 | 0.052 | 14.204 | 0.000 | *** |

m | −5.615 | 0.613 | −9.167 | 0.000 | *** |

${\theta}^{+}$ | 0.060 | 0.022 | 2.663 | 0.008 | *** |

${\omega}_{2}^{+}$ | 2.064 | 0.570 | 3.618 | 0.000 | *** |

${\theta}^{-}$ | 0.021 | 0.025 | 0.827 | 0.408 | |

${\omega}_{2}^{-}$ | 1.465 | 0.761 | 1.925 | 0.054 | * |

**Notes**: The table reports the estimates of the univariate DAGM models used in the M4, M5 and M6 specifications. Column Std. Error reports the Quasi-Maximum Likelihood based standard errors. The MIDAS variables are represented by the first difference of the Google trends. Sample Period: June 2016 to December 2020. Number of daily observations: 1671. *, ** and *** represent the significance at levels $10\%,5\%,1\%$, respectively.

M1 | M2 | M3 | M4 | M5 | M6 | M7 | |
---|---|---|---|---|---|---|---|

a | 0.017 ** | 0.038 | 0.086 ** | 0.018 *** | 0.034 | 0.089 *** | |

(0.008) | (0.031) | (0.044) | (0.006) | (0.161) | (0.022) | ||

b | 0.982 *** | 0.931 *** | 0.913 *** | 0.981 *** | 0.937 * | 0.91 *** | |

(0.008) | (0.089) | (0.054) | (0.007) | (0.483) | (0.019) | ||

$\omega $ | 1.001 | 1.001 | |||||

(1.793) | (11.109) | ||||||

Euclidean | 4.496 | 4.503 | 4.515 | 4.413 | 4.419 | 4.447 | 6.683 |

Sq. Frobenius | 2.630 | 2.630 | 2.630 | 2.556 | 2.556 | 2.556 | 3.650 |

RMSE | 1.526 | 1.531 | 1.528 | 1.515 | 1.522 | 1.519 | 1.563 |

$LL(1)$ | 11.941 *** | 14.631 *** | 8.657 *** | 5.659 | 6.281 | 5.448 | 0.004 |

$LL(2)$ | 12.487 *** | 15.525 *** | 8.744 ** | 5.756 | 6.521 | 5.525 | 0.004 |

**Notes**: Top panel reports the estimates for the correlation models. Numbers in parentheses are the Quasi-Maximum Likelihood standard errors. Bottom panel reports the average losses, according to the three loss functions in the first column. Sq. Frobenius stands for Squared Frobenius and RMSE for Root Mean Squared Error. The reported averages for the Euclidean, Squared Frobenius, and RMSE have been multiplied by 1000, 1000, and 10, respectively. Shades of gray denote inclusion in the SSM at significance level $\alpha =0.10$. $LL(1)$ and $LL(2)$ report the $LL$ test statistics, whose null is of no conditional heteroscedasticity. *, ** and *** represent the significance at levels $10\%,5\%,1\%$, respectively. Estimation period: June 2016 to December 2020, number of daily observations: 1671. Evaluation period: January 2017 to December 2020, number of daily observations: 1457.

M1 | M2 | M3 | M4 | M5 | M6 | M7 | |
---|---|---|---|---|---|---|---|

$\nu $ | 2.224 | 2.249 | 2.204 | 2.369 | 2.380 | 2.368 | 2.199 |

$QL(0.01)$ | 0.208 | 0.2 | 0.209 | 0.185 | 0.183 | 0.184 | 0.904 |

$QL(0.05)$ | 0.576 | 0.562 | 0.584 | 0.542 | 0.537 | 0.542 | 1.201 |

$QL(0.10)$ | 0.803 | 0.788 | 0.813 | 0.781 | 0.776 | 0.783 | 1.323 |

**Notes**: Top panel reports the estimates for the degrees of freedom ν of the Student’s t distribution, used to calculate the VaR series. Bottom panel reports the average QL losses, according to the three τ levels in the first column. Shades of gray denote inclusion in the SSM at significance level α = 0.10. Estimation period: June 2016 to December 2020, number of daily observations: 1671. Evaluation period: January 2017 to December 2020, number of daily observations: 1457.

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**MDPI and ACS Style**

Candila, V.
Multivariate Analysis of Cryptocurrencies. *Econometrics* **2021**, *9*, 28.
https://doi.org/10.3390/econometrics9030028

**AMA Style**

Candila V.
Multivariate Analysis of Cryptocurrencies. *Econometrics*. 2021; 9(3):28.
https://doi.org/10.3390/econometrics9030028

**Chicago/Turabian Style**

Candila, Vincenzo.
2021. "Multivariate Analysis of Cryptocurrencies" *Econometrics* 9, no. 3: 28.
https://doi.org/10.3390/econometrics9030028