# Volatility Co-Movement between Bitcoin and Stablecoins: BEKK–GARCH and Copula–DCC–GARCH Approaches

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- How can stablecoins stabilize cryptocurrencies? What potential role will stablecoins perform in the violent price fluctuations in conventional cryptocurrencies?
- (2)
- How do they play the roles of the gold- or USD-pegged stablecoins as safe-havens, diversifiers, or hedges against conventional cryptocurrencies during the financial crisis?

## 2. Literature Review

## 3. Methodology and Econometric Model

#### 3.1. Range-Based Volatility for Various Variance Estimators

#### 3.2. The BEKK–GARCH Model with Low, High, and Closing Prices

**A**,

**B**represent the square coefficient matrices that are diagonalized to ensure covariance stationarity, and $\mathit{C}$ denotes an upper triangular matrix. Therefore, the representative matrices of parameters are $\mathit{C}$,

**A**, and

**B**and can be written as follows:

**A**and

**B**also depict the estimators indicating the impacts of the ARCH effect (short-term volatility shocks) and the GARCH effect (long-run volatility shocks), respectively. Let us postulate that the conditional variance for each price changes, ${h}_{ij,t}^{}$ follows a bivariate GARCH process, and can be expressed as

#### 3.3. The Copula–DCC–GARCH Model

#### 3.4. VAR Granger-Causal Perspective

**Y**

_{t}as

**Y**

_{t}= (BTC

_{t}, USDT

_{t}, USDC

_{t}, BUSD

_{t}, TUSD

_{t}, DAI

_{t})′, we use the vector autoregression (VAR) model as follows:

**Y**

_{t}=

**c**+

**A**

_{1}

**Y**

_{t}

_{−1}+ ⋯ +

**A**

_{p}

**Y**

_{t}

_{−}

_{p}+

**u**

_{t},

**A**

_{1}, …,

**A**

_{p}represent 6 × 6 parameter matrices, and

**u**

_{t}is the residual term distributed as

**u**

_{t}∼ (0,

**Σ**

_{u}). Where

**Σ**

_{u}is the corresponding covariance matrix. In addition,

**c**denotes the constant term including a 6 × 1 vector.

## 4. Empirical Results and Portfolio Implications

#### 4.1. Data Description and Results Analysis

#### 4.2. Results of Range-Based Volatility Approaches

#### 4.3. The BEKK–GARCH Model Results

**Figure 1.**Results for range-based volatility estimators among Bitcoin and stablecoins markets.

**Note:**The ticker is denoted for Bitcoin (BTC), Tether (USDT), USD Coin (USDC), Binance USD (BUSD), Terra USD (UST), and Dai (DAI).

BTC | USDT | USDC | |||||||

${\widehat{\mathit{\sigma}}}_{\mathit{P}\mathit{K}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{R}\mathit{S}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{G}\mathit{K}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{P}\mathit{K}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{R}\mathit{S}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{G}\mathit{K}\mathbf{}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{P}\mathit{K}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{R}\mathit{S}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{G}\mathit{K}}^{\mathbf{2}}$ | |

Mean | 0.027852 | 0.039245 | 0.026614 | 0.003601 | 0.008880 | 0.007804 | 0.006659 | 0.008199 | 0.008770 |

Median | 0.021601 | 0.029224 | 0.020030 | 0.001627 | 0.006115 | 0.003756 | 0.004761 | 0.005316 | 0.006336 |

Maximum | 0.293944 | 0.502468 | 0.260957 | 0.085431 | 0.135782 | 0.165381 | 0.513387 | 0.852999 | 0.604452 |

Minimum | 0.001509 | 0.002208 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

Std. Dev. | 0.024004 | 0.035389 | 0.024461 | 0.005599 | 0.010579 | 0.011057 | 0.016240 | 0.025962 | 0.019211 |

Skewness | 2.629541 | 2.805815 | 3.057525 | 4.288796 | 3.808302 | 4.657458 | 25.18825 | 28.40215 | 24.75231 |

Kurtosis | 15.66039 | 19.39226 | 18.48525 | 36.56806 | 30.51883 | 46.13120 | 775.5576 | 916.5785 | 756.4662 |

Jarque-Bera | 21824.92 | 34860.31 | 32188.31 | 127091.2 | 86319.75 | 206145.8 | 30643454 | 42835202 | 29149533 |

Probability | 0.0000 *** | 0.0000 *** | 0.0000 *** | 0.0000 *** | 0.0000 *** | 0.0000 *** | 0.0000 *** | 0.0000 *** | 0.0000 *** |

ADF Prob. | 0.000 *** | 0.000 *** | 0.000 *** | 0.000 *** | 0.000 *** | 0.000 *** | 0.000 *** | 0.0001 *** | 0.000 *** |

Observations | 2787 | 2541 | 1227 | ||||||

BUSD | UST | DAI | |||||||

${\widehat{\mathit{\sigma}}}_{\mathit{P}\mathit{K}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{R}\mathit{S}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{G}\mathit{K}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{P}\mathit{K}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{R}\mathit{S}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{G}\mathit{K}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{P}\mathit{K}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{R}\mathit{S}}^{\mathbf{2}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{G}\mathit{K}}^{\mathbf{2}}$ | |

Mean | 0.005365 | 0.007254 | 0.006639 | 0.006856 | 0.009021 | 0.008113 | 0.008600 | 0.011459 | 0.010367 |

Median | 0.002472 | 0.004455 | 0.003222 | 0.004513 | 0.006033 | 0.005369 | 0.003829 | 0.005728 | 0.004364 |

Maximum | 0.116571 | 0.142981 | 0.147835 | 0.139373 | 0.207474 | 0.127793 | 0.784088 | 0.923166 | 1.294481 |

Minimum | 0.00000 | 8.28E-05 | 0.000000 | 0.000000 | 0.000761 | 0.000231 | 0.00000 | 8.29E-05 | 0.000173 |

Std. Dev. | 0.007848 | 0.009522 | 0.009652 | 0.010094 | 0.014627 | 0.010172 | 0.029895 | 0.035559 | 0.048217 |

Skewness | 5.831186 | 6.171779 | 5.831963 | 7.907933 | 9.580432 | 5.761683 | 22.11235 | 21.38384 | 23.87537 |

Kurtosis | 65.26500 | 70.74781 | 65.83852 | 89.76793 | 118.6213 | 54.09976 | 560.7605 | 535.2409 | 622.9322 |

J.-B. | 147141.2 | 173878.1 | 149773.3 | 145204.5 | 256394.8 | 51220.84 | 10656834 | 9705600 | 13160374 |

Probability | 0.0000 *** | 0.0000 *** | 0.0000 *** | 0.0000 *** | 0.0000 *** | 0.0000 *** | 0.0000 *** | 0.0000 *** | 0.0000 *** |

ADF Prob. | 0.000 *** | 0.000 *** | 0.000 *** | 0.000 *** | 0.000 *** | 0.000 *** | 0.000 *** | 0.001 *** | 0.000 *** |

Observations | 880 | 448 | 817 |

#### 4.4. VAR Granger Causality Test Results

BTC-USDT | BTC-USDC | BTC-BUSD | BTC-UST | BTC-DAI | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Parameter | Coeff. | p-Value | Coeff. | p-Value | Coeff. | p-Value | Coeff. | p-Value | Coeff. | p-Value |

Conditional | mean | |||||||||

μ | 0.00073 | 0.000 *** | 0.00146 | 0.000 *** | 0.00104 | 0.000 *** | 0.00326 | 0.000 *** | 0.00168 | 0.000 *** |

ϕ_{i} | 0.00014 | 0.9396 | −0.00047 | 0.0173 ** | −0.00264 | 0.1122 | 0.02362 | 0.0011 *** | −0.00019 | 0.5559 |

u | ||||||||||

Conditional | variance | |||||||||

c_{11} | 1.98 × 10^{−5} | 0.000 *** | 1.89 × 10^{−5} | 0.0001 *** | 1.46 × 10^{−5} | 0.002 *** | 0.00012 | 0.1854 | 0.0163 | 0.000 *** |

c_{12} | −1.86 × 10^{−7} | 0.2196 | −4.48 × 10^{−7} | 0.4467 | −1.397 × 10^{−7} | 0.5259 | 8.15 × 10^{−6} | 0.1515 | 0.2752 | 0.000 *** |

c_{22} | 5.65 × 10^{−8} | 0.000 *** | 4.767 × 10^{−7} | 0.000 *** | 7.76 × 10^{−8} | 0.004 *** | 9.75 × 10^{−6} | 0.0801 | 1.12 × 10^{−6} | 0.000 *** |

α_{11} | 0.0607 | 0.000 *** | 0.083 | 0.000 *** | 0.0391 | 0.000 *** | 0.13378 | 0.0038 *** | 0.2738 | 0.000 *** |

α_{12} | 0.0948 | 0.000 *** | 0.26914 | 0.000 *** | 0.1197 | 0.000 *** | 0.25731 | 0.12392 | 1.2316 | 0.000 *** |

α_{22} | 0.1482 | 0.000 *** | 0.87104 | 0.000 *** | 0.3673 | 0.000 *** | 0.49485 | 0.0009 *** | 0.9714 | 0.10523 |

β_{11} | 0.9031 | 0.000 *** | 0.87421 | 0.000 *** | 0.9104 | 0.000 *** | 0.81946 | 0.000 *** | 0.909397 | 0.10066 |

β_{12} | 0.8949 | 0.000 *** | 0.68456 | 0.000 *** | 0.8242 | 0.000 *** | 0.66368 | 0.000 *** | 0.874962 | 0.06280 |

β_{22} | 0.8867 | 0.000 *** | 0.53607 | 0.000 *** | 0.7462 | 0.000 *** | 0.53756 | 0.000 *** | 0.5947 | 0.000 *** |

Durbin-Watson stat. | 0.8115 | 1.3655 | 1.16120 | 1.5853 | 1.5647 | |||||

Schwarz criterion | −22.921 | −13.364 | −14.113 | −13.439 | −13.282 | |||||

Log likelihood | 14129.27 | 8239.93 | 6250.68 | 3046.99 | 5459.43 | |||||

Q^{2} (h) | 17.052 | 12.028 | 187.953 | 11.478 | 9.221 | |||||

(0.5195) | (0.444) | (0.000 ^{+}) | (0.0217) | (0.684) |

BTC-USDT | BTC-USDC | BTC-BUSD | BTC-UST | BTC-DAI | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Parameter | Coeff. | p-Value | Coeff. | p-Value | Coeff. | p-Value | Coeff. | p-Value | Coeff. | p-Value |

a | 0.0036 | 0.0000 *** | 0.0067 | 0.0000 *** | 0.0054 | 0.0000 *** | 0.0049 | 0.0000 *** | 0.0086 | 0.0000 *** |

b | 0.4079 | 0.0050 *** | 0.1353 | 0.0000 *** | 0.2967 | 0.0918 * | 0.2538 | 0.0011 *** | 0.0787 | 0.0000 *** |

c | 0.001882 | 0.0000 *** | 0.001479 | 0.0000 *** | 0.000504 | 0.0000 *** | 0.000569 | 0.0001 *** | 0.000271 | 0.0000 *** |

α | 0.878333 | 0.0000 *** | 0.686432 | 0.0000 *** | 0.877398 | 0.0000 *** | 0.574520 | 0.0000 *** | 0.820375 | 0.0000 *** |

β | 0.090530 | 0.0000 *** | 0.256479 | 0.0000 *** | 0.106864 | 0.0000 *** | 0.399782 | 0.0021 *** | 0.275693 | 0.0000 *** |

Log likelihood | 12861.02 | 5128.84 | 4547.35 | 2138.93 | 3627.32 | |||||

Unconditional correlations | 0.4312 | 0.1331 | 0.3094 | 0.2664 | 0.0638 |

**Figure 2.**Conditional correlations between the Bitcoin and stablecoins markets (VAR-BEEK-GARCH model). Notes: Bitcoin–stablecoins correlation via the VAR–BEKK–GARCH model.

**Figure 3.**Conditional variance for the Bitcoin and top five stablecoins based on the VAR–BEEK–GARCH model.

#### 4.5. Robustness Test

**Figure 4.**Conditional volatility and correlation between Bitcoin and stablecoins (Copula–DCC–GARCH model). (

**A**) Bitcoin vs. Tether. (

**B**) Bitcoin vs. USD Coin. (

**C**) Bitcoin vs. Binance USD. (

**D**) Bitcoin vs. Terra USD. (

**E**) Bitcoin vs. Dai. Notes: As plotted in Figure 4 the upper, and bottom panels denote conditional volatility and correlation series, respectively.

Dependent Variable: BTC | |||

Excluded | Chi-sq | df | Prob. |

USDT | 0.822534 | 2 | 0.6628 |

USDC | 0.145836 | 2 | 0.9297 |

BUSD | 7.261248 | 2 | 0.0265 |

UST | 1.533981 | 2 | 0.4644 |

DAI | 0.204514 | 2 | 0.9028 |

All | 10.78848 | 10 | 0.3742 |

Dependent Variable: USDT | |||

Excluded | Chi-sq | df | Prob. |

BTC | 4.429129 | 2 | 0.1092 |

USDC | 10.61064 | 2 | 0.0050 * |

BUSD | 14.12211 | 2 | 0.0009 * |

UST | 10.78127 | 2 | 0.0046 * |

DAI | 7.878435 | 2 | 0.0195 |

All | 68.78196 | 10 | 0.0000 |

Dependent Variable: USDC | |||

Excluded | Chi-sq | df | Prob. |

BTC | 0.714031 | 2 | 0.6998 |

USDT | 1.171648 | 2 | 0.5566 |

BUSD | 1.814377 | 2 | 0.4037 |

UST | 0.142163 | 2 | 0.9314 |

DAI | 0.545036 | 2 | 0.7615 |

All | 3.698988 | 10 | 0.9599 |

Dependent Variable: BUSD | |||

Excluded | Chi-sq | df | Prob. |

BTC | 6.120387 | 2 | 0.0469 |

USDT | 2.271693 | 2 | 0.3212 |

USDC | 4.761130 | 2 | 0.0925 |

UST | 1.800812 | 2 | 0.4064 |

DAI | 3.509030 | 2 | 0.1730 |

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Sidorenko, E.L. Stablecoin as a new financial instrument. In Digital Age: Chances, Challenges and Future; Ashmarina, S., Vochozka, M., Mantulenko, V., Eds.; Springer International Publishing: Cham, Switzerland, 2020; pp. 54–96. [Google Scholar]
- Wang, G.J.; Ma, X.Y.; Wu, H.Y. Are stablecoins truly diversifiers, hedges, or safe havens against traditional cryptocurrencies as their name suggests? Res. Int. Bus. Financ.
**2020**, 54, 101225. [Google Scholar] [CrossRef] - Xie, Y.; Kang, S.B.; Zhao, J. Are stablecoins safe havens for traditional cryptocurrencies? An empirical study during the COVID-19 pandemic. Appl. Financ. Lett.
**2021**, 10, 2–9. [Google Scholar] [CrossRef] - Ito, K.; Mita, M.; Ohsawa, S.; Tanaka, H. What is Stablecoin? A Survey on Its Mechanism and Potential as Decentralized Payment Systems. Int. J. Serv. Knowl. Manag.
**2020**, 4, 71–86. [Google Scholar] [CrossRef] - Lyons, R.K.; Viswanath-Natraj, G. What Keeps Stablecoins Stable? (No. w27136); National Bureau of Economic Research: Cambridge, MA, USA, 2020. [Google Scholar]
- Griffin, J.M.; Shams, A. Is Bitcoin really untethered? J. Financ.
**2020**, 75, 1913–1964. [Google Scholar] [CrossRef] - Wei, W.C. The impact of Tether grants on Bitcoin. Econ. Lett.
**2018**, 171, 19–22. [Google Scholar] [CrossRef] - Kristoufek, L. On the role of stablecoins in the cryptoassets pricing dynamics. Financ. Innov.
**2022**, 8, 37. [Google Scholar] [CrossRef] - Lyons, R.K.; Viswanath-Natraj, G. Stable Coins Don’t Inflate Crypto Markets VOX CEPR Policy Portal. 2020. Available online: https://voxeu.org/article/stable-coins-dont-inflate-crypto-markets (accessed on 5 December 2020).
- Hoang, L.T.; Baur, D.G. How stable are stablecoins? Eur. J. Financ.
**2021**. [Google Scholar] [CrossRef] - Grobys, K.; Junttila, J.; Kolari James, W.; Sapkota, N. On the stability of stablecoins. J. Empir. Financ.
**2021**, 64, 207–223. [Google Scholar] [CrossRef] - Baba, Y.; Engle, R.F.; Kraft, D.F.; Kroner, K.F. Multivariate Simultaneous Generalized ARCH (Working Paper); Department of Economics, University of California at San Diego: San Diego, CA, USA, 1990. [Google Scholar]
- Engle, R.F.; Kroner, K.F. Multivariate simultaneous generalized ARCH. Econom. Theory
**1995**, 11, 122–150. [Google Scholar] [CrossRef] - Fiszeder, P. Low and high prices can improve covariance forecasts: The evidence based on currency rates. J. Forecast.
**2018**, 37, 641–649. [Google Scholar] [CrossRef] - Tan, S.K.; Chan, J.S.K.; Ng, K.H. On the speculative nature of cryptocurrencies: A study on Garman and Klass volatility measure. Financ. Res. Lett.
**2020**, 32, 101075. [Google Scholar] [CrossRef] - Brandt, M.W.; Diebold, F.X. A No-Arbitrage Approach to Range-Based Estimation of Return Covariances and Correlations. J. Bus.
**2006**, 79, 61–74. [Google Scholar] [CrossRef] - Li, H.; Hong, Y. Financial volatility forecasting with range-based autoregressive volatility model. Financ. Res. Lett.
**2011**, 8, 69–76. [Google Scholar] [CrossRef] - Fiszeder, P.; Fałdziński, M.; Molnár, P. Range-based DCC models for covariance and value-at-risk forecasting. J. Empir. Financ.
**2019**, 54, 58–76. [Google Scholar] [CrossRef] - Molnár, P. High-low range in GARCH models of stock return volatility. Appl. Econ.
**2016**, 48, 4977–4991. [Google Scholar] [CrossRef] - Jacob, J. Estimation and forecasting of stock volatility with range-based estimators. J. Futures Mark.
**2008**, 28, 561–581. [Google Scholar] [CrossRef] - Todorova, N.; Husmann, S. A comparative study of range-based stock return volatility estimators for the German market. J. Futures Mark.
**2012**, 32, 560–586. [Google Scholar] [CrossRef] - Bauwens, L.; Hafner, C.M.; Laurent, S. Handbook of Volatility Models and Their Applications; Wiley: Hoboken, NJ, USA, 2012. [Google Scholar]
- Block, A.S.; Righi, M.B.; Schlender, S.G.; Coronel, D.A. Investigating dynamic correlation between crude oil and fuels in non-linear framework: The financial and economic role of structural breaks. Energy Econ.
**2015**, 49, 23–32. [Google Scholar] [CrossRef] - Mensi, W.; Hammoudeh, S.; Nguyen, D.K.; Yoon, S.M. Dynamic spillovers among major energy and cereal commodity prices. Energy Econ.
**2014**, 43, 225–243. [Google Scholar] [CrossRef] - Parkinson, M. The extreme value method for estimating the variance of the rate of return. J. Bus.
**1995**, 53, 61–65. [Google Scholar] [CrossRef] - Garman, M.B.; Klass, M.J. On the estimation of security price volatilities from historical data. J. Bus.
**1980**, 53, 67–78. [Google Scholar] [CrossRef] - Rogers LC, G.; Satchell, S.E. Estimating variance from high, low and closing prices. Ann. Appl. Probab.
**1991**, 1, 504–512. [Google Scholar] [CrossRef] - Katsiampa, P. Volatility co-movement between Bitcoin and Ether. Financ. Res. Lett.
**2019**, 30, 221–227. [Google Scholar] [CrossRef] [Green Version] - Engle, R.F. Dynamic conditional correlation—A simple class of multivariate GARCH models. J. Bus. Econ. Stat.
**2002**, 20, 339–350. [Google Scholar] [CrossRef] - Righi, M.; Ceretta, P. Global risk evolution and diversification: A Copula- DCC- GARCH model approach. Rev. Bras. Financ.
**2012**, 10, 529–550. [Google Scholar] [CrossRef] - Chen, K.S.; Huang, Y.C. Detecting Jump Risk and Jump-Diffusion Model for Bitcoin Options Pricing and Hedging. Mathematics
**2021**, 9, 2567. [Google Scholar] [CrossRef] - Zięba, D.; Kokoszczyński, R.; Śledziewska, K. Shock transmission in the cryptocurrency market. Is Bitcoin the most influential? Int. Rev. Financ. Anal.
**2019**, 64, 102–125. [Google Scholar] [CrossRef] - Katsiampa, P.; Corbet, S.; Lucey, B. Volatility spillover effects in leading cryptocurrencies: A BEKK-MGARCH analysis. Financ. Res. Lett.
**2019**, 29, 68–74. [Google Scholar] [CrossRef] [Green Version] - Kristoufek, L. Tethered, or Untethered? On the interplay between stablecoins and major cryptoassets. Financ. Res. Lett.
**2021**, 43, 101991. [Google Scholar] [CrossRef]

# Rank | Name | Ticker | Coinmark | Price | 24 h % | 7 d % | Market Cap |
---|---|---|---|---|---|---|---|

1 | Bitcoin | BTC | $44,575.2 | 9.65% | 5.45% | $845,117,683,687 | |

3 | Tether | USDT | $1.00 | 0.01% | 0.01% | $78,515,395,493 | |

5 | USD Coin | USDC | $0.9998 | 0.06% | 0.03% | $52,536,446,749 | |

13 | Binance USD | BUSD | $1.00 | 0.06% | 0.00% | $17,805,102,313 | |

17 | Terra USD | UST | $0.9999 | 0.08% | 0.21% | $11,619,278,219 | |

19 | Dai | DAI | $0.9993 | 0.05% | 0.03% | $10,373,205,726 |

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**MDPI and ACS Style**

Chen, K.-S.; Chang, S.-H.
Volatility Co-Movement between Bitcoin and Stablecoins: BEKK–GARCH and Copula–DCC–GARCH Approaches. *Axioms* **2022**, *11*, 259.
https://doi.org/10.3390/axioms11060259

**AMA Style**

Chen K-S, Chang S-H.
Volatility Co-Movement between Bitcoin and Stablecoins: BEKK–GARCH and Copula–DCC–GARCH Approaches. *Axioms*. 2022; 11(6):259.
https://doi.org/10.3390/axioms11060259

**Chicago/Turabian Style**

Chen, Kuo-Shing, and Shen-Ho Chang.
2022. "Volatility Co-Movement between Bitcoin and Stablecoins: BEKK–GARCH and Copula–DCC–GARCH Approaches" *Axioms* 11, no. 6: 259.
https://doi.org/10.3390/axioms11060259