# Tail Dependence between Crude Oil Volatility Index and WTI Oil Price Movements during the COVID-19 Pandemic

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## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. Data

#### 2.2. Copula

#### 2.3. Time-Varying Copula

## 3. Tail Dependence between OVX and WTI Oil Price—Empirical Results

## 4. Value at Risk Forecasting—Empirical Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Crude Oil Future Contract—CLF (USD per barrel) and CBOE Crude Oil Volatility Index—OVX in 11 May 2007–30 March 2021.

**Figure 2.**Common exceedances of extreme values. The left panel shows all data, the right panel data without outliers. Note: Extreme values mean exceedances of the 5th and 95th quantiles of Crude Oil Future Contract returns (rCL) and the 95th quantile of changes in the CBOE Crude Oil Volatility Index (dOVX), and in volatility (dVOL).

**Figure 3.**Estimation of the conditional correlation coefficient for the relationship between Crude Oil Future Contract returns (rCL) and changes in the CBOE Crude Oil Volatility Index (dOVX) in the entire period.

**Figure 4.**Estimation of the tail dependence coefficient for the relationship between Crude Oil Future Contract returns (rCL)/negative returns (−rCL) and changes in the CBOE Crude Oil Volatility Index (dOVX) in the entire period.

**Table 1.**Copula parameters and tail dependence coefficient for Crude Oil Future Contract returns (rCL) and changes in the CBOE Crude Oil Volatility Index (dOVX) in the entire period and sub-periods.

Parameter | Estimate | Std. Error | Parameter | Estimate | Std. Error |
---|---|---|---|---|---|

Entire period | |||||

t-copula: -rCL vs. dOVX | t-copula: rCL vs. dOVX | ||||

rho | 0.440 | 0.017 | rho | −0.440 | 0.017 |

df | 2.230 | 0.125 | df | 2.230 | 0.125 |

lambda | 0.339 | lambda | 0.058 | ||

Joe copula: -rCL vs. dOVX | Joe copula: rCL vs. dOVX | ||||

theta | 1.700 | 0.030 | theta | 1.000 | 0.006 |

lambda | 0.497 | lambda | 0.000 | ||

GFC period | |||||

t-copula: -rCL vs. dOVX | t-copula: rCL vs. dOVX | ||||

rho | 0.227 | 0.048 | rho | −0.227 | 0.048 |

df | 2.392 | 0.341 | df | 2.392 | 0.341 |

lambda | 0.230 | lambda | 0.093 | ||

Joe copula: -rCL vs. dOVX | Joe copula: rCL vs. dOVX | ||||

theta | 1.349 | 0.057 | theta | 1.000 | 0.021 |

lambda | 0.328 | lambda | 0.000 | ||

COVID-19 period | |||||

t-copula: -rCL vs. dOVX | t-copula: rCL vs. dOVX | ||||

rho | 0.350 | 0.120 | rho | −0.350 | 0.120 |

df | 1.632 | 0.470 | df | 1.632 | 0.470 |

lambda | 0.352 | lambda | 0.113 | ||

Joe copula: -rCL vs. dOVX | Joe copula: rCL vs. dOVX | ||||

theta | 1.653 | 0.194 | theta | 1.024 | 0.072 |

lambda | 0.479 | lambda | 0.033 |

**Table 2.**Copula parameters and tail dependence coefficient for Crude Oil Future Contract returns (rCL) and one-day lagged changes in the CBOE Crude Oil Volatility Index (dOVX) in the entire period and sub-periods.

Parameter | Estimate | Std. Error | Parameter | Estimate | Std. Error |
---|---|---|---|---|---|

Entire period | |||||

t-copula: -rCL vs. dOVX lag1 | t-copula: rCL vs. dOVX lag1 | ||||

rho | 0.008 | 0.020 | rho | −0.008 | 0.020 |

df | 3.003 | 0.198 | df | 3.003 | 0.198 |

lambda | 0.118 | lambda | 0.114 | ||

GFC period | |||||

t-copula: -rCL vs. dOVX lag1 | t-copula: rCL vs. dOVX lag1 | ||||

rho | −0.008 | 0.048 | rho | 0.008 | 0.048 |

df | 4.096 | 0.908 | df | 4.096 | 0.908 |

lambda | 0.071 | lambda | 0.074 | ||

COVID-19 period | |||||

t-copula: -rCL vs. dOVX lag1 | t-copula: rCL vs. dOVX lag1 | ||||

rho | 0.193 | 0.129 | rho | −0.193 | 0.129 |

df | 2.118 | 0.704 | df | 2.118 | 0.704 |

lambda | 0.239 | lambda | 0.118 |

**Table 3.**Copula parameters and tail dependence coefficient for Crude Oil Future Contract returns (rCL) and changes in the volatility of Crude Oil Future Contract returns (dVOL) in the entire period and sub-periods.

Parameter | Estimate | Std. Error | Parameter | Estimate | Std. Error |
---|---|---|---|---|---|

Entire period | |||||

t-copula: -rCL vs. dVOL | t-copula: rCL vs. dVOL | ||||

rho | −0.038 | 0.020 | rho | 0.038 | 0.020 |

df | 2.518 | 0.143 | df | 2.518 | 0.143 |

lambda | 0.133 | lambda | 0.155 | ||

GFC period | |||||

t-copula: -rCL vs. dVOL | t-copula: rCL vs. dVOL | ||||

rho | −0.011 | 0.052 | rho | 0.011 | 0.052 |

df | 2.052 | 0.249 | df | 2.052 | 0.249 |

lambda | 0.174 | lambda | 0.181 | ||

COVID-19 period | |||||

t-copula: -rCL vs. dVOL | t-copula: rCL vs. dVOL | ||||

rho | −0.251 | 0.128 | rho | 0.251 | 0.128 |

df | 3.093 | 1.711 | df | 3.093 | 1.711 |

lambda | 0.058 | lambda | 0.191 |

**Table 4.**Copula parameters and tail dependence coefficient for Crude Oil Future Contract returns (rCL) and one-day lagged changes in the volatility of Crude Oil Future Contract returns (dVOL) in the entire period and sub-periods.

Parameter | Estimate | Std. Error | Parameter | Estimate | Std. Error |
---|---|---|---|---|---|

Entire period | |||||

t-copula: -rCL vs. dVOL lag1 | t-copula: rCL vs. dVOL lag1 | ||||

rho | 0.006 | 0.020 | rho | −0.006 | 0.020 |

df | 2.616 | 0.156 | df | 2.616 | 0.156 |

lambda | 0.139 | lambda | 0.136 | ||

GFC period | |||||

t-copula: -rCL vs. dVOL lag1 | t-copula: rCL vs. dVOL lag1 | ||||

rho | −0.082 | 0.049 | rho | 0.082 | 0.049 |

df | 3.057 | 0.513 | df | 3.057 | 0.513 |

lambda | 0.093 | lambda | 0.136 | ||

COVID-19 period | |||||

t-copula: -rCL vs. dVOL lag1 | t-copula: rCL vs. dVOL lag1 | ||||

rho | −0.104 | 0.129 | rho | 0.104 | 0.129 |

df | 4.184 | 2.766 | df | 4.184 | 2.766 |

lambda | 0.051 | lambda | 0.093 |

**Table 5.**Estimation of the t-copula-GARCH model for Crude Oil Future Contract returns (rCL) and changes in the CBOE Crude Oil Volatility Index (dOVX) in the entire period.

Parameter | Estimate | Std. Error | t-Stat |
---|---|---|---|

GJR-GARCH(1,1) sstd for rCL | |||

rCL omega | 0.2054 *** | 0.0450 | 4.567 |

rCL alpha | 0.0951 *** | 0.0158 | 6.009 |

rCL beta | 0.8104 *** | 0.0198 | 40.911 |

rCL gamma | 0.1377 *** | 0.0317 | 4.345 |

rCL skew | 0.9140 *** | 0.0195 | 46.754 |

rCL shape (df) | 6.3974 *** | 0.9783 | 6.540 |

ARMA(1,1)-GARCH(1,1) sstd for dOVX | |||

dOVX ar1 | 0.7903 *** | 0.0405 | 19.518 |

dOVX ma1 | −0.8599 *** | 0.0327 | −26.322 |

dOVX omega | 0.0003 *** | 0.0001 | 4.530 |

dOVX alpha | 0.1791 *** | 0.0185 | 9.684 |

dOVX beta | 0.8199 *** | 0.0171 | 48.043 |

dOVX skew | 1.2465 *** | 0.0268 | 46.578 |

dOVX shape (df) | 3.7636 *** | 0.2290 | 16.437 |

Joint DCC(1,1) | |||

Joint dcc a1 | 0.0815 *** | 0.0132 | 6.156 |

Joint dcc b1 | 0.8856 *** | 0.0220 | 40.165 |

Joint mshape (df) | 5.5179 *** | 0.3655 | 15.097 |

Quantile | ET | T_{1} | UC | CC | UD | DQ | ADmean | Loss | T_{1} | UC | CC | UD | DQ | ADmean | Loss |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

p | p | p | p | ADmax | p | p | p | p | ADmax | ||||||

Panel A: Left tail for entire period | Panel B: Right tail for entire period | ||||||||||||||

0.95 | 149 | 147 | 0.039 | 0.048 | 0.230 | 5.959 | 3.354 | 0.368 | 143 | 0.288 | 0.292 | 1.030 | 7.217 | 1.237 | 0.251 |

0.843 | 0.976 | 0.632 | 0.652 | 292.494 | 0.591 | 0.864 | 0.310 | 0.513 | 16.058 | ||||||

0.975 | 74 | 72 | 0.099 | 0.484 | 1.134 | 0.642 | 5.603 | 0.266 | 65 | 1.344 | 4.237 | 0.007 | 1.321 | 1.503 | 0.150 |

0.752 | 0.785 | 0.287 | 1.000 | 288.081 | 0.246 | 0.120 | 0.936 | 0.995 | 13.641 | ||||||

0.99 | 29 | 37 | 1.598 | 2.091 | 5.068 | 2.002 | 9.233 | 0.185 | 23 | 1.733 | 2.090 | 0.750 | 1.582 | 2.339 | 0.078 |

0.206 | 0.352 | 0.024 | 0.981 | 281.978 | 0.188 | 0.352 | 0.387 | 0.991 | 10.264 | ||||||

Panel C: Left tail for pre-COVID-19 period | Panel D: Right tail for pre-COVID-19 period | ||||||||||||||

0.95 | 132 | 131 | 0.023 | 0.396 | 0.081 | 9.849 | 1.105 | 0.223 | 123 | 0.764 | 1.069 | 0.108 | 20.044 | 1.069 | 0.208 |

0.879 | 0.820 | 0.776 | 0.276 | 7.757 | 0.382 | 0.586 | 0.743 | 0.010 | 11.580 | ||||||

0.975 | 66 | 62 | 0.299 | 3.218 | 0.932 | 14.839 | 1.266 | 0.134 | 53 | 2.956 | 5.117 | 0.048 | 19.414 | 1.344 | 0.123 |

0.585 | 0.200 | 0.334 | 0.062 | 7.209 | 0.086 | 0.077 | 0.826 | 0.013 | 10.976 | ||||||

0.99 | 26 | 31 | 0.718 | 1.451 | 3.553 | 20.319 | 1.272 | 0.067 | 20 | 1.779 | 2.083 | 1.394 | 7.813 | 2.139 | 0.064 |

0.397 | 0.484 | 0.059 | 0.009 | 6.531 | 0.182 | 0.353 | 0.238 | 0.452 | 10.264 | ||||||

Panel E: Left tail for COVID-19 period | Panel F: Right tail for COVID-19 period | ||||||||||||||

0.95 | 16 | 16 | 0.027 | 1.591 | 0.596 | 0.725 | 21.763 | 1.521 | 20 | 0.668 | 3.234 | 9.643 | 1.300 | 2.270 | 0.587 |

0.869 | 0.451 | 0.440 | 0.999 | 292.494 | 0.414 | 0.198 | 0.002 | 0.996 | 16.058 | ||||||

0.975 | 8 | 10 | 0.325 | 1.432 | 0.157 | 0.821 | 32.491 | 1.314 | 12 | 1.467 | 2.367 | 4.866 | 1.369 | 2.205 | 0.364 |

0.569 | 0.489 | 0.692 | 0.999 | 288.081 | 0.226 | 0.306 | 0.027 | 0.995 | 13.641 | ||||||

0.99 | 3 | 6 | 1.747 | 4.694 | 1.504 | 2.614 | 50.364 | 1.123 | 3 | 0.034 | 0.089 | 1.410 | 0.086 | 3.674 | 0.187 |

0.186 | 0.096 | 0.220 | 0.956 | 281.978 | 0.853 | 0.957 | 0.235 | 1.000 | 9.864 |

_{1})—expected (actual) number of exceedances, UC—Kupiec test statistic, CC—Christoffersen’s test statistic, UD—Christoffersen and Pelletier’s test statistic, DQ—Engle and Manganelli’s test statistic (the DQ test included constant, VaR, first five lagged exceedances and one lagged squared return), p—p-value, ADmean (ADmax)—mean (maximum) absolute deviation between the returns and the quantiles if exceedances occur [87], Loss—loss function Q [88], bold—rejection of the null hypothesis at the significance level of 0.05. The calculations were performed with the rugarch and GAS packages in R [89,90].

Quantile | ET | T_{1} | UC | CC | UD | DQ | ADmean | Loss | T_{1} | UC | CC | UD | DQ | ADmean | Loss |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

p | p | p | p | ADmax | p | p | p | p | ADmax | ||||||

Panel A: Left tail for entire period | Panel B: Right tail for entire period | ||||||||||||||

0.95 | 149 | 146 | 0.080 | 0.291 | 0.789 | 4.198 | 3.375 | 0.368 | 141 | 0.500 | 0.519 | 0.864 | 5.922 | 1.242 | 0.251 |

0.778 | 0.865 | 0.072 | 0.839 | 292.494 | 0.479 | 0.771 | 0.353 | 0.656 | 16.342 | ||||||

0.975 | 74 | 72 | 0.099 | 0.484 | 0.786 | 0.624 | 5.625 | 0.266 | 62 | 2.340 | 4.969 | 0.120 | 2.216 | 1.567 | 0.150 |

0.752 | 0.785 | 0.375 | 1.000 | 288.081 | 0.126 | 0.083 | 0.729 | 0.974 | 13.980 | ||||||

0.99 | 29 | 37 | 1.598 | 2.091 | 6.487 | 1.984 | 9.258 | 0.185 | 23 | 1.733 | 2.090 | 0.750 | 1.582 | 2.339 | 0.078 |

0.206 | 0.352 | 0.011 | 0.981 | 281.978 | 0.188 | 0.352 | 0.387 | 0.991 | 10.399 | ||||||

Panel C: Left tail for pre-COVID-19 period | Panel D: Right tail for pre-COVID-19 period | ||||||||||||||

0.95 | 132 | 130 | 0.058 | 1.129 | 0.003 | 8.051 | 1.114 | 0.224 | 121 | 1.118 | 1.523 | 0.060 | 20.237 | 1.074 | 0.208 |

0.809 | 0.569 | 0.954 | 0.429 | 7.682 | 0.290 | 0.467 | 0.806 | 0.009 | 11.557 | ||||||

0.975 | 66 | 62 | 0.299 | 3.218 | 0.600 | 14.346 | 1.297 | 0.135 | 50 | 4.511 | 6.432 | 0.389 | 21.846 | 1.412 | 0.124 |

0.585 | 0.200 | 0.439 | 0.073 | 7.115 | 0.034 | 0.040 | 0.533 | 0.005 | 10.941 | ||||||

0.99 | 26 | 32 | 1.065 | 1.846 | 4.544 | 19.970 | 1.258 | 0.067 | 20 | 1.779 | 2.083 | 1.394 | 7.818 | 2.112 | 0.064 |

0.302 | 0.397 | 0.033 | 0.010 | 6.445 | 0.182 | 0.353 | 0.238 | 0.451 | 10.209 | ||||||

Panel E: Left tail for COVID-19 period | Panel F: Right tail for COVID-19 period | ||||||||||||||

0.95 | 16 | 16 | 0.027 | 1.591 | 0.596 | 0.725 | 21.763 | 1.521 | 20 | 0.668 | 3.234 | 9.643 | 1.300 | 2.270 | 0.587 |

0.869 | 0.451 | 0.440 | 0.999 | 292.494 | 0.414 | 0.198 | 0.002 | 0.996 | 16.058 | ||||||

0.975 | 8 | 10 | 0.325 | 1.432 | 0.157 | 0.821 | 32.492 | 1.314 | 12 | 1.467 | 2.367 | 4.866 | 1.369 | 2.205 | 0.364 |

0.569 | 0.489 | 0.692 | 0.999 | 288.081 | 0.226 | 0.306 | 0.027 | 0.995 | 13.641 | ||||||

0.99 | 3 | 6 | 1.747 | 4.694 | 1.504 | 2.614 | 50.364 | 1.123 | 3 | 0.034 | 0.089 | 1.410 | 0.086 | 3.674 | 0.187 |

0.186 | 0.096 | 0.220 | 0.956 | 281.978 | 0.853 | 0.957 | 0.235 | 1.000 | 9.863 |

_{1})—expected (actual) number of exceedances, UC—Kupiec test statistic, CC—Christoffersen’s test statistic, UD—Christoffersen and Pelletier’s test statistic, DQ—Engle and Manganelli’s test statistic (the DQ test included constant, VaR, first five lagged exceedances and one lagged squared return), p—p-value, ADmean (ADmax)—mean (maximum) absolute deviation between the returns and the quantiles if exceedances occur [87], Loss—loss function Q [88], bold—rejection of the null hypothesis at the significance level of 0.05. The calculations were performed with the rugarch and GAS packages in R [89,90].

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## Share and Cite

**MDPI and ACS Style**

Echaust, K.; Just, M.
Tail Dependence between Crude Oil Volatility Index and WTI Oil Price Movements during the COVID-19 Pandemic. *Energies* **2021**, *14*, 4147.
https://doi.org/10.3390/en14144147

**AMA Style**

Echaust K, Just M.
Tail Dependence between Crude Oil Volatility Index and WTI Oil Price Movements during the COVID-19 Pandemic. *Energies*. 2021; 14(14):4147.
https://doi.org/10.3390/en14144147

**Chicago/Turabian Style**

Echaust, Krzysztof, and Małgorzata Just.
2021. "Tail Dependence between Crude Oil Volatility Index and WTI Oil Price Movements during the COVID-19 Pandemic" *Energies* 14, no. 14: 4147.
https://doi.org/10.3390/en14144147