# Multivariate Analysis of Energy Commodities during the COVID-19 Pandemic: Evidence from a Mixed-Frequency Approach

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### Model Selection

## 3. Empirical Application

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

US | United States |

WTI | West Texas Intermediate |

DCC | Dynamic conditional correlation (Engle 2002) |

cDCC | Corrected DCC (Aielli 2013) |

DECO | Dynamic equicorrelation (Engle and Kelly 2012) |

MIDAS | MIxing-Data Sampling |

GM | GARCH-MIDAS (Engle et al. 2013) |

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

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

SSM | Set of Superior Models |

## Note

1 | The literature has proposed alternative distributions to capture fat tails and skewness of returns, such as the multivariate Student’s t distribution. However, according to Pesaran and Pesaran (2010), the use of the $MVN$ allows us to maintain the two-step process used to estimate the DCC model. Moreover, in order to take into consideration possible deviations from the normality assumption, all the reported standard errors are based on Quasi-Maximum Likelihood (Bollerslev and Wooldridge 1992) methods. |

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**Figure 1.**Log−returns of commodities.

**Notes:**Plots of log-returns of commodities Sample period: April 2020–June 2021. Number of observations: 276.

**Figure 2.**Weekly deaths related to COVID-19 infections in the US.

**Notes:**Plot of the weekly deaths related to COVID-19 infections in the US. Sample period: March 2020–June 2021. The labels of the x-axis indicate the week and the year of the observations. The time series starts the first week of March 2020 (9 March 2020).

**Figure 3.**Volatility structural breaks.

**Notes:**The figure shows the GARCH(1,1) estimated volatility (black lines), the structural breaks (green lines), and the related 95% confidence intervals (red lines).

**Figure 4.**The plots show the estimated volatilities on the main diagonal and correlation estimated with the model GARCH-cDCC. Sample period: June 2020–June 2021.

**Figure 5.**The plots show the estimated volatilities on the main diagonal and correlation estimated with the model DAGM-cDCC. Sample period: June 2020–June 2021.

**Figure 6.**The plots show the estimated volatilities on the main diagonal and the correlation estimated with the model DAGM-DCC-MIDAS. Red lines represent the estimated long-run correlation. Sample period: June 2020 to June 2021.

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

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

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

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

$\begin{array}{cc}{Q}_{i,t}=& (1-a-b)\mathsf{\Psi}+a\left({\mathit{\xi}}_{i-1,t}{\mathit{\xi}}_{i-1,t}^{{}^{\prime}}\right)+\hfill \\ & b{Q}_{i-1,t}\hfill \end{array}$ | |

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

$\mathsf{\Psi}=E\left(\right)open="("\; close=")">{\mathit{\xi}}_{i,t}{\mathit{\xi}}_{i,t}^{{}^{\prime}}$ | |

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

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

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

$\begin{array}{cc}{Q}_{i,t}=& (1-a-b){\overline{R}}_{i,t}\left(\omega \right)+a\left({\mathit{\xi}}_{i-1,t}{\mathit{\xi}}_{i-1,t}^{{}^{\prime}}\right)+\hfill \\ & b{Q}_{i-1,t}\hfill \end{array}$ | |

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

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

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

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

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

**Notes**: The table shows the different specifications of the correlation models used in this paper. ${\overline{R}}_{i,t}\left(\omega \right)$ is the long-run correlation of Colacito et al. (2011), ${J}_{n}$ and $\iota $ are, respectively, a matrix and a ($n\times 1$)-vector of ones.

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

WTI | −0.082 | 0.246 | 0.006 | 0.033 | 2.45 | 15.718 |

Europe Brent | −0.087 | 0.222 | 0.006 | 0.029 | 1.582 | 11.723 |

Heating Oil | −0.139 | 0.112 | 0.004 | 0.029 | −0.004 | 3.686 |

Propane | −0.082 | 0.172 | 0.004 | 0.03 | 0.785 | 4.604 |

Gasoline | −0.087 | 0.114 | 0.005 | 0.028 | 0.131 | 2.039 |

Kerosene | −0.13 | 0.153 | 0.005 | 0.033 | 0.553 | 3.889 |

**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: March 2020–June 2021. Number of observations: 271.

WTI | Brent Oil | Heating Oil | Propane | Gasoline | Kerosene |
---|---|---|---|---|---|

Mar. 2020 | Mar. 2020 | Mar. 2020 | Mar. 2020 | Mar. 2020 | Mar. 2020 |

June 2020 | June 2020 | June 2020 | June 2020 | June 2020 | June 2020 |

Dec. 2020 | Oct. 2020 |

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

WTI | |||||

$\omega $ | 0.000 | 0.000 | 2.057 | 0.040 | ** |

$\alpha $ | 0.138 | 0.044 | 3.169 | 0.002 | *** |

$\beta $ | 0.786 | 0.053 | 14.814 | 0.000 | *** |

Europe Brent | |||||

$\omega $ | 0.000 | 0.000 | 2.316 | 0.021 | ** |

$\alpha $ | 0.138 | 0.053 | 2.591 | 0.010 | *** |

$\beta $ | 0.807 | 0.041 | 19.646 | 0.000 | *** |

Heating Oil | |||||

$\omega $ | 0.000 | 0.000 | 2.744 | 0.006 | *** |

$\alpha $ | 0.136 | 0.043 | 3.133 | 0.002 | *** |

$\beta $ | 0.793 | 0.046 | 17.408 | 0.000 | *** |

Propane | |||||

$\omega $ | 0.000 | 0.000 | 0.736 | 0.462 | |

$\alpha $ | 0.069 | 0.050 | 1.378 | 0.168 | |

$\beta $ | 0.867 | 0.122 | 7.120 | 0.000 | *** |

Gasoline | |||||

$\omega $ | 0.000 | 0.000 | 1.558 | 0.119 | |

$\alpha $ | 0.100 | 0.055 | 1.819 | 0.069 | * |

$\beta $ | 0.856 | 0.066 | 13.005 | 0.000 | *** |

Kerosene | |||||

$\omega $ | 0.000 | 0.000 | 2.478 | 0.013 | ** |

$\alpha $ | 0.108 | 0.029 | 3.786 | 0.000 | *** |

$\beta $ | 0.826 | 0.041 | 20.200 | 0.000 | *** |

**Notes:**The table reports the estimates of the GARCH model. The column “Std. Error” shows the Quasi-Maximum Likelihood-based standard errors. Sample Period: March 2020–June 2021, 271 daily observations. *, **, and *** indicate the significance at levels 10%, 5%, and 1%, respectively.

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

WTI | |||||

$\omega $ | 0.000 | 0.000 | 2.336 | 0.019 | ** |

$\alpha $ | 0.104 | 0.044 | 2.380 | 0.017 | ** |

$\beta $ | 0.797 | 0.040 | 19.705 | 0.000 | *** |

$\gamma $ | 0.101 | 0.077 | 1.303 | 0.193 | |

Europe Brent | |||||

$\omega $ | 0.000 | 0.000 | 2.308 | 0.021 | ** |

$\alpha $ | 0.133 | 0.068 | 1.956 | 0.050 | * |

$\beta $ | 0.807 | 0.041 | 19.706 | 0.000 | *** |

$\gamma $ | 0.018 | 0.075 | 0.238 | 0.812 | |

Heating Oil | |||||

$\omega $ | 0.000 | 0.000 | 2.543 | 0.011 | ** |

$\alpha $ | 0.125 | 0.056 | 2.244 | 0.025 | ** |

$\beta $ | 0.781 | 0.052 | 15.083 | 0.000 | *** |

$\gamma $ | 0.057 | 0.091 | 0.624 | 0.532 | |

Propane | |||||

$\omega $ | 0.000 | 0.000 | 0.919 | 0.358 | |

$\alpha $ | 0.046 | 0.042 | 1.103 | 0.270 | |

$\beta $ | 0.876 | 0.059 | 14.788 | 0.000 | *** |

$\gamma $ | 0.099 | 0.081 | 1.228 | 0.219 | |

Gasoline | |||||

$\omega $ | 0.000 | 0.000 | 1.217 | 0.224 | |

$\alpha $ | 0.100 | 0.060 | 1.675 | 0.094 | * |

$\beta $ | 0.884 | 0.076 | 11.648 | 0.000 | *** |

$\gamma $ | −0.055 | 0.047 | −1.169 | 0.242 | |

Kerosene | |||||

$\omega $ | 0.000 | 0.000 | 1.919 | 0.055 | * |

$\alpha $ | 0.089 | 0.036 | 2.467 | 0.014 | ** |

$\beta $ | 0.835 | 0.042 | 19.696 | 0.000 | *** |

$\gamma $ | 0.058 | 0.068 | 0.860 | 0.390 |

**Notes:**The table reports the estimates of the GJR model. The column “Std. Error” shows the Quasi-Maximum Likelihood-based standard errors. Sample Period: March 2020–June 2021, 271 daily observations. *, **, and *** indicate the significance at levels 10%, 5%, and 1%, respectively.

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

WTI | |||||

$\alpha $ | 0.000 | 1.663 | 0.000 | 1.000 | |

$\gamma $ | 0.211 | 0.137 | 1.546 | 0.122 | |

$\beta $ | 0.848 | 2.026 | 0.419 | 0.675 | |

m | −8.383 | 1.518 | −5.524 | 0.000 | *** |

${\theta}^{+}$ | 4.644 | 3.801 | 1.222 | 0.222 | |

${\omega}_{2}^{+}$ | 1.051 | 1.223 | 0.859 | 0.390 | |

${\theta}^{-}$ | −14.014 | 11.070 | −1.266 | 0.206 | |

${\omega}_{2}^{-}$ | 2.698 | 7.587 | 0.356 | 0.722 | |

Europe Brent | |||||

$\alpha $ | 0.000 | 0.190 | 0.001 | 1.000 | |

$\gamma $ | 0.246 | 0.218 | 1.130 | 0.258 | |

$\beta $ | 0.875 | 0.187 | 4.677 | 0.000 | *** |

m | −5.164 | 2.723 | −1.896 | 0.058 | * |

${\theta}^{+}$ | −1.107 | 21.944 | −0.050 | 0.960 | |

${\omega}_{2}^{+}$ | 5.226 | 9.251 | 0.565 | 0.572 | |

${\theta}^{-}$ | −8.340 | 16.254 | −0.513 | 0.608 | |

${\omega}_{2}^{-}$ | 5.078 | 2.592 | 1.959 | 0.050 | * |

Heating Oil | |||||

$\alpha $ | 0.000 | 0.043 | 0.002 | 0.998 | |

$\gamma $ | 0.087 | 0.030 | 2.874 | 0.004 | *** |

$\beta $ | 0.951 | 0.050 | 18.912 | 0.000 | *** |

m | −7.865 | 0.669 | −11.754 | 0.000 | *** |

${\theta}^{+}$ | 2.761 | 1.195 | 2.310 | 0.021 | ** |

${\omega}_{2}^{+}$ | 1.001 | 0.597 | 1.678 | 0.093 | * |

${\theta}^{-}$ | −11.238 | 3.981 | −2.823 | 0.005 | *** |

${\omega}_{2}^{-}$ | 3.580 | 1.585 | 2.258 | 0.024 | ** |

**Notes:**The table reports the estimates of the DAGM model for WTI, Europe Brent, and Heating Oil. The column “Std. Error” shows the Quasi-Maximum Likelihood-based standard errors. Sample Period: March 2020–June 2021, 271 daily observations. *, **, and *** indicate the significance at levels 10%, 5%, and 1%, respectively.

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

WTI | |||||

$\alpha $ | 0.000 | 0.059 | 0.002 | 0.999 | |

$\gamma $ | 0.320 | 0.103 | 3.100 | 0.002 | *** |

$\beta $ | 0.838 | 0.084 | 9.947 | 0.000 | *** |

m | −4.009 | 0.788 | −5.088 | 0.000 | *** |

$\theta $ | −3.332 | 1.459 | −2.283 | 0.022 | ** |

${\omega}_{2}$ | 5.715 | 1.984 | 2.881 | 0.004 | *** |

Europe Brent | |||||

$\alpha $ | 0.000 | 0.033 | 0.003 | 0.998 | |

$\gamma $ | 0.220 | 0.089 | 2.475 | 0.013 | ** |

$\beta $ | 0.888 | 0.064 | 13.842 | 0.000 | *** |

m | −4.855 | 0.916 | −5.301 | 0.000 | *** |

$\theta $ | −4.902 | 1.454 | −3.372 | 0.001 | *** |

${\omega}_{2}$ | 5.599 | 1.415 | 3.958 | 0.000 | *** |

Heating Oil | |||||

$\alpha $ | 0.079 | 0.226 | 0.350 | 0.726 | |

$\gamma $ | 0.471 | 0.096 | 4.900 | 0.000 | *** |

$\beta $ | 0.685 | 0.260 | 2.636 | 0.008 | *** |

m | −2.799 | 0.533 | −5.255 | 0.000 | *** |

$\theta $ | −3.200 | 4.227 | −0.757 | 0.449 | |

${\omega}_{2}$ | 2.092 | 1.562 | 1.339 | 0.180 |

**Notes:**The table reports the estimates of the GM model for WTI, Europe Brent, and Heating Oil. The column “Std. Error” shows the Quasi-Maximum Likelihood-based standard errors. Sample Period: March 2020–June 2021, 271 daily observations. **, and *** indicate the significance at levels 10%, 5%, and 1%, respectively.

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

Propane | |||||

$\alpha $ | 0.050 | 0.052 | 0.970 | 0.332 | |

$\gamma $ | 0.150 | 0.078 | 1.920 | 0.055 | * |

$\beta $ | 0.861 | 0.067 | 12.801 | 0.000 | *** |

m | −5.966 | 1.141 | −5.229 | 0.000 | *** |

$\theta $ | −0.851 | 2.370 | −0.359 | 0.720 | |

${\omega}_{2}$ | 3.139 | 1.205 | 2.606 | 0.009 | *** |

Gasoline | |||||

$\alpha $ | 0.000 | 0.028 | 0.004 | 0.997 | |

$\gamma $ | 0.062 | 0.025 | 2.507 | 0.012 | ** |

$\beta $ | 0.968 | 0.031 | 30.750 | 0.000 | *** |

m | −6.317 | 0.261 | −24.159 | 0.000 | *** |

$\theta $ | −4.478 | 0.876 | −5.115 | 0.000 | *** |

${\omega}_{2}$ | 7.042 | 1.054 | 6.680 | 0.000 | *** |

Kerosene | |||||

$\alpha $ | 0.040 | 0.089 | 0.445 | 0.656 | |

$\gamma $ | 0.178 | 0.050 | 3.528 | 0.000 | *** |

$\beta $ | 0.869 | 0.096 | 9.061 | 0.000 | *** |

m | −4.811 | 0.581 | −8.277 | 0.000 | *** |

$\theta $ | −1.159 | 0.929 | −1.247 | 0.212 | |

${\omega}_{2}$ | 23.798 | 612.792 | 0.039 | 0.969 |

**Notes**: The table reports the estimates of the GM model for Propane, Gasoline, and Kerosene. The column “Std. Error” shows the Quasi-Maximum Likelihood-based standard errors. Sample Period: March 2020–June 2021, 271 aily observations. *, **, and *** indicate the significance at levels 10%, 5%, and 1%, respectively.

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

Propane | |||||

$\alpha $ | 0.183 | 0.179 | 1.022 | 0.307 | |

$\gamma $ | 0.015 | 0.472 | 0.032 | 0.974 | |

$\beta $ | 0.422 | 0.663 | 0.637 | 0.524 | |

m | −7.692 | 0.399 | −19.277 | 0.000 | *** |

${\theta}^{+}$ | 2.024 | 5.973 | 0.339 | 0.735 | |

${\omega}_{2}^{+}$ | 1.001 | 7.015 | 0.143 | 0.887 | |

${\theta}^{-}$ | −7.448 | 2.914 | −2.556 | 0.011 | ** |

${\omega}_{2}^{-}$ | 1.001 | 5.849 | 0.171 | 0.864 | |

Gasoline | |||||

$\alpha $ | 0.000 | 0.024 | 0.004 | 0.997 | |

$\gamma $ | 0.064 | 0.030 | 2.160 | 0.031 | ** |

$\beta $ | 0.967 | 0.021 | 44.981 | 0.000 | *** |

m | −6.498 | 0.349 | −18.613 | 0.000 | *** |

${\theta}^{+}$ | −2.695 | 1.786 | −1.509 | 0.131 | |

${\omega}_{2}^{+}$ | 6.397 | 1.406 | 4.550 | 0.000 | *** |

${\theta}^{-}$ | −6.136 | 2.476 | −2.478 | 0.013 | ** |

${\omega}_{2}^{-}$ | 7.803 | 1.216 | 6.416 | 0.000 | *** |

Kerosene | |||||

$\alpha $ | 0.000 | 0.111 | 0.001 | 0.999 | |

$\gamma $ | 0.075 | 0.053 | 1.401 | 0.161 | |

$\beta $ | 0.946 | 0.164 | 5.755 | 0.000 | *** |

m | −7.929 | 0.549 | −14.440 | 0.000 | *** |

${\theta}^{+}$ | 3.586 | 1.378 | 2.602 | 0.009 | *** |

${\omega}_{2}^{+}$ | 1.001 | 0.838 | 1.194 | 0.232 | |

${\theta}^{-}$ | −12.261 | 3.098 | −3.957 | 0.000 | *** |

${\omega}_{2}^{-}$ | 1.930 | 0.819 | 2.357 | 0.018 | ** |

**Notes:**The table reports the estimates of the DAGM model for Propane, Gasoline, and Kerosene. The column “Std. Error” shows the Quasi-Maximum Likelihood-based standard errors. Sample Period: March 2020 to June 2021, 271 daily observations. **, and *** indicate the significance at levels 10%, 5%, and 1%, respectively.

GARCH | GJR | GM | DAGM | ||
---|---|---|---|---|---|

cDCC | a | 0.141*** | 0.155 *** | 0.139 *** | 0.132 *** |

(0.045) | (0.045) | (0.039) | (0.033) | ||

b | 0.579 *** | 0.586 *** | 0.45 *** | 0.473 *** | |

(0.097) | (0.092) | (0.154) | (0.156) | ||

DCCMIDAS | a | 0.021 | 0.023 | 0.023 | 0.025 ** |

(0.022) | (0.023) | (0.017) | (0.012) | ||

b | 0.956 *** | 0.955 *** | 0.958 *** | 0.961 *** | |

(0.024) | (0.025) | (0.018) | (0.014) | ||

${\omega}_{2}$ | 1.001 | 1.001 | 1.001 | 1.001 | |

(0.747) | (0.652) | (0.718) | (0.896) | ||

DECO | a | 0.216 *** | 0.227 *** | 0.197 * | 0.165 ** |

(0.069) | (0.072) | (0.118) | (0.071) | ||

b | 0.439 *** | 0.424 *** | 0.353 | 0.286 | |

(0.155) | (0.15) | (0.281) | (0.415) |

**Notes:**The table reports the estimated coefficients of the correlation models (first column) according the univariate specifications reported in columns from three to six. Numbers in parentheses represent the Quasi-Maximum Likelihood-based standard errors. *, **, and *** indicate the significance at levels 10%, 5%, and 1%, respectively.

Univ. | GARCH | GARCH | GARCH | GJR | GJR | GJR | GM | GM | GM | DAGM | DAGM | DAGM |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mult. | DCCM | cDCC | DECO | DCCM | cDCC | DECO | DCCM | cDCC | DECO | DCCM | cDCC | DECO |

RMSE | 5.744 | 5.799 | 5.859 | 5.75 | 5.817 | 5.874 | 5.827 | 5.93 | 5.971 | 5.655 | 5.736 | 5.784 |

FROB | 3.38 | 3.27 | 3.35 | 3.37 | 3.27 | 3.35 | 3.38 | 3.32 | 3.38 | 3.27 | 3.19 | 3.27 |

EUCL | 2.26 | 2.20 | 2.24 | 2.25 | 2.20 | 2.24 | 2.27 | 2.25 | 2.28 | 2.20 | 2.16 | 2.19 |

**Notes:**The table reports the average losses multiplied by 100,000. Label “DCCM” stands for DCC-MIDAS. Shades of gray denote inclusion in the SSM at significance level $\alpha =0.25$.

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## Share and Cite

**MDPI and ACS Style**

Andreani, M.; Candila, V.; Morelli, G.; Petrella, L.
Multivariate Analysis of Energy Commodities during the COVID-19 Pandemic: Evidence from a Mixed-Frequency Approach. *Risks* **2021**, *9*, 144.
https://doi.org/10.3390/risks9080144

**AMA Style**

Andreani M, Candila V, Morelli G, Petrella L.
Multivariate Analysis of Energy Commodities during the COVID-19 Pandemic: Evidence from a Mixed-Frequency Approach. *Risks*. 2021; 9(8):144.
https://doi.org/10.3390/risks9080144

**Chicago/Turabian Style**

Andreani, Mila, Vincenzo Candila, Giacomo Morelli, and Lea Petrella.
2021. "Multivariate Analysis of Energy Commodities during the COVID-19 Pandemic: Evidence from a Mixed-Frequency Approach" *Risks* 9, no. 8: 144.
https://doi.org/10.3390/risks9080144