# Application of Multifractal Analysis in Estimating the Reaction of Energy Markets to Geopolitical Acts and Threats

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. Materials and Methods

#### 3.1. Data Description

#### 3.2. Multifractal Detrended Cross-Correlation Analysis (MF-DCCA)

## 4. Empirical Results

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Daily Index Values (

**Left**) and daily percentage changes (

**Right**) in GPR, GPR Acts and GPR Threats.

**Figure 2.**Evolution of daily prices over time, where the price unit of WTI and Brent is USD per Barrel, the natural-gas price unit is USD per MMBtu while heating oil is expressed in USD per gallon.

GPR | GPRAct | GPRThreat | WTI Crude | Brent Oil | Natural Gas | Heating Oil | |
---|---|---|---|---|---|---|---|

Data Range | 1 January 1985–30 August 2021 | 1 January 1985–30 August 2021 | 1 January 1985–30 August 2021 | 1 January 1985–30 August 2021 | 28 June 1988–30 August 2021 | 5 April 1990–30 August 2021 | 1 January 1985–30 August 2021 |

N | 13,336 | 13,336 | 13,336 | 9174 | 8377 | 7883 | 9277 |

Mean | 0.1018 | 0.1888 | 0.1590 | 0.0004 | 0.0005 | 0.0007 | 0.0004 |

Median | −0.0071 | −0.0045 | −0.0108 | 0.0009 | 0.0003 | −0.0004 | 0.0007 |

Min | −0.9511 | −0.9273 | −0.9001 | −0.3300 | −0.3477 | −0.3132 | −0.3236 |

Max | 15.4331 | 13.8347 | 15.0303 | 0.2510 | 0.2102 | 0.3831 | 0.1502 |

S.D. | 0.5660 | 0.8719 | 0.7754 | 0.0255 | 0.0230 | 0.0344 | 0.0233 |

Kurtosis | 67.1510 | 37.2597 | 51.9419 | 14.5694 | 16.3020 | 8.1301 | −0.8772 |

Skewness | 4.5892 | 4.4100 | 4.8600 | −0.1225 | −0.6401 | 0.6870 | 0.4738 |

**Table 2.**Unit root tests (RU) in the presence of structural breaks for the variables under analysis.

Asset | IO Test | AO Test | ||
---|---|---|---|---|

SB Test | UR t-Test | SB Test | UR t-Test | |

WTI | −0.149 | −33.697 ** | −0.123 | −37.319 ** |

Brent | −1.267 | −38.988 ** | −1.035 | −28.526 ** |

Natural gas | −1.03 | −34.781 ** | −0.785 | −24.152 ** |

Heating oil | −0.173 | −36.464 ** | −0.214 | −27.508 ** |

Q | GPR/WTI | GPRAct/WTI | GPRThreat/WTI | GPR/Brent | GPRAct/Brent | GPRThreat /Brent | GPR/Natural Gas | GPRAct/Natural Gas | GPRThreat/Natural Gas | GPR/Heating Oil | GPRAct/Heating Oil | GPRThreat/Heating Oil |
---|---|---|---|---|---|---|---|---|---|---|---|---|

−10 | 0.5671 | 0.6849 | 0.6062 | 0.5984 | 0.7284 | 0.6226 | 0.5441 | 0.6559 | 0.6056 | 0.5801 | 0.6858 | 0.6167 |

−9 | 0.5641 | 0.6793 | 0.6023 | 0.5935 | 0.7216 | 0.6185 | 0.5401 | 0.649 | 0.5994 | 0.5769 | 0.6807 | 0.6128 |

−8 | 0.5612 | 0.6731 | 0.5983 | 0.5883 | 0.7137 | 0.6141 | 0.5359 | 0.6412 | 0.5925 | 0.5737 | 0.6752 | 0.6087 |

−7 | 0.5584 | 0.6662 | 0.5941 | 0.5827 | 0.7047 | 0.6095 | 0.5316 | 0.6325 | 0.5851 | 0.5705 | 0.6692 | 0.6044 |

−6 | 0.5559 | 0.6586 | 0.5897 | 0.5768 | 0.6944 | 0.6047 | 0.5274 | 0.6229 | 0.577 | 0.5675 | 0.6627 | 0.5998 |

−5 | 0.5538 | 0.6506 | 0.5851 | 0.5707 | 0.6825 | 0.5996 | 0.5233 | 0.6125 | 0.5683 | 0.5648 | 0.6558 | 0.595 |

−4 | 0.5521 | 0.6424 | 0.5802 | 0.5645 | 0.6692 | 0.5943 | 0.5195 | 0.6014 | 0.5591 | 0.5625 | 0.6486 | 0.5898 |

−3 | 0.5508 | 0.6346 | 0.5744 | 0.5581 | 0.6548 | 0.5883 | 0.5163 | 0.5898 | 0.5492 | 0.5606 | 0.6412 | 0.5839 |

−2 | 0.5495 | 0.6276 | 0.5671 | 0.5515 | 0.6401 | 0.5813 | 0.5136 | 0.578 | 0.5384 | 0.5589 | 0.6337 | 0.5766 |

−1 | 0.5469 | 0.6213 | 0.5568 | 0.5445 | 0.6262 | 0.5726 | 0.5113 | 0.5664 | 0.5265 | 0.5566 | 0.6262 | 0.5671 |

0 | 0.5364 | 0.6122 | 0.5369 | 0.5359 | 0.6146 | 0.5594 | 0.5066 | 0.5544 | 0.5118 | 0.5489 | 0.6166 | 0.5505 |

1 | 0.527 | 0.6042 | 0.5188 | 0.5281 | 0.6044 | 0.5473 | 0.5022 | 0.5437 | 0.4981 | 0.542 | 0.6083 | 0.5354 |

2 | 0.5015 | 0.5876 | 0.4853 | 0.517 | 0.5943 | 0.5305 | 0.4883 | 0.5297 | 0.4813 | 0.5225 | 0.5947 | 0.5097 |

3 | 0.464 | 0.5635 | 0.4417 | 0.5028 | 0.5814 | 0.5119 | 0.4624 | 0.5108 | 0.4628 | 0.4916 | 0.5757 | 0.4763 |

4 | 0.4218 | 0.5349 | 0.3954 | 0.4863 | 0.5648 | 0.4932 | 0.4275 | 0.4866 | 0.4434 | 0.4539 | 0.5522 | 0.4388 |

5 | 0.384 | 0.5069 | 0.3542 | 0.4692 | 0.5462 | 0.4756 | 0.3925 | 0.4606 | 0.4242 | 0.4181 | 0.5278 | 0.4029 |

6 | 0.3538 | 0.4829 | 0.3209 | 0.4533 | 0.528 | 0.4602 | 0.363 | 0.4367 | 0.4065 | 0.3888 | 0.5055 | 0.3721 |

7 | 0.3306 | 0.4632 | 0.2947 | 0.4396 | 0.5117 | 0.447 | 0.3397 | 0.4162 | 0.3911 | 0.3659 | 0.4866 | 0.3468 |

8 | 0.3126 | 0.4474 | 0.274 | 0.428 | 0.4977 | 0.436 | 0.3215 | 0.3994 | 0.3779 | 0.3481 | 0.471 | 0.3264 |

9 | 0.2985 | 0.4345 | 0.2576 | 0.4182 | 0.4858 | 0.4267 | 0.3071 | 0.3855 | 0.3669 | 0.334 | 0.4581 | 0.31 |

10 | 0.2872 | 0.4239 | 0.2444 | 0.41 | 0.4758 | 0.419 | 0.2954 | 0.374 | 0.3576 | 0.3227 | 0.4474 | 0.2966 |

ΔH | 0.2799 | 0.261 | 0.3618 | 0.1884 | 0.2526 | 0.2036 | 0.2487 | 0.2819 | 0.248 | 0.2574 | 0.2384 | 0.3201 |

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**MDPI and ACS Style**

Aslam, F.; Ferreira, P.; Ali, H.; José, A.E.
Application of Multifractal Analysis in Estimating the Reaction of Energy Markets to Geopolitical Acts and Threats. *Sustainability* **2022**, *14*, 5828.
https://doi.org/10.3390/su14105828

**AMA Style**

Aslam F, Ferreira P, Ali H, José AE.
Application of Multifractal Analysis in Estimating the Reaction of Energy Markets to Geopolitical Acts and Threats. *Sustainability*. 2022; 14(10):5828.
https://doi.org/10.3390/su14105828

**Chicago/Turabian Style**

Aslam, Faheem, Paulo Ferreira, Haider Ali, and Ana Ercília José.
2022. "Application of Multifractal Analysis in Estimating the Reaction of Energy Markets to Geopolitical Acts and Threats" *Sustainability* 14, no. 10: 5828.
https://doi.org/10.3390/su14105828