# Nonlinear Causality between Crude Oil Prices and Exchange Rates: Evidence and Forecasting

## Abstract

**:**

## 1. Introduction

- First, we found strong nonlinear causal relationships between crude oil prices and most investigated exchange rates;
- Second, we showed that the significance of the detected relationships has changed in recent years;
- Third, we applied SVR models of different kernels and regressors to verify if it is possible to exploit the detected relationships for effective forecasting.

## 2. Materials and Methods

#### 2.1. Data

#### 2.2. Nonlinear Causality Tests

#### 2.3. Support Vector Regression

- Linear: $K\left({x}_{t},x\right)={x}_{t}{}^{T}x$;
- Radial Basis Function (RBF): $K\left({x}_{t},x\right)=\mathrm{exp}\left(-\gamma \parallel {x}_{t}-{x\parallel}^{2}\right)$;
- Polynomial: $K\left({x}_{t},x\right)={\left(1+{x}_{t}{}^{T}x\right)}^{p}$; $p=2,3,\dots $

## 3. Results

#### 3.1. Nonlinear Granger Causality Testing

#### 3.2. Forecasting

- The autoregressive model of type (25) with the linear kernel (SVR_ar_lin);
- The autoregressive model of type (25) with the RBF kernel (SVR_ar_rbf);
- The extended model of type (26) with the linear kernel (SVR_reg_lin);
- The extended model of type (26) with the RBF kernel (SVR_reg_rbf).

## 4. Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Mean | Min | Max | SD | Skew | Kurt | LB(10) | |
---|---|---|---|---|---|---|---|

3 January 2011–31 December 2020 | |||||||

Crude oil | −0.024 | −64.370 | 41.202 | 2.920 | −3.270 | 121.79 | 0.000 |

EUR/USD | −0.002 | −2.948 | 2.962 | 0.526 | −0.087 | 2.123 | 0.382 |

GBP/USD | −0.005 | −9.505 | 3.130 | 0.572 | −1.809 | 31.663 | 0.263 |

JPY/USD | −0.009 | −3.466 | 4.136 | 0.564 | 0.113 | 5.298 | 0.655 |

3 January 2011–31 December 2015 (Period 1) | |||||||

Crude oil | −0.074 | −8.245 | 8.508 | 1.696 | −0.062 | 3.175 | 0.051 |

EUR/USD | −0.015 | −2.230 | 2.962 | 0.595 | −0.012 | 1.524 | 0.485 |

GBP/USD | −0.003 | −1.649 | 1.490 | 0.462 | −0.085 | 0.562 | 0.250 |

JPY/USD | −0.031 | −3.466 | 3.032 | 0.577 | −0.221 | 4.110 | 0.402 |

4 January 2016–31 December 2020 (Period 2) | |||||||

Crude oil | 0.026 | −64.370 | 41.202 | 3.756 | −3.081 | 87.576 | 0.000 |

EUR/USD | 0.010 | −2.948 | 1.803 | 0.448 | −0.186 | 2.507 | 0.718 |

GBP/USD | −0.007 | −9.505 | 3.130 | 0.664 | −2.271 | 34.228 | 0.339 |

JPY/USD | 0.013 | −2.653 | 4.136 | 0.551 | 0.503 | 6.583 | 0.244 |

$\mathit{\epsilon}$ | Test | Number of Lags lx = ly | |||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||

Brent$\to $EUR/USD (Period 1) | |||||||||

1 | H-J | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

D-P | 0.0001 | 0.0000 | 0.0004 | 0.0022 | 0.0039 | 0.0055 | 0.0170 | 0.0257 | |

1.5 | H-J | 0.0005 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

D-P | 0.0007 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0002 | |

Brent$\to $EUR/USD (Period 2) | |||||||||

1 | H-J | 0.1128 | 0.3092 | 0.1869 | 0.3024 | 0.3411 | 0.1714 | 0.1495 | 0.1228 |

D-P | 0.1183 | 0.3502 | 0.2668 | 0.4334 | 0.5253 | 0.2647 | 0.3567 | 0.2685 | |

1.5 | H-J | 0.0724 | 0.2511 | 0.1410 | 0.2270 | 0.2121 | 0.2871 | 0.2542 | 0.2738 |

D-P | 0.0737 | 0.2548 | 0.1481 | 0.2267 | 0.2526 | 0.3231 | 0.2894 | 0.2821 | |

EUR/USD$\to $Brent(Period 1) | |||||||||

1 | H-J | 0.0017 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

D-P | 0.0028 | 0.0000 | 0.0015 | 0.0013 | 0.0031 | 0.0192 | 0.0175 | 0.0175 | |

1.5 | H-J | 0.0242 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

D-P | 0.0240 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0001 | 0.0003 | |

EUR/USD$\to $Brent(Period 2) | |||||||||

1 | H-J | 0.2438 | 0.0659 | 0.2864 | 0.3981 | 0.6039 | 0.4586 | 0.5040 | 0.4584 |

D-P | 0.2785 | 0.0811 | 0.3206 | 0.4528 | 0.5708 | 0.4194 | 0.5161 | 0.4944 | |

1.5 | H-J | 0.0812 | 0.0564 | 0.3510 | 0.2544 | 0.4281 | 0.3952 | 0.4743 | 0.5211 |

D-P | 0.0798 | 0.0665 | 0.3918 | 0.2936 | 0.4838 | 0.4705 | 0.5652 | 0.5735 |

$\mathit{\epsilon}$ | Test | Number of Lags lx = ly | |||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||

Brent$\to $GBP/USD (Period 1) | |||||||||

1 | H-J | 0.0154 | 0.0017 | 0.0006 | 0.0000 | 0.0000 | 0.0002 | 0.0003 | 0.0089 |

D-P | 0.0200 | 0.0059 | 0.0079 | 0.0064 | 0.0138 | 0.0290 | 0.0474 | 0.1421 | |

1.5 | H-J | 0.1482 | 0.0029 | 0.0026 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

D-P | 0.1692 | 0.0047 | 0.0069 | 0.0012 | 0.0009 | 0.0003 | 0.0002 | 0.0009 | |

Brent$\to $GBP/USD (Period 2) | |||||||||

1 | H-J | 0.0132 | 0.2492 | 0.1748 | 0.0290 | 0.0134 | 0.0075 | 0.0130 | 0.0657 |

D-P | 0.0154 | 0.3152 | 0.2010 | 0.0493 | 0.0344 | 0.0410 | 0.0453 | 0.1207 | |

1.5 | H-J | 0.0047 | 0.0308 | 0.0162 | 0.0023 | 0.0015 | 0.0005 | 0.0003 | 0.0006 |

D-P | 0.0046 | 0.0399 | 0.0192 | 0.0028 | 0.0023 | 0.0012 | 0.0008 | 0.0020 | |

GBP/USD$\to $Brent(Period 1) | |||||||||

1 | H-J | 0.0477 | 0.0017 | 0.0030 | 0.0007 | 0.0014 | 0.0079 | 0.0140 | 0.0206 |

D-P | 0.0574 | 0.0041 | 0.0130 | 0.0166 | 0.0339 | 0.0544 | 0.0562 | 0.0785 | |

1.5 | H-J | 0.0669 | 0.0026 | 0.0087 | 0.0006 | 0.0012 | 0.0033 | 0.0061 | 0.0029 |

D-P | 0.0716 | 0.0027 | 0.0129 | 0.0035 | 0.0069 | 0.0163 | 0.0235 | 0.0149 | |

GBP/USD$\to $Brent(Period 2) | |||||||||

1 | H-J | 0.0186 | 0.1343 | 0.5399 | 0.1944 | 0.1731 | 0.1291 | 0.0985 | 0.1465 |

D-P | 0.0212 | 0.1805 | 0.5493 | 0.2012 | 0.1657 | 0.1406 | 0.1829 | 0.2603 | |

1.5 | H-J | 0.0315 | 0.0628 | 0.4332 | 0.1904 | 0.2265 | 0.1261 | 0.0164 | 0.0104 |

D-P | 0.0311 | 0.0644 | 0.4081 | 0.1787 | 0.2515 | 0.1579 | 0.0269 | 0.0181 |

$\mathit{\epsilon}$ | Test | Number of Lags lx = ly | |||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||

Brent$\to $JPY/USD (Period 1) | |||||||||

1 | H-J | 0.3999 | 0.3858 | 0.6696 | 0.3276 | 0.2312 | 0.0538 | 0.0321 | 0.0092 |

D-P | 0.4608 | 0.3947 | 0.6248 | 0.2041 | 0.2074 | 0.1337 | 0.0980 | 0.0925 | |

1.5 | H-J | 0.5198 | 0.6875 | 0.8965 | 0.9217 | 0.9594 | 0.8297 | 0.5170 | 0.3306 |

D-P | 0.5192 | 0.7003 | 0.9109 | 0.9355 | 0.9686 | 0.8229 | 0.4534 | 0.3344 | |

Brent$\to $JPY/USD (Period 2) | |||||||||

1 | H-J | 0.0405 | 0.3544 | 0.3852 | 0.4075 | 0.3559 | 0.4972 | 0.7010 | 0.7573 |

D-P | 0.0579 | 0.4407 | 0.5160 | 0.5377 | 0.4822 | 0.5730 | 0.6791 | 0.6939 | |

1.5 | H-J | 0.0293 | 0.2695 | 0.2326 | 0.1326 | 0.1741 | 0.3503 | 0.5965 | 0.5193 |

D-P | 0.0328 | 0.3097 | 0.2666 | 0.1528 | 0.2097 | 0.3706 | 0.6288 | 0.5560 | |

JPY/USD$\to $Brent(Period 1) | |||||||||

1 | H-J | 0.5404 | 0.7095 | 0.5646 | 0.1271 | 0.0053 | 0.0042 | 0.0066 | 0.0039 |

D-P | 0.5813 | 0.6788 | 0.4553 | 0.1219 | 0.0326 | 0.0267 | 0.0437 | 0.0567 | |

1.5 | H-J | 0.5028 | 0.8275 | 0.8776 | 0.5392 | 0.3383 | 0.3392 | 0.2614 | 0.1260 |

D-P | 0.5102 | 0.8482 | 0.8728 | 0.4764 | 0.2832 | 0.2534 | 0.1995 | 0.1070 | |

JPY/USD$\to $Brent(Period 2) | |||||||||

1 | H-J | 0.0133 | 0.0027 | 0.0136 | 0.1000 | 0.2297 | 0.4628 | 0.2631 | 0.1618 |

D-P | 0.0192 | 0.0049 | 0.0242 | 0.1400 | 0.2772 | 0.4756 | 0.2513 | 0.1392 | |

1.5 | H-J | 0.0042 | 0.0004 | 0.0002 | 0.0008 | 0.0034 | 0.0129 | 0.0141 | 0.0142 |

D-P | 0.0037 | 0.0003 | 0.0002 | 0.0006 | 0.0034 | 0.0160 | 0.0207 | 0.0239 |

Modeled Relationship | $\mathit{l}\mathit{x}=\mathit{l}\mathit{y}$ |
---|---|

Brent$\to $EUR/USD | 3 |

EUR/USD$\to $Brent | 2 |

Brent$\to $GBP/USD | 6 |

GBP/USD$\to $Brent | 8 |

Brent$\to $JPY/USD | 8 |

JPY/USD$\to $Brent | 8 |

Modeled Relation | Period | Model | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

WN | SVR_ar_lin | SVR_ar_rbf | SVR_reg_lin | SVR_reg_rbf | |||||||

MAE | MSE | MAE | MSE | MAE | MSE | MAE | MSE | MAE | MSE | ||

Brent$\to $EUR/USD | Period 1 | 0.430 | 0.352 | 0.434 | 0.363 | 0.431 | 0.352 | 0.431 | 0.353 | 0.431 | 0.353 |

Period 2 | 0.304 | 0.162 | 0.308 | 0.164 | 0.303 | 0.161 | 0.313 | 0.180 | 0.303 | 0.161 | |

EUR/USD$\to $Brent | Period 1 | 1.353 | 3.924 | 1.359 | 3.943 | 1.359 | 3.943 | 1.359 | 3.943 | 1.359 | 3.943 |

Period 2 | 2.512 | 27.948 | 2.513 | 27.616 | 2.501 | 27.749 | 2.519 | 28.540 | 2.515 | 27.934 | |

Brent$\to $GBP/USD | Period 1 | 0.332 | 0.202 | 0.332 | 0.203 | 0.331 | 0.202 | 0.345 | 0.217 | 0.331 | 0.202 |

Period 2 | 0.453 | 0.389 | 0.452 | 0.389 | 0.453 | 0.388 | 0.466 | 0.450 | 0.455 | 0.391 | |

GBP/USD$\to $Brent | Period 1 | 1.353 | 3.924 | 1.360 | 3.972 | 1.361 | 3.929 | 1.371 | 4.041 | 1.361 | 3.958 |

Period 2 | 2.512 | 27.948 | 2.526 | 27.925 | 2.529 | 28.096 | 2.558 | 29.170 | 2.518 | 27.987 | |

Brent$\to $JPY/USD | Period 1 | 0.358 | 0.250 | 0.360 | 0.256 | 0.360 | 0.252 | 0.368 | 0.269 | 0.359 | 0.251 |

Period 2 | 0.303 | 0.221 | 0.326 | 0.341 | 0.305 | 0.222 | 0.323 | 0.264 | 0.304 | 0.222 | |

JPY/USD$\to $Brent | Period 1 | 1.353 | 3.924 | 1.369 | 3.997 | 1.353 | 3.933 | 1.358 | 3.962 | 1.356 | 3.944 |

Period 2 | 2.512 | 27.948 | 2.504 | 27.846 | 2.717 | 41.748 | 2.513 | 27.834 | 2.599 | 30.809 |

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## Share and Cite

**MDPI and ACS Style**

Orzeszko, W.
Nonlinear Causality between Crude Oil Prices and Exchange Rates: Evidence and Forecasting. *Energies* **2021**, *14*, 6043.
https://doi.org/10.3390/en14196043

**AMA Style**

Orzeszko W.
Nonlinear Causality between Crude Oil Prices and Exchange Rates: Evidence and Forecasting. *Energies*. 2021; 14(19):6043.
https://doi.org/10.3390/en14196043

**Chicago/Turabian Style**

Orzeszko, Witold.
2021. "Nonlinear Causality between Crude Oil Prices and Exchange Rates: Evidence and Forecasting" *Energies* 14, no. 19: 6043.
https://doi.org/10.3390/en14196043