# Analyzing Crude Oil Prices under the Impact of COVID-19 by Using LSTARGARCHLSTM

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{e}) tests, and secondly, it suggests the Logistic Smooth Transition Autoregressive Generalised Autoregressive Conditional Heteroskedasticity long-short term memory (LSTARGARCHLSTM) method. If the SE and (L

_{e}) tests determine chaotic or nonlinear behavior, we propose a hybrid modelling technique developed by combining the LSTARGARCH [7] with LSTM. The LSTARGARCH model has STAR type nonlinearity in both the conditional mean and variance and allows the smooth transitions between the regimes to be governed by a logistic function. To evaluate the success or accuracy of our proposed method, we compare with GARCHLSTM and traditional methods; GARCH and LSTARGARCH. Finally, in this paper, since forecasting performance is accepted as a measure of the success of applied methods, we compare the forcasting performance of all the models.

## 2. Related Work

## 3. Methodology

#### The Proposed Hybrid LSTARGARCHLSTM Model

## 4. Data and Results

#### 4.1. Data

_{t}, ldop

_{t}and lwop

_{t}show the volatilities of the Brent, Dubai and WTI crude oil prices, respectively. They were calculated as lbop

_{t}= $\mathrm{ln}({\mathrm{brent}\text{}\mathrm{oil}\text{}\mathrm{price}}_{\mathrm{t}}/{\mathrm{brent}\text{}\mathrm{oil}\text{}\mathrm{price}}_{\mathrm{t}-1}$

_{)}, ldop

_{t}= $\mathrm{ln}({\mathrm{Dubai}\text{}\mathrm{oil}\text{}\mathrm{price}}_{\mathrm{t}}/{\mathrm{Dubai}\text{}\mathrm{oil}\text{}\mathrm{price}}_{\mathrm{t}-1}$

_{)}and lwop

_{t}= $\mathrm{ln}({\mathrm{WTI}\text{}\mathrm{oil}\text{}\mathrm{price}}_{\mathrm{t}}/{\mathrm{WTI}\text{}\mathrm{oil}\text{}\mathrm{price}}_{\mathrm{t}-1}$).

#### 4.2. Results

- Firstly, some descriptive statistics were obtained. The Augmented Dickey-Fuller (ADF) unit root test [70,71] and Kapetanios, Shin, and Snell (KSS) unit root test [72] were applied. The ADF test is dependent upon a linear assumption that can cause the false results. Bigman et al. [73] showed that traditional unit root tests tends to produce “spurious regressions”. In this condition, for confirmation, we used the KSS test.
- Secondly, Tsay and Hsieh’s tests and the Brock–Dechert–Scheinkman (BDS) test were applied. These tests determined the presence of nonlinear structure, but they are not sufficient to determine the existence of chaotic behavior.
- Thirdly, SE and L
_{e}tests were applied. L_{e}is a convenient means to decide on the presence of chaotic behavior. - The LSTARGARCHLSTM method determines ARCH and GARCH effects. To evaluate the performance of our proposed method, we compared our proposed method with GARCHLSTM and traditional methods: GARCH and LSTARGARCH. For this purpose, GARCH and LSTARGARCH, and GARCHLSTM models were estimated and the most succesful model was determined.
- In the final step, the forecast accuracies of all of the models were determined.

#### 4.2.1. Some Descriptive Statistics and Tsay and Hsieh’s Tests

#### 4.2.2. BDS Test, Tsay Tests and Hsieh’s Coefficients Results

_{t,}ldop

_{t}and lwop

_{t}variables indicate evidence of chaotic structure or nonlinear stochastic processes.

#### 4.2.3. Lyapunov Exponent and Kolmogorov Entropy Tests

_{e}. L

_{e}is used to measure the average divergence from or convergence to the initial point of a dynamical system. If the L

_{e}has a large value, it indicates high sensivity to initial conditions. While a positive Lyapunov coefficient typically signifies chaotic structure, a negative coefficient typically shows convergence to initial conditions [21,82]. The positive value of L

_{e}indicates the presence of chaotic structure in the oil prices. Table 4 shows the results. Additionally, Adrangi and Chatrath [20], Bildirici and Sonustun [82], and Lahmiri [21] also determined the existence of chaotic structure for oil prices.

_{t}. Entropy may be understood as the degree of the distortion of the market information reflected in the price system. Positive values of the entropy accent that the oil price information can still be used to understand the oil market dynamics. For example, since the reciprocal of the entropy for Brent is $1/0.9617$, the corresponding time scale of an effective and rational forecast for that entropy value must be within ~11 days. He [23] found 36 days by using the Kolmogorov entropy, and Bildirici [84] found 68 days.

#### 4.2.4. Results with the GARCH, LSTARGARCH, LSTARGARCHLSTM and GARCHLSTM Models

#### 4.2.5. The Architecture of the GARCH–LSTM and LSTARGARCH–LSTM Models

#### 4.2.6. The Results of the GARCHLSTM and LSTARGARCHLSTM Method

## 5. Forecast Results

#### 5.1. In–Sample Forecast Results

#### 5.2. Out-of-Sample Forecast Results

#### 5.3. To Test for Forecast Accuracy

_{0}hypothesis of these tests assumes that the models have the same level of accuracy. For most cases, since the p-value is <0.05, the H

_{0}hypothesis is rejected. For both tests, the p-value is >0.05 only for the RMSE comparison of the GARCH and GARCHLSTM models. Hence, these two models are comparable in terms of RMSE performance.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Table 1.**Descriptive statistics and unit root tests. lbop

_{t,}ldop

_{t}and lwop

_{t}are the volatilities of the Brent, Dubai and WTI crude oil prices, respectively. ARCH shows the Engle’s [75] ARCH test statistic. White [76] shows White’s heteroscedasticity test statistic, RESET show the RESET statistic, and this test adds the second power of the fitted value as an additional regressor.

Lbop_{t} | Ldop_{t} | Lwop_{t} | |
---|---|---|---|

Kurtosis | 17.583 | 21.69 | 18.763 |

Skewness | −1.0417 | −1.5635 | −0.45 |

JB | 346.64 | 282.65 | 208.5756 |

ARCH effect | 17.89 | 27.67 | 19.25 |

White | 14.36 | 13.73 | 10.88 |

RESET | 13.58 | 10.72 | 2.18 |

Unit Root Tests | |||

- | Level | Level | Level |

ADF | −56.86 | −15.069 | −68.44 |

KSS | −54.25 | −11.38 | −53.58 |

Decision | I(0) | I(0) | I(0) |

Z Statistics | |||
---|---|---|---|

Dimension | Lbop_{t} | Ldop_{t} | Lwop_{t} |

2 | 34.40350 | 35.9270 | 18.999077 |

3 | 35.62271 | 38.6123 | 22.72316 |

4 | 38.02877 | 41.3846 | 24.84834 |

5 | 41.58379 | 44.5027 | 26.97560 |

6 | 46.53435 | 49.7965 | 28.99879 |

**Table 3.**Tsay test and Hsieh’s coefficients. r

_{ij}’s are Hsieh’s [81] third-order moment coefficients. Hsieh’s [81] third-order moment coefficients were obtained as $\left[\sum {x}_{t}{x}_{t-i}{x}_{t-j}/T\right]/{\left[\sum {x}_{t}^{2}/T\right]}^{1.5}$. Only coefficients for r(1, 1) and r(1, 2) were given.

Hsieh’s Coefficients | Tsay’s Nonlinearity Test Statistic | |||||||
---|---|---|---|---|---|---|---|---|

r_{ij} are Hsich’s Third-Order Moment Coefficients for Lags i and j | Tsay’s Nonlinearity Test Statistic | |||||||

Lbop_{t} | Ldop_{t} | Lwop_{t} | Lbop_{t} | Ldop_{t} | Lwop_{t} | |||

r(1) | r(2) | r(1) | r(2) | r(1) | r(2) | 133.41 | 100.58 | 102.001 |

0.1 | −0.42 | −0.35 | 0.12 | −0.124 | 0.45 | - | - | - |

Lyapunov Exponent Method | Shannon Entropy Method | ||||
---|---|---|---|---|---|

Lbop_{t} | Ldop_{t} | Lwop_{t} | Lbop_{t} | Ldop_{t} | Lwop_{t} |

0.9504 | 0.9071 | 0.8481 | 0.9617 | 0.983 | 0.9121 |

^{1}Shannon entropy(SE) is defined as SE(x) = $-\sum _{i=1}^{n}{p}_{i}\mathrm{log}\left({p}_{i}\right)$ where $p$

_{i}is the probability mass of the ith discrete level such that $\sum _{i}{p}_{i}=1$ [83]. The R software DChaos package for calculating ${\mathrm{L}}_{\mathrm{e}}$ and R Software Entropy package for calculating SE are employed.

**Table 5.**Baseline models. LogL: Log-likelihood, ARCH(p): pth order ARCH–LM test, AIC: Akaike information criterion, SIC: Schwarz information criterion, HQ: Hannan–Quinn information criterion.

Lwop_{t} | Lbop_{t} | Ldop_{t} | |||||||
---|---|---|---|---|---|---|---|---|---|

- | GARCH | LSTARGARCH | GARCH | LSTARGARCH | GARCH | LSTARGARCH | |||

- | Regime 1 | Regime 2 | Regime 1 | Regime 2 | Regime 1 | Regime 2 | |||

Cst(M) | 0.0214 (2.13) (0.0) ^{1} | 00633 (3428) (00) ^{1} | 0116 (576) (00) ^{1} | 00215 (187) (00) ^{1} | 0015 (456) (00) ^{1} | 000520 (625) (00) ^{1} | 0.0754 (1.94) (0.0) ^{1} | 0.0004 (2.17) (0.0) ^{1} | 0.05377 (7.61) (0.0) ^{1} |

Cst(V) | 0.255 (1.91) (0.0) ^{1} | 1109 (488) (00) ^{1} | 0287 (836) (00) ^{1} | 0312 (193) (00) ^{1} | 0338 (263) (00) ^{1} | 0089 (558) (00) ^{1} | 0.02905 (2.05) (0.0) ^{1} | 0.178 (1.93) (0.0) ^{1} | 0.205 (1.94) (0.0) ^{1} |

ARCH | 0.19 (3.61) (0.0) ^{1} | 0128 (8689) (00) ^{1} | 0089 (1427) (00) ^{1} | 0189 (278) (00) ^{1} | 00287 (385) (00) ^{1} | 01052 (1246) (00) ^{1} | 0.19899 (2.67) (0.0) ^{1} | 0.201 (7.25) (0.0) ^{1} | 0.112 (2.27) (0.0) ^{1} |

GARCH | 0.67 (4.78) (0.0) ^{1} | 0722 (686) (00) ^{1} | 0903 (516) (00) ^{1} | 0611 (1887) (00) ^{1} | 09401 (8136) (00) ^{1} | 08795 (1056) (00) ^{1} | 0.61922 (4.051) (0.0) ^{1} | 0.769 (2.105) (0.0) ^{1} | 0.872 (9.26) (0.0) ^{1} |

LogL | 10536.28 | 271389 | 118421 | 316852 | 112345 | 388873 | |||

AIC: | 10.353 | −3913 | 77224 | −33126 | 84167 | −44956 | |||

SIC: | 10.042 | −389 | 77154 | −32997 | 84869 | −44836 | |||

HQ: | 10.054 | −385 | 77199 | −33078 | 83940 | −44612 | |||

ARCH (1–2): | 0.035 | 0097 | 0056 | 0042 | 0083 | 0064 | |||

ARCH (1–5): | 0.039 | 0095 | 0058 | 0041 | 0078 | 0062 |

^{1}shows p–values.

**Table 6.**Generalised Autoregressive Conditional Heteroskedasticity long-short term memory (GARCHLSTM) and Logistic Smooth Transition Autoregressive (LSTAR) GARCHLSTM.

Lbop_{t} | Ldop_{t} | Lwop_{t} | ||||
---|---|---|---|---|---|---|

GARCHLSTM | LSTARGARCHLSTM | GARCHLSTM | LSTARGARCHLSTM | GARCHLSTM | LSTARGARCHLSTM | |

Training rho ^{1} | 0.89 | 0.92 | 0.88 | 0.93 | 0.91 | 0.95 |

Test rho | 0.88 | 0.90 | 0.87 | 0.92 | 0.90 | 0.91 |

Training RMSE | 0.24 | 0.04 | 0.21 | 0.08 | 0.33 | 0.03 |

Training MAE | 0.23 | 0.03 | 0.20 | 0.07 | 0.31 | 0.03 |

Test RMSE | 0.22 | 0.022 | 0.1 | 0.07 | 0.29 | 0.06 |

Test MAE | 0.21 | 0.022 | 0.09 | 0.06 | 0.289 | 0.059 |

^{1}Rho represents the training and test sample correlation coefficient, MAE and RMSE are the Mean Absolute Error and the Root Mean Squared Error, respectively.

Lwop_{t} | Lbop_{t} | Ldop_{t} | |||||||
---|---|---|---|---|---|---|---|---|---|

- | GARCH LSTM | LSTARGARCHLSTM | GARCH LSTM | LSTARGARCHLSTM | GARCH LSTM | LSTARGARCHLSTM | |||

- | - | Regime 1 | Regime 2 | - | Regime 1 | Regime 2 | - | Regime 1 | Regime 2 |

Cst(M) | 00618 (256) (00) ^{1} | 0986 (212) (00) ^{1} | 0651 (474) (00) ^{1} | 025 (281) (00) ^{1} | 0156 (276) (00) ^{1} | 0554 (288) (00) ^{1} | 0173 (262) (00) ^{1} | 0263 (281) (00) ^{1} | 0361 (453) (00) ^{1} |

Cst(V) | 0985 (265) (00) ^{1} | 0431 (226) (00) ^{1} | 0562 (382) (00) ^{1} | 0861 (288) (00) ^{1} | 0297 (263) (00) ^{1} | 0441 (376) (00) ^{1} | 0565 (288) (00) ^{1} | 0428 (287) (00) ^{1} | 0397 (432) (00) ^{1} |

ARCH | 0207 (316) (00) ^{1} | 0102 (977) (00) ^{1} | 0023 (229) (00) ^{1} | 0127 (458) (00) ^{1} | 0111 (803) (00) ^{1} | 0095 (297) (00) ^{1} | 0198 (448) (00) ^{1} | 0118 (675) (00) ^{1} | 0037 (236) (00) ^{1} |

GARCH | 0721 (571) (00) ^{1} | 0881 (558) (00) ^{1} | 0962 (356) (00) ^{1} | 0811 (631) (00) ^{1} | 0878 (558) (00) ^{1} | 0901 (356) (00) ^{1} | 0781 (756) (00) ^{1} | 0844 (287) (00) ^{1} | 0942 (522) (00) ^{1} |

LogL | 2849.2 | 2038.21 | 2669.3 | 1984.18 | 2986.2 | 1989.75 | |||

AIC: | 2.981 | −1.413 | 2.661 | −1.513 | 2.875 | −1.897 | |||

SIC: | 2.816 | −1.391 | 2.514 | −1.489 | 2.867 | −1.791 | |||

HQ: | 2.807 | −1.388 | 2.507 | −1.417 | 2.821 | −1.745 | |||

ARCH (1–2): | 0.123 | 0.089 | 0.107 | 0.076 | 0.109 | 0.081 | |||

ARCH (1–5): | 0.124 | 0.090 | 0.114 | 0.071 | 0.112 | 0.082 |

^{1}shows p–values.

- | GARCH | GARCHLSTM | LSTARGARCH | LSTARGARCHLSTM | |
---|---|---|---|---|---|

lbop_{t} | RMSE | 0.995 | 0.088 | 0.04 | 0.001 |

MAE | 0.84 | 0.072 | 0.027 | 0.0009 | |

ldop_{t} | RMSE | 0.937 | 0.097 | 0.029 | 0.005 |

MAE | 0.79 | 0.079 | 0.014 | 0.0039 | |

lwop_{t} | RMSE | 0.49 | 0.034 | 0.0291 | 0.006 |

MAE | 0.35 | 0.022 | 0.0216 | 0.0055 |

Lbopt | ||||||
---|---|---|---|---|---|---|

GARCH | LSTARGARCH | |||||

- | T + 1 | T + 10 | T + 20 | T + 1 | T + 10 | T + 20 |

RMSE | 0.5126 | 0.538 | 0.547 | 0.0107 | 0.0213 | 0.038 |

MAE | 0.5028 | 0.536 | 0.51 | 0.0106 | 0.0209 | 0.0375 |

GARCHLSTM | LSTARGARCHLSTM | |||||

RMSE | 0.052 | 0.0459 | 0.01438 | 0.0031 | 0.0038 | 0.0051 |

MAE | 0.049 | 0.0448 | 0.01399 | 0.0029 | 0.0036 | 0.0049 |

ldop_{t} | ||||||

GARCH | LSTARGARCH | |||||

RMSE | 0.5187 | 0.4896 | 0.626 | 0.01125 | 0.0308 | 0.0397 |

MAE | 0.5098 | 0.4891 | 0.621 | 0.01117 | 0.0299 | 0.0394 |

GARCHLSTM | LSTARGARCHLSTM | |||||

RMSE | 0.059 | 0.051 | 0.0495 | 0.005 | 0.006 | 0.0068 |

MAE | 0.058 | 0.0501 | 0.0471 | 0.0038 | 0.0043 | 0.0052 |

lwop_{t} | ||||||

GARCH | LSTARGARCH | |||||

RMSE | 0.4472 | 0.4526 | 0.5066 | 0.0131 | 0.0313 | 0.034 |

MAE | 0.4463 | 0.4511 | 0.5012 | 0.0122 | 0.0310 | 0.032 |

GARCHLSTM | LSTARGARCHLSTM | |||||

RMSE | 0.041 | 0.039 | 0.0385 | 0.0022 | 0.0024 | 0.0025 |

MAE | 0.040 | 0.037 | 0.0381 | 0.0021 | 0.0022 | 0.0024 |

WS TEST | ||||||
---|---|---|---|---|---|---|

RMSE_{GARCH} | RMSE_{GARCHLSTM} | RMSE_{LSTARGARCH} | RMSE_{LSTARGARCHLSTM} | |||

DM Test | RMSE_{GARCH} | - | 0.00 | 0.00 | 0.00 | |

lbop_{t} | RMSE_{GARCHLSTM} | 0.00 | - | 0.00 | 0.00 | |

RMSE_{LSTARGARCH} | 0.00 | 0.00 | - | 0.00 | ||

RMSE_{LSTARGARCHLSTM} | 0.00 | 0.00 | 0.00 | - | ||

RMSE_{GARCH} | - | 0.00 | 0.00 | 0.00 | ||

ldop_{t} | RMSE_{GARCHLSTM} | 0.00 | - | 0.00 | 0.00 | |

RMSE_{LSTARGARCH} | 0.00 | 0.00 | - | 0.00 | ||

RMSE_{LSTARGARCHLSTM} | 0.00 | 0.00 | 0.00 | - | ||

RMSE_{GARCH} | - | 0.00 | 0.00 | 0.00 | ||

lwop_{t} | RMSE_{GARCHLSTM} | 0.00 | - | 0.00 | 0.00 | |

RMSE_{LSTARGARCH} | 0.00 | 0.00 | - | 0.00 | ||

RMSE_{LSTARGARCHLSTM} | 0.00 | 0.00 | 0.00 | - |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bildirici, M.; Guler Bayazit, N.; Ucan, Y.
Analyzing Crude Oil Prices under the Impact of COVID-19 by Using LSTARGARCHLSTM. *Energies* **2020**, *13*, 2980.
https://doi.org/10.3390/en13112980

**AMA Style**

Bildirici M, Guler Bayazit N, Ucan Y.
Analyzing Crude Oil Prices under the Impact of COVID-19 by Using LSTARGARCHLSTM. *Energies*. 2020; 13(11):2980.
https://doi.org/10.3390/en13112980

**Chicago/Turabian Style**

Bildirici, Melike, Nilgun Guler Bayazit, and Yasemen Ucan.
2020. "Analyzing Crude Oil Prices under the Impact of COVID-19 by Using LSTARGARCHLSTM" *Energies* 13, no. 11: 2980.
https://doi.org/10.3390/en13112980