# A Nonlinear Autoregressive Distributed Lag (NARDL) Analysis of West Texas Intermediate Oil Prices and the DOW JONES Index

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## Abstract

**:**

## 1. Introduction

## 2. The Links between Oil Prices and Stock Markets

## 3. Econometric Model—The Nardl Approach

## 4. Results of the Analysis

#### 4.1. Preliminary Analysis

#### 4.2. Nardl Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Plots of real values of Dow Jones Index and West Texas Intermediate (WTI) Crude Oil Index.

Table 1A: Real Monthly Value of Dow Jones Index. | |||

Summary Statistics, 2000:02–2019:02 | |||

for RDOW (228 valid observations) | |||

Mean | Median | Minimum | Maximum |

62.179 | 57.520 | 33.205 | 105.07 |

Std. Dev. | C.V. | Skewness | Ex. kurtosis |

15.061 | 0.24221 | 1.0567 | 0.74540 |

5% perc. | 95% perc. | IQ Range | Missing obs. |

42.976 | 97.810 | 16.672 | 1 |

Table 1B: Real Monthly Value of WTI Crude Oil Index. | |||

Summary Statistics, 2000:02–2019:02 | |||

for RWTI (228 valid observations) | |||

Mean | Median | Minimum | Maximum |

0.28619 | 0.27188 | 0.10930 | 0.61564 |

Std. Dev. | C.V. | Skewness | Ex. kurtosis |

0.11102 | 0.38791 | 0.45737 | −0.65485 |

5% perc. | 95% perc. | IQ Range | Missing obs. |

0.14651 | 0.45587 | 0.19142 | 1 |

ADF Test with Constant | ADF Test with Constant and Trend | |
---|---|---|

RDOW | 0.319565 (0.98) | −1.45083 (0.84) |

LRTWI | −2.71641 (0.07) | −2.65379 (0.26) |

Coefficient | Std. Error | t-Ratio | p-Value | |
---|---|---|---|---|

Test with constant | ||||

Constant | 67.37 | 2.76 | 24.45 | 0.00 |

LRTWI | −18.14 | 8.97 | −2.02 | 0.044 |

Test for unit root residuals | −0.45 | 0.97 | ||

Test with Constant and Trend | ||||

Constant | 55.10 | 1.90 | 28.93 | 0.00 |

LRTWI | −47–49 | 6.01 | −7.9 | 0.00 |

Time | 0.18 | 0.01 | 17.83 | 0.00 |

Test for unit root residuals | −2.36 | 0.60 |

ESTIMATES | LONG RUN COEFFICIENTS | |||||
---|---|---|---|---|---|---|

Coefficients | Estimate | St.Error | tValue | Estimate | St.Error | tValue |

Const | 3.42705 | 1.01640 | 3.372 *** | |||

RDOW_1 | −0.05658 | 0.01726 | −3.277 *** | |||

LRTWI_p_1 | −2.27671 | 1.79029 | −1.272 | −40.2408 | 12.2512 | −3.2846 *** |

LRTWI_n_1 | −3.98850 | 1.96948 | −2.025 *** | −70.4967 | 21.4666 | −3.2840 *** |

D.RDOW_1 | −0.05659 | 0.06823 | −0.830 | −1.0003 | 0.3054 | −3.2745 *** |

D.RDOW_2 | −0.12431 | 0.06922 | −1.796 * | −2.1971 | 0.670542 | −3.2767 *** |

D.RDOW_3 | −0.05703 | 0.07173 | −0.795 | −1.0079 | 0.3078 | −3.2745 *** |

D.RDOW_4 | 0.01684 | 0.07198 | 0.234 | 0.2977 | 0.0907 | 3.2793 *** |

D.LRTWI_p_1 | 4.70325 | 14.50059 | 0.324 | 83.1301 | 25.4168 | 3.2707 *** |

D.LRTWI_p_2 | −25.00977 | 14.34213 | −1.744 * | −442.0481 | 134.8485 | −3.2781 *** |

D.LRTWI_p_3 | −15.05225 | 14.44845 | −1.042 | −266.0488 | 81.1257 | −3.2795 *** |

D.LRTWI_p_4 | −30.79094 | 14.43954 | −2.132 ** | −544.2303 | 165.9888 | −3.2787 *** |

D.LRTWI_n_1 | 6.27525 | 11.65583 | 0.538 | 110.9151 | 33.8112 | 3.2804 *** |

D.LRTWI_n_2 | 21.93929 | 12.40570 | 1.768 * | 387.7773 | 118.2812 | 3.2784 *** |

D.LRTWI_n_3 | 2.36444 | 12.75230 | 0.185 | 41.7916 | 12.6578 | 3.3016 *** |

D.LRTWI_n_4 | 20.18801 | 11.87684 | 1.700 * | 356.8234 | 108.7410 | 3.2814 *** |

Adjusted R-Squared | 0.06617 | |||||

F-statistic | 2.044 ** | |||||

Model Diagnostic tests | ||||||

JB test | 41.50852 *** | |||||

LM test ( 4 lags ) | 1.587934 | |||||

ARCH test (4 lags) | 8.74709 | |||||

Long-Run Asymmetry test | ||||||

F. Statistic | 17.9658 *** |

© 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/).

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**MDPI and ACS Style**

Allen, D.E.; McAleer, M.
A Nonlinear Autoregressive Distributed Lag (NARDL) Analysis of West Texas Intermediate Oil Prices and the DOW JONES Index. *Energies* **2020**, *13*, 4011.
https://doi.org/10.3390/en13154011

**AMA Style**

Allen DE, McAleer M.
A Nonlinear Autoregressive Distributed Lag (NARDL) Analysis of West Texas Intermediate Oil Prices and the DOW JONES Index. *Energies*. 2020; 13(15):4011.
https://doi.org/10.3390/en13154011

**Chicago/Turabian Style**

Allen, David E., and Michael McAleer.
2020. "A Nonlinear Autoregressive Distributed Lag (NARDL) Analysis of West Texas Intermediate Oil Prices and the DOW JONES Index" *Energies* 13, no. 15: 4011.
https://doi.org/10.3390/en13154011