# Nonlinear Relationships between Oil Prices and Implied Volatilities: Providing More Valuable Information

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology and Data

#### 2.1. Autoregressive Distributed Lag (ARDL) Models

_{t}and x

_{t}have a long-run comovement relationship, which clearly states cointegration between the variables exists at the usual level of significance. If the independent variable is I (d) $(0\le d\le 1)$, the boundary between two asymptotic critical values provides the cointegration test, which assumes that the lower limit of the independent variable is I(0) and the upper limit is I(1). If the F value of the test statistic exceeds the upper limit, there will be long-term equilibrium cointegration between two variables; if the test statistic is lower than the lower limit, the null hypothesis of cointegration will be rejected, but if the test statistic is between the upper and lower limits, it will be impossible to make a judgment. For the mixed I(0) and I(1) case, the F statistics were calculated and compared with two different sets of critical values provided by [30].

#### 2.2. Nonlinear Granger Causality Test

_{t}when the following conditions are satisfied:

^{*}for each y

^{*}, so the modified H

_{0}indicates that the following relation is established:

_{m}is the bandwidth parameter associated with the number of samples $(m)$. When a local density function is estimated, we obtain the following test statistic:

#### 2.3. Data

## 3. Empirical Result and Analysis

#### 3.1. Unit Root Test Results

#### 3.2. Results of the ARDL Test

_{OVX}leads to a 0.973% increase (decrease) in oil prices, while the estimated long-run coefficient is –1.46, which says that a 1% decline (rise) in the ${L}_{VIX}^{}$ leads to a 1.46% increase (decrease) in oil prices. Obviously, the change in oil price was more negatively impacted by ${L}_{VIX}^{}$ than ${L}_{OVX}^{}$. Moreover, since the oil price was cointegrated with OVX and VIX (at the 5% level), respectively, the short-run error correction coefficient of OVX representing the adjustment toward long-run equilibrium was larger than that of VIX, which were −0.006 and −0.004, respectively.

#### 3.3. Linear Granger Causality Tests

#### 3.4. Brock–Dechert–Scheinkma (BDS) Test Results

#### 3.5. Results of Structural Breaks

#### 3.6. Nonlinear Autoregressive Distributed Lag (NARDL)-Bound Test Results

_{LR}, rejected the null hypothesis of long-run symmetric fluctuations in the oil price. With an estimated long-run coefficient of −0.919 (−0.900), there was a significant, long-run negative relationship between the oil price and a rise in OVX, ${L}_{OVX}^{+}$ and a decline in OVX, ${L}_{OVX}^{-}$, thereby indicating that an increase of 1% in OVX results in a 0.919% decline in the oil price, and a 1% decrease in OVX causes oil prices to rise by 0.9%. At the same time, there was also a significant, long-run negative relationship between the oil price and a rise in VIX, ${L}_{VIX}^{+}$ and a decline in VIX, ${L}_{VIX}^{-}$ with the estimated long-run coefficients being −1.147 and −1.125, respectively, This indicates that a 1% increase in VIX leads to a 1.147% decline in the oil price, while a 1% decline in VIX leads to a 1.125% increase in oil prices.

_{SR}) of OVX (VIX) rejected the null hypothesis of symmetric fluctuations. This means that when OVX (VIX) rises, current OVX (VIX) negatively impacts the oil price, but one-period lag and two-period lag OVX (VIX) positively impact the oil price; when OVX (VIX) declines, current OVX (VIX) negatively impacts the oil price, but only one-period lag OVX (VIX) positively impacts the oil price (Table 6, row 9–10). However, in the long-run equilibrium analysis (Table 6, row 11–12), OVX (VIX) negatively impacts the oil price regardless of rising OVX (VIX) or declining OVX (VIX), but the impact of rising OVX (VIX) on the oil price is slightly greater than the impact of their decline on the oil price, which emphasizes a long-run, asymmetric co-integration between them.

#### 3.7. Nonlinear Granger Causality Result

^{-2/7}, 1.5) based on the DP(2006) model). Table 7 shows the results of the nonlinear Granger causal relationship between the oil price and OVX (VIX). It is clear that there was bidirectional causality between the oil price and OVX in dimensions lx = ly = 1, 2, and 3, though that for VIX was simply in lx = ly = 1, 2. Apparently, the nonlinear causality results in Table 7 are rather different from the linear causality results in Table 3. In order to compare the difference between these two methods (linear vs. nonlinear Granger causality), we put the results together in the same table (Table 8), where we will analyze their causal relationships in some detail.

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The series graphs of oil price, oil price volatility index (OVX), and stock index options volatility index (VIX).

Level | ADF | PP | KPSS |
---|---|---|---|

OP | −1.598 | −1.655 | 1.843 *** |

OVX | −3.046 ** | −2.903 ** | 0.068 |

VIX | −4.452 *** | −4.305 *** | 0.072 |

OVX ⟶ OP | VIX ⟶ OP | ||||
---|---|---|---|---|---|

Variable | Coefficient | t-stat | Variable | Coefficient | t-stat |

Constant | 0.05 *** | 4.026 | Constant | 0.042 *** | 4.370 |

$O{P}_{t-1}$ | 0.993 *** | 600.772 | $O{P}_{t-1}$ | 0.995 *** | 744.150 |

$OVX$ | −0.118 *** | −14.183 | $VIX$ | −0.057 *** | −10.647 |

$OV{X}_{t-1}^{}$ | 0.071 *** | 6.382 | $VI{X}_{t-1}^{+}$ | 0.213 *** | 2.962 |

$OV{X}_{t-2}^{}$ | 0.042 *** | 4.935 | $VI{X}_{t-2}^{+}$ | 0.029 *** | 5.492 |

${L}_{OVX}^{}$ | −0.973 *** | −4.668 | ${L}_{VIX}^{}$ | −1.460 *** | −3.335 |

ECM(−1) | −0.006 *** | −3.717 | ECM(−1) | −0.004 *** | −3.257 |

B.T _{Trend} | 8.037 ** | B.T _{Trend} | 10.106 *** | ||

$Ad{j}_{.}{R}^{2}$ | 0.996 | $Ad{j}_{.}{R}^{2}$ | 0.996 |

_{Trend}indicates that the bound test model has a linear trend term.

Causality Direction | F-Statistic | Causality Direction | F-Statistic | Optimal Lag |
---|---|---|---|---|

OP→OVX | 1.884 | OVX→OP | 11.637 *** | 2 |

OP→VIX | 0.113 | VIX→OP | 24.780 *** | 2 |

Dimension | BDS Statistic | ||
---|---|---|---|

OP | OVX | VIX | |

2 | 0.0157 *** (0.00) | 0.0143 *** (0.00) | 0.0112 *** (0.00) |

3 | 0.0343 *** (0.00) | 0.0258 *** (0.00) | 0.0284 *** (0.00) |

4 | 0.0484 *** (0.00) | 0.0324 *** (0.00) | 0.0408 *** (0.00) |

5 | 0.0593 *** (0.00) | 0.0353 *** (0.00) | 0.0470 *** (0.00) |

6 | 0.0649 *** (0.00) | 0.0357 *** (0.00) | 0.0484 *** (0.00) |

Parameter: Z_{t} = [1]; q = 1; p = 1; M = 5; h = 15 | ||
---|---|---|

Structural Break | Break | |

1 | 4.221 *** | 29 December 2008 |

2 | 4.704 *** | 1 December 2010 |

3 | 4.645 *** | 4 April 2013 |

4 | 3.862 *** | 3 December 2014 |

5 | 4.010 *** | 2 September 2016 |

Adjusted R ^{2}: 0.75F-statistic: 1675.950 (0.00) |

Panel A | Panel B | ||||
---|---|---|---|---|---|

OVX $\stackrel{\mathit{N}\mathit{A}\mathit{R}\mathit{D}\mathit{L}}{\to}$ OP | VIX $\stackrel{\mathit{N}\mathit{A}\mathit{R}\mathit{D}\mathit{L}}{\to}$ OP | ||||

Variable | Coefficient | t-stat | Variable | Coefficient | t-stat |

Constant | 0.031 *** | 3.809 | Constant | 0.025 *** | 3.611 |

OP_{t−1} | 0.994 *** | 596.254 | OP_{t−1} | 0.997 *** | 731.262 |

$OV{X}^{+}$ | −0.126 *** | −9.646 | $VI{X}^{+}$ | −0.071 *** | −8.451 |

$OV{X}^{+}$ | 0.067 *** | 3.552 | $VI{X}_{t-1}^{+}$ | 0.028 *** | 2.315 |

$OV{X}_{t-2}^{+}$ | 0.052 *** | 4.126 | $VI{X}_{t-2}^{+}$ | 0.038 *** | 4.660 |

$OV{X}^{-}$ | −0.107 *** | −6.550 | $VI{X}^{-}$ | −0.036 *** | −3.237 |

$OV{X}_{t-1}^{-}$ | 0.101 *** | 6.269 | $VI{X}_{t-1}^{-}$ | 0.031 *** | 2.885 |

${L}_{OVX}^{+}$ | −0.919 *** | −4.340 | ${L}_{VIX}^{+}$ | −1.147 *** | −2.847 |

${L}_{OVX}^{-}$ | −0.900 *** | −4.281 | ${L}_{VIX}^{-}$ | −1.125 *** | −2.827 |

B.T _{Level} | 5.61 * | B.T _{Level} | 4.27 * | ||

ECM(−1) | −0.006 *** | −3.577 | ECM(−1) | −0.004 *** | −3.053 |

${W}_{LR}$ | 5.281 *** | [0.00] | ${W}_{LR}$ | 4.678 ** | [0.00] |

${W}_{SR}$ | 28.111 *** | [0.00] | ${W}_{SR}$ | 8.082 *** | [0.00] |

$Ad{j}_{.}{R}^{2}$ | 0.996 | $Ad{j}_{.}{R}^{2}$ | 0.996 |

lx = ly | H_{0}: OP Does Not Cause VIX | H_{0}: OVX Does Not Cause OP | H_{0}: OP Does Not Cause VIX | H_{0}: VIX Does Not Cause OP |
---|---|---|---|---|

t-value | t-value | t-value | t-value | |

1 | 2.642 *** | 2.450 *** | 1.707 ** | 2.376 *** |

2 | 3.700 *** | 4.092 *** | 2.365 *** | 2.212 *** |

3 | 3.212 *** | 1.708 ** | 1.792 *** | 1.090 |

Linear Granger Causality | Results | Nonlinear Granger Causality | Results |
---|---|---|---|

H0: OP does not cause OVX | ✕ | H0: OP does not cause OVX | ✓ |

H0: OVX does not cause OP | ✓ | H0: OVX does not cause OP | ✓ |

H0: OP does not cause VIX | ✕ | H0: OP does not cause VIX | ✓ |

H0: VIX does not cause OP | ✓ | H0: VIX does not cause OP | ✓ |

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**MDPI and ACS Style**

Lin, J.-B.; Liang, C.-C.; Tsai, W. Nonlinear Relationships between Oil Prices and Implied Volatilities: Providing More Valuable Information. *Sustainability* **2019**, *11*, 3906.
https://doi.org/10.3390/su11143906

**AMA Style**

Lin J-B, Liang C-C, Tsai W. Nonlinear Relationships between Oil Prices and Implied Volatilities: Providing More Valuable Information. *Sustainability*. 2019; 11(14):3906.
https://doi.org/10.3390/su11143906

**Chicago/Turabian Style**

Lin, Jeng-Bau, Chin-Chia Liang, and Wei Tsai. 2019. "Nonlinear Relationships between Oil Prices and Implied Volatilities: Providing More Valuable Information" *Sustainability* 11, no. 14: 3906.
https://doi.org/10.3390/su11143906