# A Non-Parametric and Entropy Based Analysis of the Relationship between the VIX and S&P 500

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## Abstract

**:**

## 1. Introduction

## 2. Research Methods and Data

**Table 1.**Summary statistics, S&P 500, VIX, Daily S&P 500 returns and daily VIX returns, Jan 1990 June 2011.

S&P 500 | S&P 500 Daily Returns | VIX | VIX Daily Returns | |
---|---|---|---|---|

Min | 295.5 | −0.0947 | 9.31 | −0.3506 |

Median | 1047 | 0.0005 | 18.88 | −0.0031 |

Mean | 939 | 0.00024 | 20.34 | 0.00043 |

Maximum | 1560 | 0.1096 | 80.86 | 0.496 |

Variance | 0.0001356015 | 0.0035794839 |

**Figure 1.**Time series behaviour of the S&P 500 and VIX series from Jan 1990 to June 2011. (

**a**) S&P-500; (

**b**) Vix.

**Figure 2.**Time series behaviour of (

**a**) S&P 500 and (

**b**) VIX logarithmic return series from Jan 1990 to June 2011.

**Table 2.**OLS regression of daily continuously compounded S&P 500 returns on daily continuously compounded VIX returns.

Intercept | Slope | ||
---|---|---|---|

Coefficient | 0.000238 | −0.13527 | |

Standard Error | 0.0001138 | 0.001902 | |

t value | 2.094 * | −71.006 ** | |

Adjusted ${R}^{2}$ | 0.4829 | ||

F Value | 5056 ** |

**Figure 3.**OLS regression of daily continuously compounded S&P 500 returns on daily continuously compounded VIX returns with fitted line.

**Table 3.**Cramer–von Mises tests of the normality of the return series for the S&P 500 and the VIX for the period 1990 to June 2011.

S&P 500 Returns | VIX Returns | |
---|---|---|

Cramer-von Mise statistic | 15.0744 | 5.6701 |

Probability value | 0.00000 | 0.00000 |

#### 2.1. Entropy-Based Measures

#### 2.1.1. Maximum Likelihood Estimation

#### 2.1.2. Miller–Madow Estimator

#### 2.1.3. Bayesian Estimators

#### 2.1.4. The Chao–Shen Estimator

#### 2.1.5. Mutual Information

#### 2.2. Some Preliminary Results

**Table 4.**Common choices for the parameters of the Dirichlet prior in the Bayesian estimators of cell frequencies, and corresponding entropy estimators.

${a}_{k}$ | Cell Frequency Prior | Entropy Estimator |
---|---|---|

0 | no prior | maximum likelihood |

1/2 | Jeffreys prior ([38]) | [39] |

1 | Bayes–Laplace uniform prior | [40] |

1/p | Perks prior ([41]) | [42] |

$\sqrt{n}/p$ | minmax prior ([43]) |

S&P500 returns | VIX returns | |
---|---|---|

Maximum Likelihood Estimate | 2.1434 | 3.8269 |

Miller–Madow Estimator | 2.1462 | 3.8367 |

Jeffrey’s prior | 2.1598 | 3.8846 |

Bayes–Laplace | 2.1747 | 3.9322 |

SG | 2.1444 | 3.8279 |

Minimax | 2.1983 | 3.8780 |

Chao–Shen | 2.1494 | 3.8424 |

Mutual Information $MI$ | ||

$MI$ Dirichlet ( a = 0) | 0.3535395 | |

$MI$ Dirichlet ( a = 1/2) | 0.3465671 | |

$MI$ Empirical ($ML)$ | 0.3535395 |

#### 2.3. Non-Parametric Estimation

#### 2.3.1. Kernel Estimation of a Conditional Quantile

**Figure 5.**Nonparametric conditional PDF and CDF estimates of the joint distribution of S&P500 returns and VIX returns 1990-June 2011. (

**a**) PDF; (

**b**) CDF.

#### 2.3.2. Testing the Equality of Univariate Densities

## 3. Main Results

#### 3.1. Entropy Metrics

S&P500 2005–2006 | S&P500 2007–2008 | S&P500 2009–2010 | S&P500 2011–2012 | |
---|---|---|---|---|

Maximum Likelihood Estimate | 1.6518 | 2.4165 | 2.2611 | 2.1859 |

Miller–Madow Estimator | 1.6596 | 2.4454 | 2.2790 | 2.2211 |

Jeffrey’s prior | 1.6610 | 2.5364 | 2.2956 | 2.2516 |

Bayes–Laplace | 1.6698 | 2.6299 | 2.3262 | 2.3051 |

SG | 1.6542 | 2.4243 | 2.2649 | 2.1935 |

Minimax | 1.6967 | 2.545 | 2.331 | 2.2920 |

Chao–Shen | 1.6527 | 2.4781 | 2.2809 | 2.2282 |

Mutual Information $MI$ S&P500 and VIX | ||||

$MI$ Dirichlet ( a = 0) | 0.09489 | 0.1941 | 0.1927 | 0.2055 |

$MI$ Dirichlet ( a = 1/2) | 0.08485 | 0.1693 | 0.1702 | 0.1746 |

$MI$ Empirical ($ML)$ | 0.09489 | 0.1941 | 0.1927 | 0.2055 |

VIX 2005–2006 | VIX 2007–2008 | VIX 2009–2010 | VIX 2011–2012 | |
---|---|---|---|---|

Maximum Likelihood Estimate | 3.6441 | 4.0114 | 3.7744 | 3.8723 |

Miller–Madow Estimator | 3.7045 | 4.0911 | 3.8421 | 3.9669 |

Jeffrey’s prior | 3.9220 | 5.3321 | 4.0743 | 4.2843 |

Bayes–Laplace | 4.0811 | 5.7334 | 4.2444 | 4.4751 |

SG | 3.6541 | 4.0278 | 3.7835 | 3.8861 |

Minimax | 3.7787 | 4.2524 | 3.9054 | 4.0306 |

Chao–Shen | 3.7086 | 4.1006 | 3.8539 | 3.9573 |

**Figure 6.**Conditional Density Plots PDF and CDF for S&P500 and VIX returns 2005–2006 for our sample intervals. (

**a**) PDF 2005–2006; (

**b**) CDF 2005–2006.

#### 3.2. Non-Parametric Conditional PDF and CDF Estimation

**Figure 7.**Conditional Density Plots PDF and CDF for S&P500 and VIX returns 2005-2006 for our sample intervals. (

**a**) PDF 2007–2008; (

**b**) CDF 2007–2008.

**Figure 8.**Conditional Density Plots PDF and CDF for S&P500 and VIX returns 2009-2010 for our sample interval. (

**a**) PDF 2009–2010; (

**b**) CDF 2009–2010.

**Figure 9.**Conditional Density Plots PDF and CDF for S&P500 and VIX returns 2011-2012 for our sample interval. (

**a**) PDF 2011–2012; (

**b**) CDF 2011–2012.

#### 3.3. Quantile Regression Analysis of Hedge Ratios

**Figure 10.**Quantile regression slope coefficients of S&P 500 daily returns for sample sub-periods regressed in daily continuously compounded VIX returns, sub-samples 2005–2006 and 2007–2008. (

**a**) Quantile regression slope coefficients by decile 2005–2006 with error bands; (

**b**) Quantile regression slope coefficients by decile 2007–2008 with error bands.

**Figure 11.**Quantile regression slope coefficients of S&P500 daily returns for sample sub-periods regressed in daily continuously compounded VIX returns, sub-samples 2009–2010 and 2011–2012. (

**a**) Quantile regression slope coefficients by decile 2009–2010 with error bands; (

**b**) Quantile regression slope coefficients by decile 2011–2012 with error bands.

#### 3.4. Non-Parametric Tests of Density Equalities

**Table 8.**Entropy density equality tests for sub-samples 2005–2006, 2007–2008, 2009–2010, and 2011–2012 for daily continuously compounded S&P 500 returns and VIX returns.

2005–2006 | 2007–2008 | 2009–2010 | 2011–2012 | |
---|---|---|---|---|

Consistent Univariate Entropy Density Equality Test | ||||

Test Statistic ‘Srho’ | 0.4660 | 0.3017 | 0.3219 | 0.3817 |

Probability | 2.22e-16 *** | 2.22e-16 *** | 2.22e-16 *** | 2.22e-16 *** |

**Table 9.**Entropy density asymmetry tests sub-samples 2005–2006, 2007–2008, 2009–2010, and 2011–2012 for daily continuously compounded S&P 500 returns and VIX returns.

2005–2006 | 2007–2008 | 2009–2010 | 2011–2012 | |
---|---|---|---|---|

Consistent entropy asymmetry test | ||||

S&P 500 Test Statistic ‘Srho’ | 0.00419 | 0.034547 | 0.01239 | 0.01904 |

Probability | 0.52525 | 0.0 *** | 0.19191 | 0.09090 |

VIX Test Statistic ‘Srho’ | 0.00923 | 0.01278 | 0.02485 | 0.03018 |

Probability | 0.1818 | 0.15151 | 0.0 *** | 0.0 *** |

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Allen, D.E.; McAleer, M.; Powell, R.; Singh, A.K.
A Non-Parametric and Entropy Based Analysis of the Relationship between the VIX and S&P 500. *J. Risk Financial Manag.* **2013**, *6*, 6-30.
https://doi.org/10.3390/jrfm6010006

**AMA Style**

Allen DE, McAleer M, Powell R, Singh AK.
A Non-Parametric and Entropy Based Analysis of the Relationship between the VIX and S&P 500. *Journal of Risk and Financial Management*. 2013; 6(1):6-30.
https://doi.org/10.3390/jrfm6010006

**Chicago/Turabian Style**

Allen, David E., Michael McAleer, Robert Powell, and Abhay K. Singh.
2013. "A Non-Parametric and Entropy Based Analysis of the Relationship between the VIX and S&P 500" *Journal of Risk and Financial Management* 6, no. 1: 6-30.
https://doi.org/10.3390/jrfm6010006