# Connecting VIX and Stock Index ETF with VAR and Diagonal BEKK

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## Abstract

**:**

## 1. Introduction

## 2. VIX Literature Review

#### 2.1. Conditional Volatility

#### 2.2. Implied Volatility

## 3. Data and Variables

^{®}UCITS ETF, which tracks DAX, which is used to measure the performance of the German stock market and comprises 30 blue-chip stocks traded on the Frankfurt Stock Exchange. In addition to using the weighted market value, consideration is given to the stock index’s constituent parts to determine expected dividends. The DBXD portfolio consists of 30 blue-chip stocks and is traded in Euro. From the listing of DBXD in 2007 to the first quarter of 2015, the rate of return has been 71.68%. Differences between the rates of return for DBXD and DAX over the same period are quite small.

## 4. Empirical Results

#### 4.1. ETFs in the European Market

#### 4.2. ETFs in the US Market

#### 4.3. Testing and Correcting for Conditional Heteroskedasticity

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. VAR(p) Models

## Appendix B. Diagonal BEKK Model

## References

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1 | Using the S&P 500 index excess returns obtained after subtracting adjusted dividends from one-month Treasury bills. |

Endogenous Variables | |||

Market | ETFs | Benchmark Index | Sample Time |

US | SPY | S&P 500 | 29 January 1993~30 March 2015 |

DIA | DJIA | 20 January 1998~30 March 2015 | |

ONEQ | NASDAQ | 1 October 2003~30 March 2015 | |

Europe | FEZ | EURO STOXX50 | 21 October 2002~30 March 2015 |

DBXD | German DAX | 17 January 2007~30 March 2015 | |

XUKX | FTSE100 | 6 September~30 March 2015 | |

Exogenous variables | |||

VIX | 2 January 1990~30 March 2015 | ||

S&P 500 | 29 January 1993~30 March 2015 |

**Data source:**ETF Database, https://beta.finance.yahoo.com/ accessed 1 October 2015.

Endogenous Variables | ||||||

USA | Europe | |||||

ETF | R_SPY | R_DIA | R_ONEQ | R_FEZ | R_DBXD | R_XUKX |

Mean | 0.034 | 0.025 | 0.036 | −0.005 | 0.023 | 0.013 |

Median | 0.038 | 0.033 | 0.082 | 0.000 | 0.036 | 0.000 |

Maximum | 13.558 | 12.710 | 10.022 | 16.169 | 11.148 | 13.840 |

Minimum | −10.364 | −9.874 | −8.181 | −12.142 | −8.038 | −21.575 |

Std. Dev. | 1.175 | 1.294 | 1.409 | 2.052 | 1.473 | 1.521 |

Skewness | −0.107 | 0.386 | −0.200 | −0.008 | −0.098 | −2.221 |

Kurtosis | 13.585 | 17.709 | 8.336 | 9.355 | 8.603 | 47.260 |

Exogenous variables | ||||||

R_VIX | R_S&P500 | |||||

Mean | −0.025 | 0.017 | ||||

Median | −0.323 | 0.036 | ||||

Maximum | 40.547 | 10.957 | ||||

Minimum | −35.059 | −9.470 | ||||

Std. Dev. | 7.028 | 1.403 | ||||

Skewness | 0.595 | −0.305 | ||||

Kurtosis | 6.198 | 12.635 |

ADF Test | ||||

Market | Variables | Trend and Intercept | Intercept Only | No Trend or Intercept |

US | R_SPY | −58.106 *** | −58.106 *** | −58.029 *** |

R_DIA | −72.456 *** | −72.462 *** | −72.430 *** | |

R_ONEQ | −57.239 *** | −57.233 *** | −57.195 *** | |

Europe | R_FEZ | −63.167 *** | −63.168 *** | −63.161 *** |

R_DBXD | −45.995 *** | −45.968 *** | −45.963 *** | |

R_XUKX | −54.310 *** | −54.302 *** | −54.311 *** | |

R_VIX | −7.659 *** | −7.636 *** | −2.847 *** | |

R_SP500 | −127.464 *** | −127.467 *** | −127.362 *** | |

PP Test | ||||

Market | Variables | Trend and Intercept | Intercept Only | No Trend or Intercept |

US | R_SPY | −82.507 ** | −82.505 ** | −82.211 ** |

R_DIA | −73.122 ** | −73.124 ** | −73.002 ** | |

R_ONEQ | −57.418 ** | −57.403 ** | −57.355 ** | |

Europe | R_FEZ | −63.660 ** | −63.655 ** | −63.576 ** |

R_DBXD | −46.041 ** | −46.008 ** | −46.000 ** | |

R_XUKX | −55.659 ** | −55.528 ** | −55.526 ** | |

R_VIX | −6.001 ** | −5.984 ** | −2.005 * | |

R_SP500 | −127.477 *** | −127.480 *** | −127.353 *** |

**Note:*****, ** and * denote significance at 1%, 5%, and 10%, respectively.

Variables | R_FEZ | R_DBXD | R_XUKX |
---|---|---|---|

R_FEZ(−1) | −0.083 * | 0.117 ** | 0.173 ** |

(0.039) | (0.027) | (0.027) | |

R_DBXD(−1) | 0.097 | −0.068 | 0.018 |

(0.056) | (0.040) | (0.039) | |

R_XUKX(−1) | −0.035 | −0.132 ** | −0.392 ** |

(0.045) | (0.032) | (0.031) | |

C | −0.006 | 0.025 | 0.017 |

(0.046) | (0.032) | (0.032) | |

R_VIX(−1) | 0.020 | −0.021 ** | −0.026 ** |

(0.009) | (0.007) | (0.006) | |

R10_VIX(−1) | 0.049 | −0.016 | −0.054 * |

(0.037) | (0.026) | (0.026) | |

R20_VIX(−1) | −0.038 | −0.049 | −0.006 |

(0.055) | (0.039) | (0.039) |

**Note:**** and * denote significance at 1% and 5%, respectively. Standard errors are in parentheses.

Variables | R_FEZ | R_DBXD | R_XUKX |
---|---|---|---|

R_FEZ(−1) | −0.047 | −0.038 | −0.064 |

(0.052) | (0.037) | (0.035) | |

R_DBXD(−1) | 0.104 | −0.050 | 0.042 |

(0.056) | (0.039) | (0.038) | |

R_XUKX(−1) | −0.031 | −0.145 ** | −0.415 ** |

(0.045) | (0.032) | (0.031) | |

C | −0.001 | 0.020 | 0.007 |

(0.046) | (0.032) | (0.031) | |

R_SP500(−1) | −0.137 ** | 0.346 ** | 0.502 ** |

(0.069) | (0.049) | (0.047) | |

R10_SP500(−1) | −0.332 | −0.006 | 0.228 |

(0.176) | (0.124) | (0.120) | |

R20_SP500(−1) | 0.135 | 0.071 | −0.145 |

(0.246) | (0.172) | (0.167) |

**Note:**** denote significance at 1%. Standard errors are in parentheses.

Variables | R_DIA | R_ONEQ |
---|---|---|

R_DIA(−1) | −0.139 ** | 0.035 |

(0.039) | (0.044) | |

R_ONEQ(−1) | 0.052 | −0.087 * |

(0.034) | (0.039) | |

C | 0.033 | 0.038 |

(0.020) | (0.023) | |

R_VIX(−1) | 0.002 | −0.004 |

(0.005) | (0.005) | |

R10_VIX(−1) | 0.027 | 0.019 |

(0.018) | (0.020) | |

R20_VIX(−1) | −0.019 | −0.027 |

(0.026) | (0.030) |

**Note:**** and * denote significance at 1% and 5%, respectively. Standard errors are in parentheses.

Europe | ${\widehat{\mathsf{\epsilon}}}_{R\_\mathrm{FEZt}}$ | ${\widehat{\mathsf{\epsilon}}}_{R\_DBXDt}$ | ${\widehat{\mathsf{\epsilon}}}_{R\_XUKXt}$ |

LM statistic | 38.819 * | 28.254 * | 321.64 * |

(0) | (0) | (0) | |

US | ${\widehat{\mathsf{\epsilon}}}_{R\_DIAt}$ | ${\widehat{\mathsf{\epsilon}}}_{R\_ONEQt}$ | |

LM statistic | 97.495 * | 129.692 * | |

(0) | (0) |

**Note:*** denotes significance at the 1% level. p-values are in parentheses.

**Table 8.**Mean Equation for European Market ETF and VIX (whole period) (10 September 2007 to 30 March 2015).

Variables | R_FEZ | R_DBXD | R_XUKX |
---|---|---|---|

R_FEZ(−1) | −0.218 *** | −0.005 | −0.038 ** |

(0.031) | (0.023) | (0.019) | |

R_DBXD(−1) | 0.183 *** | 0.010 | −0.045 * |

(0.038) | (0.026) | (0.021) | |

R_XUKX(−1) | 0.019 | −0.085 ** | −0.080 ** |

(0.037) | (0.021) | (0.020) | |

CONSTANT | 0.063 *** | 0.087 *** | 0.083 *** |

(0.024) | (0.024) | (0.019) | |

R_VIX(−1) | 0.000 | −0.028 *** | −0.037 *** |

(0.006) | (0.004) | (0.004) | |

R10_VIX(−1) | −0.003 | −0.017 | −0.032 ** |

(0.024) | (0.019) | (0.015) | |

R20_VIX(−1) | 0.057 | −0.039 | 0.046 * |

(0.037) | (0.031) | (0.025) |

**Note:*****, ** and * denote significance at 1%, 5% and 10%, respectively. Standard errors are in parentheses.

**Table 9.**Diagonal BEKK for Europe Market ETF returns (whole period) (10 September 2007 to 30 March 2015).

Variables | C | A | B | ||
---|---|---|---|---|---|

R_FEZ | 0.263 *** | 0.275 *** | 0.951 *** | ||

(0.023) | (0.013) | (0.004) | |||

R_DBXD | 0.173 *** | 0.176 *** | 0.292 *** | 0.940 *** | |

(0.017) | (0.010) | (0.014) | (0.004) | ||

R_XUKX | 0.246 *** | 0.139 *** | 0.000 | 0.499 *** | 0.859 *** |

(0.021) | (0.023) | (0.059) | (0.017) | (0.008) |

**Note:***** denote significance at 1%. Standard errors are in parentheses.

**Table 10.**Mean Equation for European Market ETF and VIX (Before GFC) (10 September 2007 to 30 October 2007).

Variables | R_FEZ | R_DBXD | R_XUKX |
---|---|---|---|

R_FEZ(−1) | −0.034 *** | 0.201 *** | −0.198 *** |

(0.008) | (0.010) | (0.001) | |

R_DBXD(−1) | 0.179 *** | −0.085 *** | 0.454 *** |

(0.032) | (0.008) | (0.003) | |

R_XUKX(−1) | 0.072 *** | −0.106 *** | −0.365 *** |

(0.019) | (0.003) | (0.002) | |

CONSTANT | 0.784 *** | 0.353 *** | 0.804 *** |

(0.018) | (0.009) | (0.002) | |

R_VIX(−1) | 0.028 *** | −0.010 *** | −0.033 *** |

(0.003) | (0.001) | (0.000) | |

R10_VIX(−1) | −0.018 | −0.022 ** | −0.012 *** |

(0.021) | (0.007) | (0.003) | |

R20_VIX(−1) | 0.523 *** | 0.236 *** | 0.502 *** |

(0.021) | (0.010) | (0.008) |

**Note:***** and ** denote significance at 1% and 5%, respectively. Standard errors are in parentheses.

**Table 11.**Diagonal BEKK for Europe Market ETF returns (Before GFC) (10 September 2007 to 30 October 2007).

Variables | C | A | B | ||
---|---|---|---|---|---|

R_FEZ | 1.022 *** | −0.311 *** | −0.000 | ||

(0.024) | (0.013) | (0.000) | |||

R_DBXD | 0.418 *** | 0.383 *** | −0.308 *** | −0.000 | |

(0.024) | (0.019) | (0.022) | (0.000) | ||

R_XUKX | 0.964 *** | 0.491 *** | −0.000 | 0.511 *** | 0.000 |

(0.033) | (0.025) | (0.00) | (0.023) | (0.000) |

**Note:***** denotes significance at 1%. Standard errors are in parentheses.

**Table 12.**Mean Equation for European Market ETF and VIX (During GFC) (1 November 2007 to 31 March 2009).

Variables | R_FEZ | R_DBXD | R_XUKX |
---|---|---|---|

R_FEZ(−1) | −0.266 ** | −0.078 | 0.030 |

(0.083) | (0.058) | (0.059) | |

R_DBXD(−1) | 0.280 ** | 0.053 | 0.081 |

(0.106) | (0.080) | (0.072) | |

R_XUKX(−1) | −0.058 | −0.152 * | −0.254 ** |

(0.085) | (0.063) | (0.070) | |

CONSTANT | −0.193 * | –0.133 * | −0.110 * |

(0.078) | (0.058) | (0.060) | |

R_VIX(−1) | 0.011 | −0.021 | −0.047 ** |

(0.020) | (0.016) | (0.017) | |

R10_VIX(−1) | −0.090 | −0.088 | −0.092 |

(0.083) | (0.074) | (0.071) | |

R20_VIX(−1) | 0.065 | −0.035 | 0.003 |

(0.111) | (0.100) | (0.094) |

**Note:**** and * denote significance at 5% and 10%, respectively. Standard errors are in parentheses.

**Table 13.**Diagonal BEKK for Europe Market ETF returns (During GFC) (1 November 2007 to 31 March 2009).

Variables | C | A | B | ||
---|---|---|---|---|---|

R_FEZ | 0.427 ** | 0.383 ** | 0.918 ** | ||

(0.024) | (0.013) | (0.000) | |||

R_DBXD | 0.402 ** | 0.295 ** | 0.278 ** | 0.928 ** | |

(0.024) | (0.019) | (0.022) | (0.000) | ||

R_XUKX | 0.576 ** | 0.371 ** | −0.000 | 0.485 ** | 0.837 ** |

(0.033) | (0.025) | (0.00) | (0.023) | (0.000) |

**Note:**** denote significance at 5%. Standard errors are in parentheses.

**Table 14.**Mean Equation for European Market ETF and VIX (After GFC) (1 April 2009 to 30 March 2015).

Variables | R_FEZ | R_DBXD | R_XUKX |
---|---|---|---|

R_FEZ(−1) | −0.184 *** | −0.004 | −0.037 * |

(0.035) | (0.026) | (0.020) | |

R_DBXD(−1) | 0.153 *** | –0.001 | –0.057 ** |

(0.047) | (0.035) | (0.030) | |

R_XUKX(–1) | 0.049 | –0.073 ** | –0.054 * |

(0.045) | (0.031) | (0.028) | |

CONSTANT | 0.089 ** | 0.102 *** | 0.087 *** |

(0.032) | (0.023) | (0.018) | |

R_VIX(−1) | −0.000 | −0.030 *** | −0.038 *** |

(0.006) | (0.005) | (0.004) | |

R10_VIX(−1) | 0.009 | −0.010 | −0.026 ** |

(0.020) | (0.015) | (0.012) | |

R20_VIX(−1) | 0.057 | −0.036 | 0.050 ** |

(0.033) | (0.024) | (0.020) |

**Note:*****, ** and * denote significance at 1%, 5% and 10%, respectively. Standard errors are in parentheses.

Variables | C | A | B | ||
---|---|---|---|---|---|

R_FEZ | 0.271 *** | 0.243 *** | 0.954 *** | ||

(0.023) | (0.013) | (0.004) | |||

R_DBXD | 0.215 *** | 0.186 *** | 0.287 *** | 0.928 *** | |

(0.023) | (0.012) | (0.020) | (0.007) | ||

R_XUKX | 0.287 *** | 0.090 *** | 0.000 | 0.504 *** | 0.834 *** |

(0.022) | (0.031) | (0.048) | (0.020) | (0.013) |

**Note:***** denote significance at 1%. Standard errors are in parentheses.

Variables | R_DIA | R_ONEQ |
---|---|---|

R_DIA(−1) | 0.041 | 0.128 ** |

(0.035) | (0.043) | |

R_ONEQ(−1) | −0.044 * | −0.117 *** |

(0.024) | (0.031) | |

CONSTANT | 0.064 *** | 0.076 *** |

(0.013) | (0.016) | |

R_VIX(−1) | 0.001 | −0.006 |

(0.003) | (0.004) | |

R10_VIX(−1) | 0.010 | 0.006 |

(0.011) | (0.014) | |

R20_VIX(−1) | 0.009 | 0.010 |

(0.018) | (0.022) |

**Note:*****, ** and * denote significance at 1%, 5% and 10%, respectively. Standard errors are in parentheses.

Variables | C | A | B | |
---|---|---|---|---|

R_DIA | 0.138 *** | 0.269 *** | 0.949 *** | |

(0.010) | (0.012) | (0.005) | ||

R_ONEQ | 0.126 *** | 0.071 *** | 0.229 *** | 0.964 *** |

(0.010) | (0.008) | (0.011) | (0.003) |

**Note:***** denote significance at 1%. Standard errors are in parentheses.

Variables | R_DIA | R_ONEQ |
---|---|---|

R_DIA(−1) | 0.033 | 0.060 * |

(0.029) | (0.034) | |

R_ONEQ(−1) | −0.024 | −0.039 |

(0.029) | (0.041) | |

CONSTANT | 0.055 *** | 0.053 ** |

(0.017) | (0.022) | |

R_VIX(−1) | −0.002 | −0.006 |

(0.004) | (0.006) | |

R10_VIX(−1) | 0.022 | 0.008 |

(0.018) | (0.024) | |

R20_VIX(−1) | 0.024 | 0.038 |

(0.029) | (0.039) |

**Note:*****, ** and * denote significance at 1%, 5% and 10%, respectively. Standard errors are in parentheses.

Variables | C | A | B | |
---|---|---|---|---|

R_DIA | 0.146 *** | 0.190 *** | 0.956 *** | |

(0.016) | (0.018) | (0.007) | ||

R_ONEQ | 0.106 *** | −0.000 | 0.158 *** | 0.979 *** |

(0.012) | (0.055) | (0.014) | (0.002) |

**Note:***** denote significance at 1%. Standard errors are in parentheses.

Variables | R_DIA | R_ONEQ |
---|---|---|

R_DIA(−1) | 0.074 * | 0.256 *** |

(0.044) | (0.038) | |

R_ONEQ(−1) | −0.130 ** | −0.299 *** |

(0.052) | (0.065) | |

CONSTANT | –0.086 | 0.089 |

(0.070) | (0.084) | |

R_VIX(–1) | 0.016 | 0.014 |

(0.017) | (0.020) | |

R10_VIX(–1) | 0.023 | 0.032 |

(0.072) | (0.084) | |

R20_VIX(−1) | 0.009 | −0.089 |

(0.094) | (0.110) |

**Note:*****, ** and * denote significance at 1%, 5% and 10%, respectively. Standard errors are in parentheses.

Variables | C | A | B | |
---|---|---|---|---|

R_DIA | 0.325 *** | 0.331 *** | 0.926 *** | |

(0.059) | (0.027) | (0.012) | ||

R_ONEQ | 0.305 *** | −0.138 *** | 0.285 *** | 0.945 *** |

(0.063) | (0.035) | (0.032) | (0.009) |

**Note:***** denote significance at 1%. Standard errors are in parentheses.

Variables | R_DIA | R_ONEQ |
---|---|---|

R_DIA(−1) | 0.070 | 0.171 *** |

(0.048) | (0.058) | |

R_ONEQ(−1) | −0.043 | −0.130 *** |

(0.035) | (0.043) | |

CONSTANT | 0.079 *** | 0.100 *** |

(0.017) | (0.021) | |

R_VIX(−1) | 0.004 | −0.004 |

(0.004) | (0.005) | |

R10_VIX(−1) | 0.004 | 0.003 |

(0.013) | (0.016) | |

R20_VIX(−1) | 0.001 | 0.010 |

(0.022) | (0.027) |

**Note:***** denote significance at 1%. Standard errors are in parentheses.

Variables | C | A | B | |
---|---|---|---|---|

R_DIA | 0.169 *** | 0.282 *** | 0.936 *** | |

(0.016) | (0.017) | (0.008) | ||

R_ONEQ | 0.166 *** | 0.102 *** | 0.241 *** | 0.950 *** |

(0.018) | (0.011) | (0.015) | (0.006) |

**Note:***** denote significance at 1%. Standard errors are in parentheses.

© 2018 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**

Chang, C.-L.; Hsieh, T.-L.; McAleer, M.
Connecting VIX and Stock Index ETF with VAR and Diagonal BEKK. *J. Risk Financial Manag.* **2018**, *11*, 58.
https://doi.org/10.3390/jrfm11040058

**AMA Style**

Chang C-L, Hsieh T-L, McAleer M.
Connecting VIX and Stock Index ETF with VAR and Diagonal BEKK. *Journal of Risk and Financial Management*. 2018; 11(4):58.
https://doi.org/10.3390/jrfm11040058

**Chicago/Turabian Style**

Chang, Chia-Lin, Tai-Lin Hsieh, and Michael McAleer.
2018. "Connecting VIX and Stock Index ETF with VAR and Diagonal BEKK" *Journal of Risk and Financial Management* 11, no. 4: 58.
https://doi.org/10.3390/jrfm11040058