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

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## 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

- Arik, Ali. 2011. Modeling Market Sentiment and Conditional Distribution of Stock Index Returns under GARCH Process. Ph.D. dissertation, Department of Economics, Claremont Graduate University, Claremont, CA, USA. [Google Scholar]
- Baba, Yoshi, Robert F. Engle, Dennis Kraft, and Kenneth F. Kroner. 1985. Multivariate Simultaneous Generalized ARCH, Unpublished manuscript. San Diego, CA, USA: Department of Economics, University of California, [published as Engle and Kroner 1995].
- Black, F. 1976. Studies of Stock Market Volatility Changes. Paper presented at the American Statistical Association, Business and Economic Statistics Section, Washington, DC, USA, August; pp. 177–81. [Google Scholar]
- Bollerslev, Tim. 1986. Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics 31: 307–27. [Google Scholar] [CrossRef]
- Bollerslev, Tim. 1990. Modelling the Coherence in Short-run Nominal Exchange Rate: A Multivariate Generalized ARCH Approach. Review of Economics and Statistics 72: 498–505. [Google Scholar] [CrossRef]
- Borovkova, Svetlana, and Ferry J. Permana. 2009. Implied Volatility in Oil Markets. Computational Statistics and Data Analysis 53: 2022–39. [Google Scholar] [CrossRef]
- Boussama, F. 2000. Asymptotic Normality for the Quasi-maximum Likelihood Estimator of a GARCH Model. Comptes Rendus de l’Academie des Sciences 331: 81–84. [Google Scholar]
- Caporin, Massimiliano, and Michael McAleer. 2012. Do We Really Need both BEKK and DCC? A Tale of Two Multivariate GARCH Models. Journal of Economic Surveys 26: 736–51. [Google Scholar] [CrossRef]
- Caporin, Massimiliano, and Michael McAleer. 2013. Ten Things You Should Know About the Dynamic Conditional Correlation Representation. Econometrics 1: 115–26. [Google Scholar] [CrossRef][Green Version]
- Chang, Chia-Lin, Michael McAleer, and Guangdong Zuo. 2017. Volatility Spillovers and Causality of Carbon Emissions, Oil and Coal Spot and Futures for the EU and USA. Sustainability 9: 1789. [Google Scholar] [CrossRef]
- Chang, Chia-Lin, Yiying Li, and Michael McAleer. 2015. Volatility Spillovers between Energy and Agricultural Markets: A Critical Appraisal of Theory and Practice. Energies 11: 1–19. [Google Scholar] [CrossRef]
- Chang, Chia-Lin, and Michael McAleer. 2015. Econometric Analysis of Financial Derivatives. Journal of Econometrics 187: 403–7. [Google Scholar] [CrossRef]
- Chang, Chia-Lin, and Michael McAleer. 2018. The Fiction of Full BEKK: Pricing Fossil Fuels and Carbon Emissions. Finance Research Letters. forthcoming. [Google Scholar] [CrossRef]
- Chang, Chia-Lin, Michael McAleer, and Yu-Ann Wang. 2018. Modelling Volatility Spillovers for Bio-ethanol, Sugarcane and Corn Spot and Futures Prices. Renewable and Sustainable Energy Reviews 81: 1002–18. [Google Scholar] [CrossRef]
- Cochran, Steven J., Iqbal Mansur, and Babatunde Odusami. 2015. Equity Market Implied Volatility and Energy Prices: A Double Threshold GARCH Approach. Energy Economics 50: 264–72. [Google Scholar] [CrossRef]
- Cox, John C., Stephen A. Ross, and Mark Rubinstein. 1979. Option Pricing: A Simplified Approach. Journal of Financial Economics 7: 229–63. [Google Scholar] [CrossRef]
- Dickey, David A., and Wayne A. Fuller. 1979. Distribution of the Estimators for Autoregressive Time Series with a Unit Root. Journal of the American Statistical Association 74: 427–31. [Google Scholar]
- Dumas, Bernard, Jeff Fleming, and Robert E. Whaley. 1998. Implied Volatility Functions: Empirical Tests. Journal of Finance 53: 2059–106. [Google Scholar] [CrossRef]
- Engle, Robert F. 1982. Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica 50: 987–1007. [Google Scholar] [CrossRef]
- Engle, Robert F. 2002. Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Hereoskedasticity Models. Journal of Business and Economic Statistics 20: 339–50. [Google Scholar] [CrossRef]
- Engle, Robert F., and Kenneth F. Kroner. 1995. Multivariate Simultaneous Generalized ARCH. Econometric Theory 11: 122–50. [Google Scholar] [CrossRef]
- Fernandes, Marcelo, Marcelo C. Medeiros, and Marcel Scharth. 2014. Modeling and Predicting the CBOE Market Volatility Index. Journal of Banking & Finance 40: 1–10. [Google Scholar]
- Giot, Pierre. 2005. Relationships between Implied Volatility Indexes and Stock Index Returns. Journal of Portfolio Management 31: 92–100. [Google Scholar] [CrossRef]
- Glosten, Lawrence R., Ravi Jagannathan, and David E. Runkle. 1993. On the Relation between the Expected Value and the Volatility Nominal Excess Return on Stocks. Journal of Finance 46: 1779–801. [Google Scholar] [CrossRef]
- Hamilton, James Douglas. 1994. Time Series Analysis. Princeton: Princeton University Press. [Google Scholar]
- Kanas, Angelos. 2013. The Risk-Return Relation and VIX: Evidence from the S&P 500. Empirical Economics 44: 1291–314. [Google Scholar]
- Ling, Shiqing, and Michael McAleer. 2003. Asymptotic Theory for a Vector ARMA-GARCH Model. Econometric Theory 19: 278–308. [Google Scholar] [CrossRef]
- Lütkepohl, Helmut. 2005. New Introduction to Multiple Time Series Analysis. New York: Springer. [Google Scholar]
- McAleer, Michael. 2005. Automated Inference and Learning in Modeling Financial Volatility. Econometric Theory 21: 232–61. [Google Scholar] [CrossRef]
- McAleer, Michael. 2014. Asymmetry and Leverage in Conditional Volatility Models. Econometrics 2: 145–50. [Google Scholar] [CrossRef][Green Version]
- McAleer, Michael, Felix Chan, Suhejla Hoti, and Offer Lieberman. 2008. Generalized Autoregressive Conditional Correlation. Econometric Theory 24: 1554–83. [Google Scholar] [CrossRef]
- McAleer, Michael, and Christian M. Hafner. 2014. A One Line Derivation of EGARCH. Econometrics 2: 92–97. [Google Scholar] [CrossRef][Green Version]
- McAleer, Michael, Suhejla Hoti, and Felix Chan. 2009. Structure and Asymptotic Theory for Multivariate Asymmetric Conditional Volatility. Econometric Reviews 28: 422–40. [Google Scholar] [CrossRef]
- Nelson, Daniel B. 1991. Conditional Heteroskedasticity in Asset Returns: A New Approach. Econometrica 59: 347–70. [Google Scholar] [CrossRef]
- Phillips, Peter C. B., and Pierre Perron. 1988. Testing for a Unit Root in Time Series Regression. Biometrika 75: 335–46. [Google Scholar] [CrossRef]
- Poon, Ser-Huang, and Clive W. J. Granger. 2003. Forecasting Volatility in Financial Markets: A Review. Journal of Economic Literature 41: 478–539. [Google Scholar] [CrossRef]
- Rosenberg, Joshua V. 2000. Implied Volatility Functions: A Reprise. Leonard N. Stern School Finance Department Working Paper Series; New York: New York University, pp. 12–14. [Google Scholar]
- Sarwar, Ghulam. 2012. Is VIX an Investor Fear Gauge in BRIC Equity Markets? Journal of Multinational Financial Management 22: 55–65. [Google Scholar] [CrossRef]
- Tsay, Ruey S. 1987. Conditional Heteroscedastic Time Series Models. Journal of the American Statistical Association 82: 590–604. [Google Scholar] [CrossRef]
- Tse, Yiu K., and Albert K. C. Tsui. 2001. A Multivariate GARCH Model with Time-varying Correlations. Journal of Business and Economic Statistics 20: 351–62. [Google Scholar] [CrossRef]

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