# Dynamic Responses of Major Equity Markets to the US Fear Index

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review of the Relevant Literature

#### 2.1. Time Series Properties

#### 2.2. Information Content of the VIX

#### 2.3. Cross Market Association between the VIX and Equities

#### 2.4. The Predictive Power of the VIX

## 3. Data

## 4. Methodology

#### Structural Vector Autoregressive Formulation

**A**is a n × n square matrix, in our case 5 × 5 because we have five endogenous variables.

**A**represents the structural model coefficients, vector

**u**comprises of structural shocks, and vector

_{t}**X**

_{t}= (VIX

_{t}, GARCAC

_{t}, GARDAX

_{t}, GARFTS

_{t}, GARSP

_{t})’ are the model variables. The GAR prefix stands for the time varying variances from a GARCH (1,1) model.

**A**represent the contemporaneous relationship among the five by one (5 × 1) elements of the vector of the model stationary endogenous variables, i.e., VIX and time varying volatility in the five indices.

**B**

_{0}is a 5 × 1 vector of intercepts.

**B**

_{i}is a 5 × 5 × s coefficient matrix of lagged endogenous variables on the right-hand-side of the Equation (1). There are 5 × 5 × s (s is the lag order) parameters to be estimated in the matrix

**B**

_{i}. We determine the lag order of the endogenous variables in SVAR be based on statistical criteria while estimating the model. The vector of white noise structural innovations (shocks) is the 5 × 1 vector

**u**

_{t}with elements that are uncorrelated with the model endogenous variables and across equations.

**A**

^{−1}produces the reduced form of the VAR, i.e.,

**G**

_{0}=

**A**

^{−1}×

**B**

_{0},

**G**i =

**A**

^{−1}×

**B**and

**e**

_{t}=

**A**

^{−1}×

**u**

_{t}. The elements of vector

**e**

_{t}, i.e., forecast errors, are a linear function of the structural innovations given by vector

**u**

_{t}.

**e**

_{t}are the forecast errors associated with the VIX, and the time varying volatility in CAC, DAX, FTS, and SP, in the reduced form of Equation (1), respectively.

**u**

_{t}, are fully recoverable from the forecast errors in the reduced form model by Equation (2).

**A**in Equation (1) are necessary.

^{2}− n)/2, imposed on the elements of matrix

**A,**where n = 5 in this study. These restrictions are sufficient to render the remaining unrestricted elements a

_{i}of matrix

**A**in Equation (3) identifiable.

**x**.

**F**equal to zero. For instance, F

_{ij}= 0 implies that the long-run accumulated impulse response of variable I to shocks to the variable j is zero.

**Ω**and

**Φ**are the vector of intercepts and the matrix of infinite structural shocks, respectively.

**Φ**in Equation (4) can be used to derive variable responses to structural shocks to other model variables. For instance, ϕ

_{ij}(0) is the instantaneous impact of a shock to innovation j on endogenous variable i and is called the impact multiplier. One-period impact of shocks to innovations j on variable i in time period t are given by ϕ

_{ij}(t). Furthermore, by performing innovation accounting, one can examine the forecast error variance or variance decomposition. If shocks to a structural innovation explain none of the forecast error variance of endogenous variable x

_{j}, then the series x

_{j}is unrelated with the remaining endogenous variables of the model.

_{t}represents percentage changes in each series. The lag length for each series is selected based on the Akaike (1974) criterion. The residual term (ε

_{t}) represents the index movements that are purged of linear relationships and seasonal influences. The conditional variance equation of the model GARCH (1,1) model is given by Equation (6).

^{2}

_{i,t}is the conditional variance, u

^{2}

_{i,t}

_{−1}is the lagged innovations, and σ

^{2}

_{i,t−}

_{1}is the lagged conditional volatility. The GARCH models are estimated by the Maximum likelihood method.

## 5. Empirical Results

**A**

^{−1}is lower triangular, we are able to derive the structural innovations vector

**u**and the impulse responses. The impulse response function is the time path of the volatility in the four equity markets following a positive shock to the VIX index. Impulse responses show the size of the impact of a shock as well as the rate at which the response tapers off. The point estimates and their two-standard error bands are shown by the solid and dotted lines in all cases. Figure 4 presents these results.

#### Nonlinear Causality Test Results

_{j}= (δ

_{j}

_{1}… δ

_{jq})’, j = 1, 2, ν

_{t}= (x

_{t−}

_{1}… x

_{t−q})’ and G(.) is a transition function. Non-causality is tested as H0: G$\equiv $0 & δ

_{1i}= 0, $i=1\dots q$. The approximation to the above equation is

_{1}… k

_{q}), and non-causality is supported by k

_{i}= 0, φ

_{ij}= 0 and ψ

_{i}= 0, i = 1 … q, j = 1 … q. Under H

_{0}, the resulting test statistic has an asymptotic χ

^{2}distribution with (q × (q + 1)/2) + 2q degrees of freedom.

_{i}= 0, φ

_{ij}= 0 and ψ

_{i}= 0. Therefore, at some lag levels of variable x the null may not be rejected. Skalin and Teräsvirta (1999) vary the lag order to detect possible causality between variables at varying lags.

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | Academic and the popular sources have argued that the VIX may suffer from computational flaws and perhaps even manipulated. For instance, there has been periods that VIX predictions and market movements have diverged during some time periods. As an example, for four months between 8 August 2017 and 8 November 2017, the VIX was up 19%, signaling rising fear among market participants. This would imply a downward trend in S&P 500. However, the S&P 500 was rising in an upward trajectory. Other periods of strong divergences between VIX and S&P 500 are April 2007 to October 2007 and December 2014 to February 2015. |

2 | Other implied volatility indices have been devised following the CBOE VIX, including: the VXN and VXD in the CBOE, the VDAX-NEW in Germany, the VX1 and VX6 in France, and the VSTOXX in the Eurex, among others. |

**Figure 1.**Anxiety and VIX Index. Source: www.investopedia.com/anxiety-index-explained.

**Figure 4.**Responses of volatility in equity indices to structural innovations in the Chicago Board of Trade Volatility Index (VIX). Estimates and two standard deviation confidence band are in solid and dotted lines. SVAR lag order is one. Long-run identification restrictions are imposed. 24 June 2013–12 August 2015.

**Figure 5.**Accumulated responses of volatility in equity indices to structural innovations in VIX. Estimates and two standard deviation confidence band are in solid and dotted lines. SVAR lag order is one. Long-run identification restrictions are imposed. 24 June 2013–12 August 2015.

**Figure 6.**Responses of volatility in equity indices to structural innovations in VIX. Estimates and two standard deviation confidence band are in solid and dotted lines. SVAR lag order is one. 13 August 2015−17 May 2016.

**Figure 7.**Responses of volatility in equity indices to structural innovations in VIX. Estimates and two standard deviation confidence band are in solid and dotted lines. SVAR lag order is one. 13 August 2015−17 May 2016.

**Figure 8.**Responses of volatility in equity indices to structural innovations in VIX. Estimates and two standard deviation confidence band are in solid and dotted lines. SVAR lag order is one. 18 May 2016−13 February 2017.

**Figure 9.**Responses of equity indices to innovations in the VIX from the vector error correction model. VECM lag order is one. 24 June 2013–12 August 2015.

Panel A: Bai Perron Test of Structural Breaks | |||||

Break Test | Scaled F-Statistic | Critical Value * | Dates | ||

0 vs. 1 ^{b} | 43.310 | 11.47 | 8/12/2015 | ||

1 vs. 2 ^{b} | 20.905 | 12.95 | 5/17/2016 | ||

2 vs. 3 ^{b} | 19.799 | 14.03 | 2/13/2017 | ||

3 vs. 4 | 1.427 | 14.85 | |||

ADF Unit Root Test with Structural Break | |||||

Based on Minimizing the Dickey-Fuller t-statistic 1st | −7.110 ^{a} | ||||

Based on Minimizing the Dickey-Fuller t-statistic 1st | −6.774 ^{a} | ||||

Based on Minimizing the Dickey-Fuller t-statistic 1st | −6.770 ^{a} | ||||

Panel B: Levels 6/24/2013–8/12/2015 | |||||

Tests | CAC | DAX | FTSE | S&P | VIX |

ADF | −2.826 | −2.409 | −4.216 ^{a} | −4.141 ^{a} | −5.058 ^{a} |

PP | −2.847 | −2.409 | −4.392 ^{a} | −4.137 ^{a} | −4.753 ^{a} |

KPPS | 0.284 ^{a} | 0.256 ^{a} | 0.109 | 0.390 ^{a} | 0.328 ^{a} |

ARCH-LM | 6.934 ^{a} | 8.788 ^{a} | 7.848 ^{a} | 13.345 ^{a} | |

Panel C: Levels 8/13/2015–5/17/2016 | |||||

Tests | CAC | DAX | FTSE | S&P | VIX |

ADF | −2.538 | −2.286 | −3.301 ^{c} | −2.394 | −3.052 |

PP | −2.600 | 2.354 | −3.282 ^{c} | −2.442 | −2.911 |

KPPS | 0.145 ^{b} | 0.152 ^{b} | 0.177 ^{b} | 0.165 ^{b} | 0.165 ^{b} |

ARCH-LM | 6.934 ^{a} | 8.788 ^{a} | 7.848 ^{a} | 13.345 ^{a} | |

Panel D: Levels 5/18/2016–2/13/2017 | |||||

Tests | CAC | DAX | FTSE | S&P | VIX |

ADF | −3.065 | −2.539 | −2.599 | −3.179 ^{c} | −4.024 ^{a} |

PP | −3.192^{c} | −2.616 | −2.581 | −3.166 ^{c} | −4.137 ^{a} |

KPPS | 0.103 | 0.208 ^{b} | 0.081 | 0.200 ^{b} | 0.114 |

ARCH-LM | 2.191 | 2.693^{c} | 13.955 ^{a} | 29.097 ^{a} | |

Panel E: Summary descriptive statistics for model variables. All variables are in level | |||||

Statistics | CAC | DAX | FTSE | S&P | VIX |

Mean | 64.730 | 79733.720 | 107.271 | 72.388 | |

Stand Dev | 8.231 | 27324.000 | 102.228 | 34.522 | |

Skewness | 0.281 | 0.977 | 0.992 | 0.899 | |

Kurtosis | 1.912 | 2.563 | 3.858 | 3.013 | |

J-B | 33.149 ^{a} | 87.081 ^{a} | 103.658 | 71.473 ^{a} | |

Panel F: Johansen-Juselius Cointegration Test, unrestricted VAR lag order = 6 | |||||

r = The number of cointegrating vectors among the four variables | |||||

vector | λ_{m} | p-Value | λ_{t} | p-Value | |

r = 0 | 40.271 ^{a} | 0.001 | 70.191 ^{a} | 0.000 | |

r ≤ 1 | 26.011 ^{a} | 0.009 | 29.919 ^{b} | 0.048 | |

r ≤ 2 | 3.860 | 0.873 | 3.908 | 0.910 | |

r ≤ 3 | 0.047 | 0.827 | 0.047 | 0.827 |

_{m}and λ

_{t}reject no or one cointegrating vector. Maximum eigenvalue and traces tests suggest two cointegrating vectors at 5% level.

^{a},

^{b}, and

^{c}, represent significance at 0.01, 0.05, and 0.10, respectively.

**Table 2.**Percentage of equity index volatility forecast error variations explained by VIX and Equity Indices, 6/24/2013–8/12/2015.

CAC | ||||||

Period | S.E. | VIX | CAC | DAX | FTSE | S&P 500 |

1 | 1.523907 | 23.06727 | 53.16470 | 17.57172 | 6.186519 | 0.009798 |

10 | 3.325619 | 19.33011 | 62.15958 | 15.28370 | 3.158607 | 0.068011 |

20 | 3.553793 | 16.78376 | 68.39097 | 12.85680 | 1.914194 | 0.054273 |

30 | 3.596813 | 15.23153 | 72.33904 | 10.97169 | 1.416521 | 0.041231 |

DAX | ||||||

Period | S.E. | VIX | CAC | DAX | FTSE | S&P 500 |

1 | 222.0597 | 15.51648 | 74.49980 | 0.168141 | 9.667806 | 0.147774 |

10 | 688.4700 | 12.91648 | 82.06945 | 0.217239 | 4.644559 | 0.152269 |

20 | 935.9210 | 10.97281 | 85.79171 | 0.446709 | 2.689165 | 0.099600 |

30 | 1090.002 | 9.727636 | 87.40137 | 0.853079 | 1.945311 | 0.072604 |

FTSE | ||||||

Period | S.E. | VIX | CAC | DAX | FTSE | S&P 500 |

1 | 1308.442 | 11.12727 | 40.32318 | 7.713062 | 33.81403 | 7.022454 |

10 | 4132.530 | 9.762820 | 47.22903 | 8.129161 | 32.40228 | 2.476710 |

20 | 5707.291 | 9.125213 | 49.48482 | 8.540962 | 30.72729 | 2.121724 |

30 | 6718.869 | 8.916086 | 50.41408 | 8.771329 | 29.85478 | 2.043726 |

SP | ||||||

Period | S.E. | VIX | CAC | DAX | FTSE | S&P 500 |

1 | 438.4022 | 0.698788 | 20.86801 | 0.167246 | 8.772745 | 69.49321 |

10 | 993.6533 | 2.033265 | 32.29965 | 2.183065 | 16.41988 | 47.06415 |

20 | 1107.649 | 2.374618 | 35.12433 | 3.529921 | 17.92936 | 41.04177 |

30 | 1141.226 | 2.445579 | 35.85994 | 4.117280 | 18.04220 | 39.53501 |

Factorization: Structural |

**Table 3.**Percentage of equity index volatility forecast error variations explained by VIX and Equity Indices, 13 August 2015–17 May 2016.

CAC | ||||||

Period | S.E. | VIX | CAC | DAX | FTSE | S&P 500 |

1 | 1.847290 | 5.629366 | 42.89547 | 8.235468 | 1.730171 | 41.50953 |

10 | 4.213181 | 20.57815 | 46.04785 | 5.384713 | 1.144778 | 26.84451 |

20 | 4.803299 | 22.96667 | 44.30306 | 5.285415 | 1.125518 | 26.31934 |

30 | 5.022063 | 23.30053 | 44.05778 | 5.270821 | 1.125361 | 26.24551 |

DAX | S.E. | VIX | CAC | DAX | FTSE | S&P 500 |

1 | 1031.178 | 8.503843 | 40.11901 | 3.069205 | 0.485771 | 47.82217 |

10 | 1540.439 | 17.22094 | 45.11183 | 3.074743 | 0.384313 | 34.20818 |

20 | 1595.023 | 17.90871 | 44.26796 | 3.222857 | 0.390217 | 34.21025 |

30 | 1602.747 | 17.95395 | 44.20409 | 3.234434 | 0.390874 | 34.21664 |

FTSE | S.E. | VIX | CAC | DAX | FTSE | S&P 500 |

1 | 4240.179 | 6.352564 | 17.10188 | 17.15250 | 9.062495 | 50.33057 |

10 | 5679.630 | 31.55825 | 32.41167 | 7.387517 | 7.149734 | 21.49282 |

20 | 5779.511 | 38.40037 | 32.18627 | 5.762504 | 6.048016 | 17.60284 |

30 | 5786.676 | 39.48003 | 31.75040 | 5.582793 | 5.804940 | 17.38184 |

S&P 500 | S.E. | VIX | CAC | DAX | FTSE | S&P 500 |

1 | 1212.531 | 27.93073 | 44.85573 | 5.416856 | 20.34589 | 1.450803 |

10 | 2840.446 | 38.13473 | 48.99117 | 1.206103 | 9.250807 | 2.417189 |

20 | 3281.835 | 39.88873 | 45.88734 | 1.480075 | 7.741952 | 5.001906 |

30 | 3377.665 | 40.11354 | 45.20379 | 1.576134 | 7.521768 | 5.584767 |

Factorization: Structural |

**Table 4.**Nonlinear Granger causality test: p-values of F statistics for the Ho of no nonlinear Granger causality.

Panel A: 24 June 2013–12 August 2015 | |||||

Causing Variable | Caused Variables | ||||

Lags | VIX | CAC | DAX | FTSE | S&P 500 |

5 | 0.307 | 0.178 | 0.880 | 0.998 | |

6 | 0.002 | 0.012 | 0.361 | 0.086 | |

7 | 0.016 | 0.062 | 0.260 | 0.097 | |

8 | 0.027 | 0.083 | 0.124 | 0.958 | |

9 | 0.039 | 0.037 | 0.001 | 0.797 | |

10 | 0.033 | 0.016 | 0.004 | 0.901 | |

Panel B: 13 August 2015–17 May 2016 | |||||

CAC | DAX | FTSE | S&P 500 | ||

5 | 0.754 | 0.968 | 0.749 | 0.662 | |

6 | 0.618 | 0.882 | 0.007 | 0.048 | |

7 | 0.041 | 0.028 | 0.077 | 0.065 | |

8 | 0.074 | 0.094 | 0.073 | 0.003 | |

9 | 0.073 | 0.857 | 0.690 | 0.004 | |

10 | 0.706 | 0.707 | 0.474 | 0.536 | |

Panel C: 18 May–13 February 2017 | |||||

CAC | DAX | FTSE | S&P 500 | ||

5 | 0.996 | 0.999 | 0.701 | 0.101 | |

6 | 0.974 | 0.868 | 0.838 | 0.169 | |

7 | 0.984 | 0.943 | 0.806 | 0.317 | |

8 | 0.838 | 0.822 | 0.772 | 0.634 | |

9 | 0.561 | 0.647 | 0.608 | 0.499 | |

10 | 0.194 | 0.512 | 0.385 | 0.536 |

_{i}= 0, φ

_{ij}= 0 and ψ

_{i}= 0. Therefore, at some lag levels of variable x the null may not be rejected. For instance, the computed p-values for the VIX causing FTSE for the 2015−2016 interval show that the former causes the latter for lags of six to eight days. The degrees of freedom in the numerator and the denominator of the F-test of causality are q × (q + 1)/2 + 2q and T − n − q × (q + 1)/2 − 2q, respectively, where q is the number of lags, n is the dimension of the gradient vector, and T is the number of observations. Degrees of freedom in the numerator of the F statistics are 25, 32, 42, 52, 63, and 75 for q = 5 through 10 respectively.

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## Share and Cite

**MDPI and ACS Style**

Adrangi, B.; Chatrath, A.; Macri, J.; Raffiee, K. Dynamic Responses of Major Equity Markets to the US Fear Index. *J. Risk Financial Manag.* **2019**, *12*, 156.
https://doi.org/10.3390/jrfm12040156

**AMA Style**

Adrangi B, Chatrath A, Macri J, Raffiee K. Dynamic Responses of Major Equity Markets to the US Fear Index. *Journal of Risk and Financial Management*. 2019; 12(4):156.
https://doi.org/10.3390/jrfm12040156

**Chicago/Turabian Style**

Adrangi, Bahram, Arjun Chatrath, Joseph Macri, and Kambiz Raffiee. 2019. "Dynamic Responses of Major Equity Markets to the US Fear Index" *Journal of Risk and Financial Management* 12, no. 4: 156.
https://doi.org/10.3390/jrfm12040156