# Dynamic Responses of Standard and Poor’s Regional Bank Index to the U.S. Fear Index, VIX

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Review of the Relevant Literature

## 3. Data

## 4. Methodology

#### 4.1. Structural Breaks and Stationarity

#### 4.2. Spectral Analysis and Co-Spectral Analysis

_{F}are known as “lag window.” In the estimation process, one increases “the bandwidth” of the estimate to derive smooth estimates of the spectrum. We will estimate the individual spectrums for various time series utilizing three different lag windows, i.e., Bartlett, Tukey, and Parzen.

_{j}= jπ/m and j = 0, 1, 2 …… m and m is the window size and ρ

_{F}is the autocorrelation coefficient of order F.

#### 4.3. Kalman Filter

_{t}in a linear state—space can be formulated by the system of equations

_{t}, SPRB in this study, is assumed to follow a first-order vector autoregression, ${\alpha}_{t}$ is an n × 1vector of unobserved state variables, ${\varphi}_{t},{X}_{t},{\gamma}_{t},{T}_{t}$, are conformable vectors and matrices, and ${u}_{t}$ and ${\upsilon}_{t}$ are vectors of Gaussian innovations with zero mean, zero autocovariance, and the contemporaneous covariance matrix:

_{t}

_{|t−1}= E

_{t}

_{−1}(α

_{t})

V

_{t}

_{|t−1}= E

_{t}

_{−1}[(α

_{t}− α

_{t}

_{|t−1}) (α

_{t}− α

_{t}

_{|t−1})’].

_{t}by

_{t}

_{|t−1}= E

_{t}

_{−1}(y

_{t}) = E(y

_{t}

_{|αt|τ}).

_{t}, the Kalman filter computes one-step ahead estimates of the state variable α

_{t}

_{|t−1}and the associated mean square error matrix, the contemporaneous or filtered state mean and variance, α

_{t}and V

_{t}, and the one-step ahead prediction, prediction error, and prediction error variance, in Equation (8) through (10), respectively.

_{t}, may also be interpreted as the MSE of the smoothed state estimate ${\widehat{\alpha}}_{t}$.

#### 4.4. Quantile Regression

_{i}(q

_{i}(τ)) is the conditional density function of the response, evaluated at the τth conditional quantile for individual i. We deploy the Powell (1984) kernel method based on residuals of the estimated model.

_{n}is a kernel bandwidth. For bandwidth specification, we employ a method suggested by Hall and Sheather (1988) and a kernel bandwidth suggested by Koenker and Ng (2005). Koenker and Machado (1999) define a pseudo R-squared for the goodness-of-fit statistic for quantile regression that is analogous to the R squared from conventional regression analysis.

## 5. Empirical Findings

#### 5.1. Structural Breaks and Stationarity

#### 5.2. Spectral and Co-Spectral Analysis

_{xy}) is between zero and one, it could be an indication of the presence of random disturbances, which are common in markets. Alternatively, it could be showing that the assumed function relating x(t) and y(t) is not linear. Another possibility may be that y(t) is dependent on x(t) as well as other input. If the coherence is equal to zero, it is an indication that x(t) and y(t) are perfectly unrelated.

#### 5.3. Kalman Filter Estimation

#### 5.4. Quantile Regression Findings

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The Investopedia Anxiety Index (IAI) and VIX. Source: https://www.investopedia.com/anxiety-index-explained/ (accessed on 9 March 2020).

**Figure 8.**Quantile Process Coefficients. Intercept is C and the remaining graphs show the coefficients of the stationary variables consumer confidence, capacity utilization, and VIX over nine quantiles.

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

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

0 vs. 1 ^{a} | 1701.59 | 8.58 | 20 January 2012 | ||

1 vs. 2 ^{a} | 68.96 | 10.13 | 10 December 2014 | ||

2 vs. 3 ^{a} | 42.26 | 11.14 | 7 July 2016 | ||

3 vs. 4 | 0.00 | 11.83 | |||

Panel B: Levels | |||||

CC | CU | SPRB | VIX | ||

ADF | −1.318 | −4.613 ^{a} | −1.211 | −6.452 ^{a} | |

PP | −1.766 | −4.414 ^{a} | −1.175 | −6.310 ^{a} | |

KPPS | 5.412 ^{a} | 2.349 ^{a} | 5.453 ^{a} | 2.517 ^{a} | |

Panel C: Hodrick-Prescott Detrended Series | |||||

ADF | −19.51 ^{a} | −17.748 ^{a} | −19.878 ^{a} | −20.411 ^{a} | |

PP | −19.036 ^{a} | −19.332 ^{a} | −52.965 ^{a} | −56.499 ^{a} | |

KPPS | 0.003 | 0.033 | 0.029 | 0.027 | |

Panel D: Summary Descriptive Statistics for Model Variables. All Variables Are in Level | |||||

Mean | 83.421 | 76.100 | 1182.885 | 17.661 | |

Stand Dev | 11.428 | 2.630 | 396.484 | 6.136 | |

Skewness | −0.204 | −1.691 | 0.514 | 1.351 | |

Kurtosis | 1.881 | 6.018 | 2.182 | 4.967 | |

J-B | 145.237 ^{a} | 2102.665 ^{a} | 176.558 ^{a} | 1143.153 ^{a} |

^{a}represent significance at 1percent level.

SPRB | C1 | CC | CU | tr_10 | VIX | LL |
---|---|---|---|---|---|---|

23.162 | 5.068 ^{a} | 16.327 ^{a} | −3.982 | −6.452 ^{a} | −9964.262 | |

(31.173) | (1.693) | (5.373) | (5.754) | (0.171) | ||

state sv1 = sv1(−1) | ||||||

state sv2 = α×sv2(−1) + β×sv3(−1) + [var = exp(c(4))] | ||||||

state sv3 = sv2(−1) |

^{a}represents significance at 1 percent level.

D(SPRB) | 1st Q | C | DCC | DCU | DVIX | Ps_R^{2} | Quasi_LR |

−15.291 ^{a} | 3.022 | 65.581 ^{a} | −6.636 ^{a} | 0.215 | 410.032 ^{a} | ||

(0.629) | (3.422) | (26.297) | (0.459) | ||||

D(SPRB) | 5th Q | ||||||

0.107 | 3.349 ^{a} | 32.492 ^{b} | −6.118 ^{a} | 0.205 | 956.007 ^{a} | ||

(0.304) | (1.325) | (15.985) | (0.294) | ||||

D(SPRB) | 9th Q | ||||||

16.508 ^{a} | 2.926 | −14.983 | −6.256 ^{a} | 0.171 | 341.955 ^{a} | ||

(0.500) | (3.507) | (30.100) | (0.168) | ||||

D(SPRB) | OLS | 0.230 | 5.158 ^{a} | 19.421 | −6.245 ^{a} | 0.350 | 440.403 ^{a} |

(0.290) | (1.501) | (13.996) | (0.172) |

^{a},

^{b}represent significance at 1, 5, percent levels, respectively. The values of R-sq and likelihood ratio functions for the OLS estimates and pseudo R-sq and quasi-likelihood ratio functions are in the last two columns. Huber sandwich standard errors and covariances. Method of Epanechnikov kernel, and Hall–Sheather residuals bandwidth (bw = 0.025648).

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**MDPI and ACS Style**

Adrangi, B.; Chatrath, A.; Kolay, M.; Raffiee, K. Dynamic Responses of Standard and Poor’s Regional Bank Index to the U.S. Fear Index, VIX. *J. Risk Financial Manag.* **2021**, *14*, 114.
https://doi.org/10.3390/jrfm14030114

**AMA Style**

Adrangi B, Chatrath A, Kolay M, Raffiee K. Dynamic Responses of Standard and Poor’s Regional Bank Index to the U.S. Fear Index, VIX. *Journal of Risk and Financial Management*. 2021; 14(3):114.
https://doi.org/10.3390/jrfm14030114

**Chicago/Turabian Style**

Adrangi, Bahram, Arjun Chatrath, Madhuparna Kolay, and Kambiz Raffiee. 2021. "Dynamic Responses of Standard and Poor’s Regional Bank Index to the U.S. Fear Index, VIX" *Journal of Risk and Financial Management* 14, no. 3: 114.
https://doi.org/10.3390/jrfm14030114