# Scalar Measures of Volatility and Dependence for the Multivariate Models with Applications to Asian Financial Markets

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## Abstract

**:**

## 1. Introduction

## 2. Generalized Variance and Collective Correlation

#### 2.1. Statistical Interpretation of Generalized Variance

#### 2.2. Statistical Properties of the Generalized Variance

**Proposition 1.**

#### 2.3. Scatter Coefficient

#### 2.4. Modifications of GVAR and CCOR

## 3. An Empirical Application

#### 3.1. Data and Descriptive Statistics

#### 3.2. Market Comovements and Volatility

#### 3.3. Instantaneous Measures of the Regional Volatility and Correlation

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The scatter plots of random numbers from the normal distributions with ellipses of 95% confidence interval. The covariance matrices ${S}_{1},{S}_{2}$, ${S}_{3}$, and ${S}_{4}$ defined in (3) were used for the graphs (

**i**), (

**ii**), (

**iii**), and (

**iv**), respectively.

**Figure 3.**The estimated generalized standard deviation (${GSD}_{t}$) and effective standard deviation (${ESD}_{t}$) of the six Asian stock returns. ${GSD}_{t}$ and ${ESD}_{t}$ were calculated by $|{H}_{t}{|}^{1/2}$ and $|{H}_{t}{|}^{1/2k}$, respectively. The conditional covariance matrices ${H}_{t}$ were estimated by DCC-GARCH.

**Figure 4.**The estimated dependence measures of the six Asian stock market returns. ${CCOR}_{t}$ and ${ECOR}_{t}$ were calculated by $\sqrt{|{R}_{t}|}$ and $\sqrt[2k]{|{R}_{t}|}$, respectively. The correlation matrices ${R}_{t}$ were estimated by DCC-GARCH.

**Figure 5.**Top graph is the total standard deviation (${TSD}_{t}={\left({TVAR}_{t}\right)}^{1/2}$) of six Asian stock returns. Total variance (${TVAR}_{t}$) was calculated by ${({h}_{1t}\cdots {h}_{6t})}^{1/6}$, where the conditional variance ${h}_{it}$$(i=1,\cdots ,6)$ was retrieved from the DCC-GARCH estimation. Bottom graph is the collective correlation ${CCOR}_{t}$, computed as $(1-|{R}_{t}{\left|\right)}^{1/2}$, where the conditional correlation matrix ${R}_{t}$ was obtained from the DCC-GARCH estimation.

**Figure 6.**The cross serial-correlation between the regional volatility and the regional dependence: $\mathrm{corr}({TSD}_{t+j},{ECOR}_{t}),j=-20,-18,\cdots ,20$.

**Table 1.**Descriptive statistics of weekly returns on Asian stock indices. The table reports descriptive statistics for six Asian stock markets, Japan (Nikkei 225), Hong Kong (HangSeng), Singapore (STI), Korea (KOSPI), Thailand (SET), and Indonesia (JSX). We used weekly (Wednesday close) returns from 16 January 1985 to 29 March 2017, yielding 1531 observations. The pre-crisis period and post-crisis period are 16 January 1985 to 24 December 1997 and 14 January 1998 to 29 March 2017, respectively, which gives 613 and 918 observations, respectively.

Japan | Hong Kong | Singapore | Korea | Thailand | Indonesia | |
---|---|---|---|---|---|---|

Full Sample Period | ||||||

mean | 0.036 | 0.193 | 0.093 | 0.127 | 0.114 | 0.190 |

standard deviation | 2.972 | 3.305 | 2.866 | 3.614 | 3.570 | 3.273 |

skewness | −0.213 | −0.519 | −0.092 | −0.126 | −0.175 | −0.076 |

kurtosis | 4.772 | 5.787 | 6.258 | 6.681 | 5.830 | 8.207 |

JB test | 213.41 ** | 567.19 ** | 682.83 ** | 872.28 ** | 521.43 ** | 1738.36 ** |

$Q\left(5\right)$ | 1.49 | 3.37 | 15.70 ** | 24.27 ** | 24.53 ** | 51.13 ** |

${Q}^{2}\left(5\right)$ | 137.97 ** | 228.65 ** | 181.13 ** | 394.65 ** | 199.25 ** | 322.97 ** |

Pre-Crisis | ||||||

mean | 0.054 | 0.355 | 0.143 | 0.115 | 0.134 | 0.146 |

standard deviation | 2.774 | 3.273 | 2.901 | 3.403 | 3.707 | 2.912 |

skewness | −0.375 | −1.047 | −0.455 | −0.274 | −0.519 | 0.426 |

kurtosis | 5.143 | 7.150 | 5.689 | 7.468 | 5.165 | 8.983 |

JB test | 133.50 ** | 557.27 ** | 208.50 ** | 523.28 ** | 149.24 ** | 941.81 ** |

$Q\left(5\right)$ | 10.83 | 8.68 | 3.07 | 9.45 | 18.03 ** | 72.60 ** |

${Q}^{2}\left(5\right)$ | 143.18 ** | 48.96 ** | 69.45 ** | 122.03 ** | 95.20 ** | 123.60 ** |

Post-Crisis | ||||||

mean | 0.025 | 0.085 | 0.060 | 0.135 | 0.101 | 0.219 |

standard deviation | 3.098 | 3.324 | 2.845 | 3.751 | 3.477 | 3.495 |

skewness | −0.132 | −0.181 | 0.164 | −0.052 | 0.102 | −0.276 |

kurtosis | 4.532 | 5.038 | 6.685 | 6.227 | 6.357 | 7.657 |

JB test | 93.62 ** | 165.58 ** | 527.53 ** | 402.10 ** | 436.15 ** | 847.27 ** |

$Q\left(5\right)$ | 3.50 | 6.63 | 14.50 * | 18.20 ** | 13.36 * | 19.62 ** |

${Q}^{2}\left(5\right)$ | 39.50 ** | 241.15 ** | 119.93 ** | 212.87 ** | 119.63 ** | 189.86 ** |

Periods | 1985–1997 | 1998–2011 | ||
---|---|---|---|---|

# of Obs | 613 | 631 | ||

Volatility | Correlation | Volatility | Correlation | |

GVAR | $4.309\times {10}^{5}$ | - | $1.002\times {10}^{5}$ | - |

GSD | 656.43 | - | 316.57 | - |

EVAR | 8.69 | - | 6.82 | - |

ESD | 2.95 | - | 2.61 | - |

CCOR | - | 0.73 | - | 0.97 |

ECOR | - | 0.12 | - | 0.38 |

$\widehat{\mathit{\mu}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{{\mathit{\omega}}_{\mathit{i}}}$ | $\widehat{{\mathit{\alpha}}_{\mathit{i}}}$ | $\widehat{{\mathit{\beta}}_{\mathit{i}}}$ | |
---|---|---|---|---|---|

Japan | 0.161 | −0.007 | 0.672 | 0.149 | 0.783 |

(2.22) | (−0.24) | (1.95) | (4.43) | (15.81) | |

Hong Kong | 0.261 | 0.024 | 0.495 | 0.135 | 0.821 |

(3.50) | (0.88) | (2.69) | (4.53) | (22.08) | |

Singapore | 0.135 | 0.054 | 0.134 | 0.096 | 0.892 |

(2.10) | (1.82) | (1.55) | (3.61) | (26.91) | |

Korea | 0.192 | 0.008 | 0.147 | 0.142 | 0.857 |

(2.89) | (0.29) | (1.77) | (4.53) | (28.99) | |

Thailand | 0.181 | 0.064 | 0.242 | 0.123 | 0.864 |

(2.16) | (2.31) | (1.95) | (3.87) | (23.93) | |

Indonesia | 0.196 | 0.132 | 0.058 | 0.104 | 0.895 |

(2.84) | (4.20) | (1.24) | (6.28) | (50.41) |

Threshold Values for TSD | 3.0 | 3.5 | ||
---|---|---|---|---|

Periods | Tranquil | Volatile | Tranquil | Volatile |

Number of Obs | 967 | 564 | 1208 | 323 |

CCOR | 0.845 | 0.950 | 0.853 | 0.968 |

ECOR | 0.189 | 0.322 | 0.195 | 0.369 |

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**MDPI and ACS Style**

Kim, S.; Bera, A.K.
Scalar Measures of Volatility and Dependence for the Multivariate Models with Applications to Asian Financial Markets. *J. Risk Financial Manag.* **2023**, *16*, 212.
https://doi.org/10.3390/jrfm16040212

**AMA Style**

Kim S, Bera AK.
Scalar Measures of Volatility and Dependence for the Multivariate Models with Applications to Asian Financial Markets. *Journal of Risk and Financial Management*. 2023; 16(4):212.
https://doi.org/10.3390/jrfm16040212

**Chicago/Turabian Style**

Kim, Sangwhan, and Anil K. Bera.
2023. "Scalar Measures of Volatility and Dependence for the Multivariate Models with Applications to Asian Financial Markets" *Journal of Risk and Financial Management* 16, no. 4: 212.
https://doi.org/10.3390/jrfm16040212