# 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

- Anderson, Theodore Wilbur. 1984. An Introduction to Multivariate Statistical Analysis, 2nd ed. New York: John Wiley & Sons. [Google Scholar]
- Bartram, Sohnke M., and Yaw-Huei Wang. 2005. Another look at the relationship between cross-market correlation and volatility. Finance Research Letters 2: 75–88. [Google Scholar] [CrossRef]
- Bauwens, Luc, Sebastien Laurent, and Jeroen V. K. Rombouts. 2006. Multivariate GARCH Models: A Survey. Journal of Applied Econometrics 21: 79–110. [Google Scholar] [CrossRef]
- Bekaert, Geert, Campbell R. Harvey, Andrea Kiguel, and Xiaozheng Wang. 2016. Globalization and Asset Returns. Annual Review of Financial Economics 8: 221–88. [Google Scholar] [CrossRef]
- Calvo, Sarah, and Carmen M. Reinhart. 1996. Capital flows to Latin America: Is there evidence of contagion effects? In Private Capital Flows to Emerging Markets After the Mexican Crisis. Edited by Guillermo Calvo, Morris Goldstein and Eduard Hochreiter. Washington, DC: Institute for International Economics. [Google Scholar]
- Caporale, Guglielmo Maria, Andrea Cipollini, and Nicola Spagnolo. 2005. Testing for contagion: A conditional correlation analysis. Journal of Empirical Finance 3: 476–89. [Google Scholar] [CrossRef]
- Chiang, Thomas C., Bang Nam Jeon, and Huimin Li. 2007. Dynamic correlation analysis of financial contagion: Evidence from Asian markets. Journal of International Money and Finance 26: 1206–28. [Google Scholar] [CrossRef]
- Corsetti, Giancarlo, Marcello Pericoli, and Massimo Sbracia. 2005. ‘Some contagion, some interdependence’: More pitfalls in tests of financial contagion. Journal of International Money and Finance 24: 1177–99. [Google Scholar] [CrossRef]
- Cramér, Harold. 1946. Mathematical Methods of Statistics. Princeton: Princeton University Press. [Google Scholar]
- Dungey, Mardi, Renee Fry, Brenda Gonzalez-Hermosillo, and Vance Martin. 2004. Empirical Modeling of Contagion: A Review of Methodologies. IMF Working Papers 04/78. Washington: International Monetary Fund. [Google Scholar]
- Engle, Robert F. 2002. Dynamic conditional correlation—A simple class of multivariate GARCH models. Journal of Business and Economic Statistics 20: 339–50. [Google Scholar] [CrossRef]
- Engle, Robert F., and Kevin Sheppard. 2001. Theoretical and Empirical Properties of Dynamic Conditional Correlation Multivariate GARCH. NBER Working Paper No. 8554. Cambridge: National Bureau of Economic Research. [Google Scholar]
- Forbes, Kristin J., and Roberto Rigobon. 2002. No contagion, Only Interdependence: Measuring Stock Market Comovements. The Journal of Fiance 57: 2223–61. [Google Scholar] [CrossRef]
- Frisch, Ragnar. 1929. Correlation and Scatter in Statistical Variables. Nordic Statistical Journal 8: 36–102. [Google Scholar]
- Giri, Narayan C. 1977. Multivariate Statistical Inference. New York: Academic Press. [Google Scholar]
- Gjika, Dritan, and Roman Horvath. 2013. Stock market comovements in Central Europe: Evidence from the asymmetric DCC model. Economic Modelling 33: 55–64. [Google Scholar] [CrossRef]
- Horn, Roger A., and Charles R. Johnson. 1985. Matrix Analysis. Cambridge: Cambridge University Press. [Google Scholar]
- Peña, Daniel, and Julio Rodriguez. 2003. Descriptive measures of multivariate scatter and linear dependence. Journal of Multivariate Analysis 85: 361–74. [Google Scholar] [CrossRef]
- Quinn, Dennis P., and Hans-Joachim Voth. 2008. A Century of Global Equity Market Correlations. American Economic Review 98: 535–40. [Google Scholar] [CrossRef]
- Rigobon, Roberto. 2003. On the Measurement of the International Propogation of Shock: Is the Transmission Stable? Journal of International Economics 61: 261–83. [Google Scholar] [CrossRef]
- Rigobon, Roberto. 2019. Contagion, Spillover, and Interdependence. Economía 19: 69–100. [Google Scholar] [CrossRef]
- Serfling, Robert J. 1980. Approximation Theorems of Mathematical Statistics. New York: John Wiley & Sons. [Google Scholar]
- Siddiqui, Taufeeque Ahmad, Mazia Fatima Khan, Mohammad Naushad, and Abdul Malik Syed. 2022. Cross-market Correlations and Financial Contagion from Developed to Emerging Economies: A Case of COVID-19 Pandemic. Economies 10: 147–58. [Google Scholar] [CrossRef]
- Silvennoinen, Annastiina, and Timo Terasvirta. 2009. Multivariate GARCH Models. In Handbook of Financial Time Series. Edited by Thomas Mikosch, Jens-Peter Kreiß, Richard A. Davis and Torben Gustav Andersen. Berlin/Heidelberg: Springer. [Google Scholar]
- Solnik, Bruno, Cyril Boucrelle, and Yann Le Fur. 1996. International Market Correlation and Volatility. Financial Analysts Journal 52: 17–34. [Google Scholar] [CrossRef]
- Tilfani, Oussama, Paulo Ferreira, Andreia Dionisio, and My Youssef El Boukfaoui. 2020. EU Stock Markets vs. Germany, UK and US: Analysis of Dynamic Comovements Using Time-Varying DCCA Correlation Coefficients. Journal of Risk and Financial Management 13: 91–113. [Google Scholar] [CrossRef]
- Tse, Yiu Kuen, and Albert K. C. Tsui. 2002. A multivariate GARCH model with time-varying correlations. Journal of Business and Economic Statistics 20: 351–62. [Google Scholar] [CrossRef]
- Wilks, Samuel S. 1932. Certain generalizations in the analysis of variance. Biometrika 24: 471–94. [Google Scholar] [CrossRef]

**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