doi: 10.3934/dcdss.2020267

Measuring financial contagion between major european stock markets via the regime-switching vine copula model

1. 

Straits Institute, Minjiang University, Fuzhou 350108, China

2. 

Newhuadu Business School, Minjiang University, Fuzhou 350108, China

3. 

Department of Mathematics, University of Sussex, Brighton BN1 9RH, UK

*Corresponding author: Yang-Chao Wang

Received  April 2019 Revised  May 2019 Published  February 2020

Using regime-switching copula models, this study investigates co-movements of financial markets, including four major stock markets of European countries (UK, French, Germany, and Italy) in stable, stressful, and extremely stressful market conditions during 2004–2017. We introduce vine copulas to describe high-dimensional dependence structure between markets in modelling. In addition, we integrate copula methods with advanced regime-switching detection to identify different volatility regimes. Because of the characteristics of stock returns, we also release the assumption of joint normality to capture markets' tail dependence. The results show increasing tail dependence and asymmetry during stressful and extremely stressful market conditions, validating the financial contagion phenomenon, the spread of market turmoil. Further, our findings show continuing risk spillover after the 2008 financial crisis. The lasting effects make markets vulnerable for a long time after the crisis, which may be a cause of the decreasing time intervals between financial crises in recent years. Moreover, before the crisis the tail dependence strengthened first in the low volatility regime, followed by joint volatility strengthening between markets. Therefore, observing changes in tail dependence may provide early warnings of the next financial crisis.

Citation: Jui-Jung Tsai, Yang-Chao Wang, Jiarong Zhang. Measuring financial contagion between major european stock markets via the regime-switching vine copula model. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020267
References:
[1]

S. Aidara, Anticipated backward doubly stochastic differential equations with non-liphschitz coefficients, Applied Mathematics and Nonlinear Sciences, 4 (2019), 9-20.  doi: 10.2478/AMNS.2019.1.00002.  Google Scholar

[2]

T. Bedford and R. M. Cooke, Probability density decomposition for conditionally dependent random variables modeled by vines, Annals of Mathematics and Artificial Intelligence, 32 (2001), 245-268.  doi: 10.1023/A:1016725902970.  Google Scholar

[3]

T. C. Chiang and D. Zheng, An empirical analysis of herd behavior in global stock markets, Journal of Banking and Finance, 34 (2010), 1911-1921.  doi: 10.1016/j.jbankfin.2009.12.014.  Google Scholar

[4]

C. Erdman and J. W. Emerson, bcp: An r package for performing a bayesian analysis of change point problems, Journal of Statistical Software, 23 (2007), 1-13.   Google Scholar

[5]

H. Fink, Y. Klimova, C. Czado and J. Stöber, Regime switching vine copula models for global equity and volatility indices, Econometrics, 5 (2017), 3. doi: 10.3390/econometrics5010003.  Google Scholar

[6]

K. J. Forbes and R. Rigobon, No contagion, only interdependence: Measuring stock market comovements, The Journal of Finance, 57 (2002), 2223-2261.  doi: 10.1111/0022-1082.00494.  Google Scholar

[7]

M. I. Garcia-Planas and T. Klymchuk, Perturbation analysis of a matrix differential equation $\dot{x} = abx$, Applied Mathematics and Nonlinear Sciences, 3 (2018), 97-104.  doi: 10.21042/AMNS.2018.1.00007.  Google Scholar

[8]

A. M. Khalid and M. Kawai, Was financial market contagion the source of economic crisis in asia?: Evidence using a multivariate var model, Journal of Asian Economics, 14 (2003), 131-156.   Google Scholar

[9]

R. KillickP. Fearnhead and I. A. Eckley, Optimal detection of changepoints with a linear computational cost, Journal of the American Statistical Association, 107 (2012), 1590-1598.  doi: 10.1080/01621459.2012.737745.  Google Scholar

[10]

G. H. Kuper and Le stano, Dynamic conditional correlation analysis of financial market interdependence: An application to thailand and indonesia, Journal of Asian Economics, 18 (2007), 670-684.  doi: 10.1016/j.asieco.2007.03.007.  Google Scholar

[11]

D. S. Matteson and N. A. James, A nonparametric approach for multiple change point analysis of multivariate data, Journal of the American Statistical Association, 109 (2014), 334-345.  doi: 10.1080/01621459.2013.849605.  Google Scholar

[12]

A. K. NikoloulopoulosH. Joe and H. Li, Vine copulas with asymmetric tail dependence and applications to financial return data, Computational Statistics and Data Analysis, 56 (2012), 3659-3673.  doi: 10.1016/j.csda.2010.07.016.  Google Scholar

[13]

A. J. Patton, Modelling asymmetric exchange rate dependence, International Economic Review, 47 (2006), 527-556.  doi: 10.1111/j.1468-2354.2006.00387.x.  Google Scholar

[14]

J. C. ReboredoM. A. Rivera-Castro and A. Ugolini, Downside and upside risk spillovers between exchange rates and stock prices, Journal of Banking and Finance, 62 (2016), 76-96.  doi: 10.1016/j.jbankfin.2015.10.011.  Google Scholar

[15]

M. L. Rizzo and G. J. Székely, Disco analysis: A nonparametric extension of analysis of variance, The Annals of Applied Statistics, 4 (2010), 1034-1055.  doi: 10.1214/09-AOAS245.  Google Scholar

[16]

J. C. Rodriguez, Measuring financial contagion: A copula approach, Journal of Empirical Finance, 14 (2007), 401-423.  doi: 10.1016/j.jempfin.2006.07.002.  Google Scholar

[17]

H. Sander and S. Kleimeier, Contagion and causality: An empirical investigation of four asian crisis episodes, Journal of International Financial Markets, Institutions and Money, 13 (2003), 171-186.   Google Scholar

[18]

J.-J. Tsai, Y.-C. Wang and K. Weng, The asymmetry and volatility of the chinese stock market caused by the "new national ten.", Emerging Markets Finance and Trade, 51 (2015), S86–S98. doi: 10.1080/1540496X.2014.998918.  Google Scholar

[19]

X. Wang and J. W. Emerson, Bayesian change point analysis of linear models on graphs, arXiv preprint, arXiv1509.00817. Google Scholar

[20]

Y.-C. Wang, J.-J. Tsai and Q. Li, Policy impact on the chinese stock market: From the 1994 bailout policies to the 2015 shanghai-hong kong stock connect, International Journal of Financial Studies, 5 (2017), 4. Google Scholar

[21]

Y.-C. WangJ.-J. Tsai and L. Lu, The impact of chinese monetary policy on co-movements between money and capital markets, Applied Economics, 51 (2019), 4939-4955.  doi: 10.1080/00036846.2019.1606407.  Google Scholar

show all references

References:
[1]

S. Aidara, Anticipated backward doubly stochastic differential equations with non-liphschitz coefficients, Applied Mathematics and Nonlinear Sciences, 4 (2019), 9-20.  doi: 10.2478/AMNS.2019.1.00002.  Google Scholar

[2]

T. Bedford and R. M. Cooke, Probability density decomposition for conditionally dependent random variables modeled by vines, Annals of Mathematics and Artificial Intelligence, 32 (2001), 245-268.  doi: 10.1023/A:1016725902970.  Google Scholar

[3]

T. C. Chiang and D. Zheng, An empirical analysis of herd behavior in global stock markets, Journal of Banking and Finance, 34 (2010), 1911-1921.  doi: 10.1016/j.jbankfin.2009.12.014.  Google Scholar

[4]

C. Erdman and J. W. Emerson, bcp: An r package for performing a bayesian analysis of change point problems, Journal of Statistical Software, 23 (2007), 1-13.   Google Scholar

[5]

H. Fink, Y. Klimova, C. Czado and J. Stöber, Regime switching vine copula models for global equity and volatility indices, Econometrics, 5 (2017), 3. doi: 10.3390/econometrics5010003.  Google Scholar

[6]

K. J. Forbes and R. Rigobon, No contagion, only interdependence: Measuring stock market comovements, The Journal of Finance, 57 (2002), 2223-2261.  doi: 10.1111/0022-1082.00494.  Google Scholar

[7]

M. I. Garcia-Planas and T. Klymchuk, Perturbation analysis of a matrix differential equation $\dot{x} = abx$, Applied Mathematics and Nonlinear Sciences, 3 (2018), 97-104.  doi: 10.21042/AMNS.2018.1.00007.  Google Scholar

[8]

A. M. Khalid and M. Kawai, Was financial market contagion the source of economic crisis in asia?: Evidence using a multivariate var model, Journal of Asian Economics, 14 (2003), 131-156.   Google Scholar

[9]

R. KillickP. Fearnhead and I. A. Eckley, Optimal detection of changepoints with a linear computational cost, Journal of the American Statistical Association, 107 (2012), 1590-1598.  doi: 10.1080/01621459.2012.737745.  Google Scholar

[10]

G. H. Kuper and Le stano, Dynamic conditional correlation analysis of financial market interdependence: An application to thailand and indonesia, Journal of Asian Economics, 18 (2007), 670-684.  doi: 10.1016/j.asieco.2007.03.007.  Google Scholar

[11]

D. S. Matteson and N. A. James, A nonparametric approach for multiple change point analysis of multivariate data, Journal of the American Statistical Association, 109 (2014), 334-345.  doi: 10.1080/01621459.2013.849605.  Google Scholar

[12]

A. K. NikoloulopoulosH. Joe and H. Li, Vine copulas with asymmetric tail dependence and applications to financial return data, Computational Statistics and Data Analysis, 56 (2012), 3659-3673.  doi: 10.1016/j.csda.2010.07.016.  Google Scholar

[13]

A. J. Patton, Modelling asymmetric exchange rate dependence, International Economic Review, 47 (2006), 527-556.  doi: 10.1111/j.1468-2354.2006.00387.x.  Google Scholar

[14]

J. C. ReboredoM. A. Rivera-Castro and A. Ugolini, Downside and upside risk spillovers between exchange rates and stock prices, Journal of Banking and Finance, 62 (2016), 76-96.  doi: 10.1016/j.jbankfin.2015.10.011.  Google Scholar

[15]

M. L. Rizzo and G. J. Székely, Disco analysis: A nonparametric extension of analysis of variance, The Annals of Applied Statistics, 4 (2010), 1034-1055.  doi: 10.1214/09-AOAS245.  Google Scholar

[16]

J. C. Rodriguez, Measuring financial contagion: A copula approach, Journal of Empirical Finance, 14 (2007), 401-423.  doi: 10.1016/j.jempfin.2006.07.002.  Google Scholar

[17]

H. Sander and S. Kleimeier, Contagion and causality: An empirical investigation of four asian crisis episodes, Journal of International Financial Markets, Institutions and Money, 13 (2003), 171-186.   Google Scholar

[18]

J.-J. Tsai, Y.-C. Wang and K. Weng, The asymmetry and volatility of the chinese stock market caused by the "new national ten.", Emerging Markets Finance and Trade, 51 (2015), S86–S98. doi: 10.1080/1540496X.2014.998918.  Google Scholar

[19]

X. Wang and J. W. Emerson, Bayesian change point analysis of linear models on graphs, arXiv preprint, arXiv1509.00817. Google Scholar

[20]

Y.-C. Wang, J.-J. Tsai and Q. Li, Policy impact on the chinese stock market: From the 1994 bailout policies to the 2015 shanghai-hong kong stock connect, International Journal of Financial Studies, 5 (2017), 4. Google Scholar

[21]

Y.-C. WangJ.-J. Tsai and L. Lu, The impact of chinese monetary policy on co-movements between money and capital markets, Applied Economics, 51 (2019), 4939-4955.  doi: 10.1080/00036846.2019.1606407.  Google Scholar

Figure 1.  Posterior transition probability for indices
Figure 2.  Contour plots of C-vine copulas by regimes
Figure 3.  Pairwise scatter plots by regimes
Table 1.  Some widely used Archimedean copulas
Family Pair Copula Function Domain Dependence
Gumbel $ \exp\{-[(-\ln v)^\theta+(-\ln z)^\theta ]^{1/\theta}\} $ $ \theta\in [1, \infty ) $ $ 2^{-1/\theta} $
Clayton $ \max[v^{-\theta}+z^{-\theta}-1, 0] $ $ \theta\in [-1, \infty )\backslash \{0\} $ $ 2-2^{-1/\theta} $
Frank $ -\frac{1}{\theta}\ln\left(1+\frac{(\exp(-\theta v)-1)(\exp(-\theta z)-1)}{\exp(-\theta)-1}\right) $ $ \theta\in (-\infty , \infty )\backslash\{0\} $ -
Clayton-Gumbel $ [1+((v^{-\theta}-1)^\gamma +(z^{-\theta}-1)^\gamma )^{-1/\gamma}]^{1/\theta} $ $ \theta\in (0, \infty ) $ $ \gamma \in [1, \infty ) $
$ \lambda _L=2^{-1/\gamma \theta} $ $ \lambda_U=2-2^{-1/\gamma} $
Joe-Clayton (BB7) $ 1-\left(\left\{[1-(1-v)^\theta]^{-\gamma}+[1-(1-z)^\theta]^{-\gamma}-1\right\}^{-1/\gamma}\right)^{1/\theta} $ $ \theta\in [1, \infty ) $ $ \gamma \in (0, \infty ) $
$ \lambda_L=2^{-1/\gamma} $ $ \lambda _U=2-2^{-1/\theta} $
Family Pair Copula Function Domain Dependence
Gumbel $ \exp\{-[(-\ln v)^\theta+(-\ln z)^\theta ]^{1/\theta}\} $ $ \theta\in [1, \infty ) $ $ 2^{-1/\theta} $
Clayton $ \max[v^{-\theta}+z^{-\theta}-1, 0] $ $ \theta\in [-1, \infty )\backslash \{0\} $ $ 2-2^{-1/\theta} $
Frank $ -\frac{1}{\theta}\ln\left(1+\frac{(\exp(-\theta v)-1)(\exp(-\theta z)-1)}{\exp(-\theta)-1}\right) $ $ \theta\in (-\infty , \infty )\backslash\{0\} $ -
Clayton-Gumbel $ [1+((v^{-\theta}-1)^\gamma +(z^{-\theta}-1)^\gamma )^{-1/\gamma}]^{1/\theta} $ $ \theta\in (0, \infty ) $ $ \gamma \in [1, \infty ) $
$ \lambda _L=2^{-1/\gamma \theta} $ $ \lambda_U=2-2^{-1/\gamma} $
Joe-Clayton (BB7) $ 1-\left(\left\{[1-(1-v)^\theta]^{-\gamma}+[1-(1-z)^\theta]^{-\gamma}-1\right\}^{-1/\gamma}\right)^{1/\theta} $ $ \theta\in [1, \infty ) $ $ \gamma \in (0, \infty ) $
$ \lambda_L=2^{-1/\gamma} $ $ \lambda _U=2-2^{-1/\theta} $
Table 2.  Summary statistics of log-returns of stock indices
Statistics FTSE100 CAC40 DAX FTMIB
Mean 0.01563 0.01102 0.03454 –0.00812
Minimum –9.26557 –9.47154 –7.43346 –13.33140
Maximum 9.38434 10.59459 10.79747 10.87425
Std. Dev. 1.15012 1.39394 1.35008 1.57505
Skewness –0.15734 –0.03905 –0.04179 –0.24440
Kurtosis 11.48727 9.65953 9.35357 8.66536
Statistics FTSE100 CAC40 DAX FTMIB
Mean 0.01563 0.01102 0.03454 –0.00812
Minimum –9.26557 –9.47154 –7.43346 –13.33140
Maximum 9.38434 10.59459 10.79747 10.87425
Std. Dev. 1.15012 1.39394 1.35008 1.57505
Skewness –0.15734 –0.03905 –0.04179 –0.24440
Kurtosis 11.48727 9.65953 9.35357 8.66536
Table 3.  Volatility regime switching
Panel A: FTSE-CAC40
Sample Obs. 0-798 799-1095 1096-1220 1221-1833 1834-1932 1933-2865 2865-3101 3102-3324
Change Point 9 Jun 2004 17 Jul 2007 12 Sep 2008 11 Mar 2009 29 Jul 2011 15 Dec 2011 11 Aug 2015 12 Jul 2016
Volatility Regime Low High Extreme High Low High Low High Low
Panel B: FTSE-DAX
Sample Obs. 0-792 793-1087 1088-1217 1218-1833 1834-1932 1933-2873 2874-3101 3102-3324
Change Point 9 Jun 2004 9 Jul 2007 2 Sep 2008 6 Mar 2009 29 Jul 2011 15 Dec 2011 11 Aug 2015 12 Jul 2016
Volatility Regime Low High Extreme High Low High Low High Low
Panel C: FTSE-FTMIB
Sample Obs. 0-798 799-1095 1096-1220 1221-1816 1817-1932 1933-3324
Change Point 9 Jun 2004 17 Jul 2007 12 Sep 2008 11 Mar 2009 6 Jul 2011 15 Dec 2011
Volatility Regime Low High Extreme High Low High Low
Panel A: FTSE-CAC40
Sample Obs. 0-798 799-1095 1096-1220 1221-1833 1834-1932 1933-2865 2865-3101 3102-3324
Change Point 9 Jun 2004 17 Jul 2007 12 Sep 2008 11 Mar 2009 29 Jul 2011 15 Dec 2011 11 Aug 2015 12 Jul 2016
Volatility Regime Low High Extreme High Low High Low High Low
Panel B: FTSE-DAX
Sample Obs. 0-792 793-1087 1088-1217 1218-1833 1834-1932 1933-2873 2874-3101 3102-3324
Change Point 9 Jun 2004 9 Jul 2007 2 Sep 2008 6 Mar 2009 29 Jul 2011 15 Dec 2011 11 Aug 2015 12 Jul 2016
Volatility Regime Low High Extreme High Low High Low High Low
Panel C: FTSE-FTMIB
Sample Obs. 0-798 799-1095 1096-1220 1221-1816 1817-1932 1933-3324
Change Point 9 Jun 2004 17 Jul 2007 12 Sep 2008 11 Mar 2009 6 Jul 2011 15 Dec 2011
Volatility Regime Low High Extreme High Low High Low
Table 4.  The estimation of C-vine decomposition by regimes
Regime 1: Low volatility regime
Pair Copulas $ (1, 4) $ $ (1, 3) $ $ (1, 2) $ $ (2, 4|1) $ $ (2, 3|1) $ $ (3, 4|1, 2) $
Upper tail par. $ \theta $ 2.31 2.65 3.11 1.69 2.13 1.12
Lower tail par. $ \gamma $ 1.39 1.24 1.43 0.59 1.08 0.27
Upper tail dep. $ \lambda_U $ 0.61 0.57 0.62 0.49 0.62 0.08
Lower tail dep. $ \lambda_L $ 0.67 0.7 0.75 0.31 0.53 0.14
Kendall's $ \tau $ 0.58 0.59 0.63 0.4 0.52 0.17
Family Type Survival BB7 Survival BB7 Survival BB7 BB7 BB7 Survival BB7
Regime 2: High volatility regime
Pair Copulas $ (1, 4) $ $ (1, 3) $ $ (1, 2) $ $ (2, 4|1) $ $ (2, 3|1) $ $ (3, 4|1, 2) $
Upper tail par. $ \theta $ 2.29 2.45 2.78 1.79 2.23 1.07
Lower tail par. $ \gamma $ 1.21 1.48 1.7 1.15 1.11 0.22
Upper tail dep. $ \lambda_U $ 0.56 0.67 0.66 0.55 0.54 0.04
Lower tail dep. $ \lambda_L $ 0.65 0.63 0.72 0.53 0.63 0.09
Kendall's 0.55 0.59 0.62 0.49 0.54 0.04
Family Type Survival BB7 BB7 Survival BB7 Survival BB7 Survival BB7 Survival BB7
Regime 3: Extremely high volatility regime
Pair Copulas $ (1, 4) $ $ (1, 3) $ $ (1, 2) $ $ (2, 4|1) $ $ (2, 3|1) $ $ (3, 4|1, 2) $
Upper tail par. $ \theta $ 2.75 3.4 4.02 1.9 1.7 1.08
Lower tail par. $ \gamma $ 2.04 1.39 2.47 0.47 0.84 0.13
Upper tail dep. $ \lambda_U $ 0.71 0.61 0.81 0.23 0.5 0.1
Lower tail dep. $ \lambda_L $ 0.71 0.77 0.76 0.56 0.44 0.01
Kendall's $ \tau $ 0.64 0.65 0.71 0.42 0.44 0.1
Family Type Survival BB7 Survival BB7 BB7 Survival BB7 BB7 BB7
Regime 4: Low volatility regime
Pair Copulas $ (1, 4) $ $ (1, 3) $ $ (1, 2) $ $ (2, 4|1) $ $ (2, 3|1) $ $ (3, 4|1, 2) $
Upper tail par. $ \theta $ 2.29 2.8 2.78 1.65 2.04 1.03
Lower tail par. $ \gamma $ 1.12 1.52 2.38 0.84 1.03 0.06
Upper tail dep. $ \lambda_U $ 0.54 0.63 0.72 0.44 0.51 0
Lower tail dep. $ \lambda_L $ 0.65 0.72 0.75 0.48 0.6 0.03
Kendall's $ \tau $ 0.54 0.62 0.65 0.43 0.51 0.04
Family Type Survival BB7 Survival BB7 BB7 Survival BB7 Survival BB7 Survival BB7
Regime 5: High volatility regime
Pair Copulas $ (1, 4) $ $ (1, 3) $ $ (1, 2) $ $ (2, 4|1) $ $ (2, 3|1) $ $ (3, 4|1, 2) $
Upper tail par. $ \theta $ 3.18 3.83 3.97 1.52 2.03 1.34
Lower tail par. $ \gamma $ 1.15 1.59 1.85 1.3 0.59 0
Upper tail dep. $ \lambda_U $ 0.76 0.8 0.81 0.42 0.31 0.33
Lower tail dep. $ \lambda_L $ 0.55 0.65 0.69 0.59 0.59 0
Kendall's $ \tau $ 0.62 0.68 0.69 0.47 0.46 0.16
Family Type BB7 BB7 BB7 BB7 Survival BB7 BB7
Regime 6: Low volatility regime
Pair Copulas $ (1, 4) $ $ (1, 3) $ $ (1, 2) $ $ (2, 4|1) $ $ (2, 3|1) $ $ (3, 4|1, 2) $
Upper tail par. $ \theta $ 1.53 2.13 2.08 1.56 2.02 1.06
Lower tail par. $ \gamma $ 0.87 0.93 1.39 0.73 0.93 0.1
Upper tail dep. $ \lambda_U $ 0.43 0.47 0.61 0.44 0.48 0.08
Lower tail dep. $ \lambda_L $ 0.45 0.61 0.61 0.39 0.59 0
Kendall's $ \tau $ 0.41 0.51 0.55 0.4 0.49 0.08
Family Type BB7 Survival BB7 BB7 BB7 Survival BB7 BB7
Goodness-of-fit Regime 1 Regime 2 Regime 3 Regime 4 Regime 5 Regime 6
KS Statistic 1.046 1.734 1.105 1.026 1.186 0.988
$ p $-value 0.965 0.635 0.75 0.925 0.635 0.945
Notes: The variables "1" to "4" denote FTSE, CAC40, DAX, and FTMIB, respectively. The last part of the table reports Kolmogorov-Smirnov (KS) statistics for the goodness-of-fit tests by regimes. The corresponding $ p $-values represent that the null hypothesis of the correct specification cannot be rejected, suggesting that the copula models are not misspecified.
Regime 1: Low volatility regime
Pair Copulas $ (1, 4) $ $ (1, 3) $ $ (1, 2) $ $ (2, 4|1) $ $ (2, 3|1) $ $ (3, 4|1, 2) $
Upper tail par. $ \theta $ 2.31 2.65 3.11 1.69 2.13 1.12
Lower tail par. $ \gamma $ 1.39 1.24 1.43 0.59 1.08 0.27
Upper tail dep. $ \lambda_U $ 0.61 0.57 0.62 0.49 0.62 0.08
Lower tail dep. $ \lambda_L $ 0.67 0.7 0.75 0.31 0.53 0.14
Kendall's $ \tau $ 0.58 0.59 0.63 0.4 0.52 0.17
Family Type Survival BB7 Survival BB7 Survival BB7 BB7 BB7 Survival BB7
Regime 2: High volatility regime
Pair Copulas $ (1, 4) $ $ (1, 3) $ $ (1, 2) $ $ (2, 4|1) $ $ (2, 3|1) $ $ (3, 4|1, 2) $
Upper tail par. $ \theta $ 2.29 2.45 2.78 1.79 2.23 1.07
Lower tail par. $ \gamma $ 1.21 1.48 1.7 1.15 1.11 0.22
Upper tail dep. $ \lambda_U $ 0.56 0.67 0.66 0.55 0.54 0.04
Lower tail dep. $ \lambda_L $ 0.65 0.63 0.72 0.53 0.63 0.09
Kendall's 0.55 0.59 0.62 0.49 0.54 0.04
Family Type Survival BB7 BB7 Survival BB7 Survival BB7 Survival BB7 Survival BB7
Regime 3: Extremely high volatility regime
Pair Copulas $ (1, 4) $ $ (1, 3) $ $ (1, 2) $ $ (2, 4|1) $ $ (2, 3|1) $ $ (3, 4|1, 2) $
Upper tail par. $ \theta $ 2.75 3.4 4.02 1.9 1.7 1.08
Lower tail par. $ \gamma $ 2.04 1.39 2.47 0.47 0.84 0.13
Upper tail dep. $ \lambda_U $ 0.71 0.61 0.81 0.23 0.5 0.1
Lower tail dep. $ \lambda_L $ 0.71 0.77 0.76 0.56 0.44 0.01
Kendall's $ \tau $ 0.64 0.65 0.71 0.42 0.44 0.1
Family Type Survival BB7 Survival BB7 BB7 Survival BB7 BB7 BB7
Regime 4: Low volatility regime
Pair Copulas $ (1, 4) $ $ (1, 3) $ $ (1, 2) $ $ (2, 4|1) $ $ (2, 3|1) $ $ (3, 4|1, 2) $
Upper tail par. $ \theta $ 2.29 2.8 2.78 1.65 2.04 1.03
Lower tail par. $ \gamma $ 1.12 1.52 2.38 0.84 1.03 0.06
Upper tail dep. $ \lambda_U $ 0.54 0.63 0.72 0.44 0.51 0
Lower tail dep. $ \lambda_L $ 0.65 0.72 0.75 0.48 0.6 0.03
Kendall's $ \tau $ 0.54 0.62 0.65 0.43 0.51 0.04
Family Type Survival BB7 Survival BB7 BB7 Survival BB7 Survival BB7 Survival BB7
Regime 5: High volatility regime
Pair Copulas $ (1, 4) $ $ (1, 3) $ $ (1, 2) $ $ (2, 4|1) $ $ (2, 3|1) $ $ (3, 4|1, 2) $
Upper tail par. $ \theta $ 3.18 3.83 3.97 1.52 2.03 1.34
Lower tail par. $ \gamma $ 1.15 1.59 1.85 1.3 0.59 0
Upper tail dep. $ \lambda_U $ 0.76 0.8 0.81 0.42 0.31 0.33
Lower tail dep. $ \lambda_L $ 0.55 0.65 0.69 0.59 0.59 0
Kendall's $ \tau $ 0.62 0.68 0.69 0.47 0.46 0.16
Family Type BB7 BB7 BB7 BB7 Survival BB7 BB7
Regime 6: Low volatility regime
Pair Copulas $ (1, 4) $ $ (1, 3) $ $ (1, 2) $ $ (2, 4|1) $ $ (2, 3|1) $ $ (3, 4|1, 2) $
Upper tail par. $ \theta $ 1.53 2.13 2.08 1.56 2.02 1.06
Lower tail par. $ \gamma $ 0.87 0.93 1.39 0.73 0.93 0.1
Upper tail dep. $ \lambda_U $ 0.43 0.47 0.61 0.44 0.48 0.08
Lower tail dep. $ \lambda_L $ 0.45 0.61 0.61 0.39 0.59 0
Kendall's $ \tau $ 0.41 0.51 0.55 0.4 0.49 0.08
Family Type BB7 Survival BB7 BB7 BB7 Survival BB7 BB7
Goodness-of-fit Regime 1 Regime 2 Regime 3 Regime 4 Regime 5 Regime 6
KS Statistic 1.046 1.734 1.105 1.026 1.186 0.988
$ p $-value 0.965 0.635 0.75 0.925 0.635 0.945
Notes: The variables "1" to "4" denote FTSE, CAC40, DAX, and FTMIB, respectively. The last part of the table reports Kolmogorov-Smirnov (KS) statistics for the goodness-of-fit tests by regimes. The corresponding $ p $-values represent that the null hypothesis of the correct specification cannot be rejected, suggesting that the copula models are not misspecified.
Table 5.  Stationary and ARCH tests for stock indices returns
Stationary test FTSE CAC40 DAX FTMIB
ADF $ -59.7002 $ $ -59.8414 $ $ -57.2153 $ $ -59.0875 $
$ p $-value for ADF $<0.001 $ $<0.001 $ $<0.001 $ $<0.001 $
Jarque-Bera 9990.3962 6143.2219 5591.9138 4478.4301
$ p $-value for JB $<0.001 $ $<0.001 $ $<0.001 $ $<0.001 $
ARCH Test FTSE CAC40 DAX FTMIB
Engle 197.7418 127.9930 95.4764 113.3684
$ p $-value for Engle $<0.001 $ $<0.001 $ $<0.001 $ $<0.001 $
Stationary test FTSE CAC40 DAX FTMIB
ADF $ -59.7002 $ $ -59.8414 $ $ -57.2153 $ $ -59.0875 $
$ p $-value for ADF $<0.001 $ $<0.001 $ $<0.001 $ $<0.001 $
Jarque-Bera 9990.3962 6143.2219 5591.9138 4478.4301
$ p $-value for JB $<0.001 $ $<0.001 $ $<0.001 $ $<0.001 $
ARCH Test FTSE CAC40 DAX FTMIB
Engle 197.7418 127.9930 95.4764 113.3684
$ p $-value for Engle $<0.001 $ $<0.001 $ $<0.001 $ $<0.001 $
Table 6.  FTSE-CAC40 Joe-Clayton copula estimation by regimes
Regime 1 2 3 4 5 6 7 8
Upper tail par. $ \theta $ 2.11 2.6 5 3.22 5 2.13 2.52 1.84
Lower tail par. $ \gamma $ 2.17 2.37 2.53 2.53 1.55 1.42 2.12 0.89
Upper tail dep. $ \lambda_U $ 0.61 0.69 0.85 0.76 0.85 0.61 0.68 0.54
Lower tail dep. $ \lambda_L $ 0.72 0.75 0.77 0.76 0.64 0.61 0.72 0.46
Kendall's $ \tau $ 0.6 0.64 0.75 0.68 0.73 0.55 0.63 0.46
KS Statistic 1.01 1.23 0.64 0.95 0.82 1.29 0.51 0.58
$ p $-value 0.29 0.16 0.59 0.25 0.31 0.12 0.66 0.59
Notes: Regimes 1 to 8 represent low, high, extremely high, low, high, low, high, and low volatility regimes, respectively.
Regime 1 2 3 4 5 6 7 8
Upper tail par. $ \theta $ 2.11 2.6 5 3.22 5 2.13 2.52 1.84
Lower tail par. $ \gamma $ 2.17 2.37 2.53 2.53 1.55 1.42 2.12 0.89
Upper tail dep. $ \lambda_U $ 0.61 0.69 0.85 0.76 0.85 0.61 0.68 0.54
Lower tail dep. $ \lambda_L $ 0.72 0.75 0.77 0.76 0.64 0.61 0.72 0.46
Kendall's $ \tau $ 0.6 0.64 0.75 0.68 0.73 0.55 0.63 0.46
KS Statistic 1.01 1.23 0.64 0.95 0.82 1.29 0.51 0.58
$ p $-value 0.29 0.16 0.59 0.25 0.31 0.12 0.66 0.59
Notes: Regimes 1 to 8 represent low, high, extremely high, low, high, low, high, and low volatility regimes, respectively.
Table 7.  FTSE-DAX Joe-Clayton copula estimation by regimes
Regime 1 2 3 4 5 6 7 8
Upper tail par. $ \theta $ 2.59 2.66 3.06 2.98 2.16 2.12 2.32 1.6
Lower tail par. $ \gamma $ 1.04 1.55 2.82 1.82 4.98 1.02 1.26 0.88
Upper tail dep. $ \lambda_U $ 0.51 0.64 0.78 0.68 0.87 0.51 0.58 0.45
Lower tail dep. $ \lambda_L $ 0.69 0.70 0.75 0.74 0.62 0.61 0.65 0.46
Kendall's $ \tau $ 0.57 0.61 0.68 0.64 0.71 0.52 0.56 0.43
KS Statistic 1.27 0.81 0.74 1.09 0.81 0.94 0.85 0.69
$ p $-value 0.1 0.28 0.55 0.17 0.34 0.62 0.23 0.78
Notes: Regimes 1 to 8 represent low, high, extremely high, low, high, low, high, and low volatility regimes, respectively.
Regime 1 2 3 4 5 6 7 8
Upper tail par. $ \theta $ 2.59 2.66 3.06 2.98 2.16 2.12 2.32 1.6
Lower tail par. $ \gamma $ 1.04 1.55 2.82 1.82 4.98 1.02 1.26 0.88
Upper tail dep. $ \lambda_U $ 0.51 0.64 0.78 0.68 0.87 0.51 0.58 0.45
Lower tail dep. $ \lambda_L $ 0.69 0.70 0.75 0.74 0.62 0.61 0.65 0.46
Kendall's $ \tau $ 0.57 0.61 0.68 0.64 0.71 0.52 0.56 0.43
KS Statistic 1.27 0.81 0.74 1.09 0.81 0.94 0.85 0.69
$ p $-value 0.1 0.28 0.55 0.17 0.34 0.62 0.23 0.78
Notes: Regimes 1 to 8 represent low, high, extremely high, low, high, low, high, and low volatility regimes, respectively.
Table 8.  FTSE-FTMIB Joe-Clayton copula estimation by regimes
Regime 1 2 3 4 5 6
Upper tail par. $ \theta $ 2.04 2.14 3.72 2.19 3.62 1.63
Lower tail par. $ \gamma $ 1.47 1.69 1.82 1.67 0.89 0.86
Upper tail dep. $ \lambda_U $ 0.59 0.62 0.80 0.63 0.79 0.47
Lower tail dep. $ \lambda_L $ 0.62 0.66 0.68 0.66 0.46 0.44
Kendall's $ \tau $ 0.55 0.57 0.68 0.58 0.64 0.43
KS Statistic 1.16 0.88 0.56 0.75 0.83 1.04
$ p $-value 0.14 0.32 0.95 0.54 0.38 0.18
Notes: Regimes 1 to 6 represent low, high, extremely high, low, high, and low volatility regimes, respectively.
Regime 1 2 3 4 5 6
Upper tail par. $ \theta $ 2.04 2.14 3.72 2.19 3.62 1.63
Lower tail par. $ \gamma $ 1.47 1.69 1.82 1.67 0.89 0.86
Upper tail dep. $ \lambda_U $ 0.59 0.62 0.80 0.63 0.79 0.47
Lower tail dep. $ \lambda_L $ 0.62 0.66 0.68 0.66 0.46 0.44
Kendall's $ \tau $ 0.55 0.57 0.68 0.58 0.64 0.43
KS Statistic 1.16 0.88 0.56 0.75 0.83 1.04
$ p $-value 0.14 0.32 0.95 0.54 0.38 0.18
Notes: Regimes 1 to 6 represent low, high, extremely high, low, high, and low volatility regimes, respectively.
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