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Open AccessOpen Access JUSTC Mathematics 15 January 2024

Measuring systemic risk for financial time series: A dynamic bivariate Dvine model

  • Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.52396/JUSTC-2023-0014
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  • Author Bio:

    Yu Chen is currently an Associate Professor at the Department of Statistics and Finance, University of Science and Technology of China (USTC). She received her Ph.D. degree in Statistics from USTC in 2006. Her research mainly focuses on extreme value theory, financial econometric models, network risk analysis, and multivariate statistical analysis theory

    Tao Xu is a Ph.D. candidate at the Department of Statistics and Finance, University of Science and Technology of China. His research mainly focuses on extreme value theory and risk measurement

  • Corresponding author: E-mail: xu0130@mail.ustc.edu.cn
  • Received Date: 15 February 2023
  • Accepted Date: 04 May 2023
    • Available Online: 15 January 2024
    Abstract

  • Abstract

    Accurate measurements of the tail risk of financial assets are major interest in financial markets. The main objective of our paper is to measure and forecast the value-at-risk (VaR) and the conditional value-at-risk (CoVaR) of financial assets using a new bivariate time series model. The proposed model can simultaneously capture serial dependence and cross-sectional dependence that exist in bivariate time series to improve the accuracy of estimation and prediction. In the process of model inference, we provide the parameter estimators of our bivariate time series model and give the estimators of VaR and CoVaR via the plug-in principle. We also establish the asymptotic properties of the Dvine model estimators. Real applications for financial stock price show that our model performs well in risk measurement and prediction.

    Graphical abstract

    The framework of the bivariate time series model.

    Abstract

    Accurate measurements of the tail risk of financial assets are major interest in financial markets. The main objective of our paper is to measure and forecast the value-at-risk (VaR) and the conditional value-at-risk (CoVaR) of financial assets using a new bivariate time series model. The proposed model can simultaneously capture serial dependence and cross-sectional dependence that exist in bivariate time series to improve the accuracy of estimation and prediction. In the process of model inference, we provide the parameter estimators of our bivariate time series model and give the estimators of VaR and CoVaR via the plug-in principle. We also establish the asymptotic properties of the Dvine model estimators. Real applications for financial stock price show that our model performs well in risk measurement and prediction.

    • Public Summary

    • We propose a dynamic bivariate Dvine model, which not only captures the nonlinear serial dependence between univariate time series via the copula function but also uses the GAS mechanism to capture the dynamic cross-sectional dependence of bivariate financial time series.
    • Compared to the traditional models for bivariate time series, the dynamic bivariate Dvine model is considered to present a new version that provides more information that exists in the nonlinear form of time series.
    • Compared to the existing asymptotic result of copula-based time series models, our paper provides the asymptotic properties of the univariate Dvine model when the unconditional marginal distribution is parametric.

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    • References(38)
  • [1]
    Sklar M. Fonctions de repartition an dimensions et leurs marges. Publications de l’Institut de Statistique de L’Université de Paris, 1959, 8: 229–231.
    [2]
    Nelsen R B. An Introduction to Copulas. New York: Springer, 1999: 414–422.
    [3]
    Whelan N. Sampling from Archimedean copulas. Quantitative Finance, 2004, 4 (3): 339. doi: 10.1088/1469-7688/4/3/009
    [4]
    Bollerslev T. Generalized autoregressive conditional hetero-skedasticity. Journal of Econometrics, 1986, 31: 307–327. doi: 10.1016/0304-4076(86)90063-1
    [5]
    Liu Y, Luger R. Efficient estimation of copula-GARCH models. Computational Statistics & Data Analysis, 2009, 53 (6): 2284–2297. doi: 10.1016/j.csda.2008.01.018
    [6]
    Ghorbel A, Hamma W, Jarboui A. Dependence between oil and commodities markets using time-varying Archimedean copulas and effectiveness of hedging strategies. Journal of Applied Statistics, 2017, 44 (9): 1509–1542. doi: 10.1080/02664763.2016.1155107
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    Carvalho M D M, Sáfadi T. Risk analysis in the Brazilian stock market: copula-APARCH modeling for value-at-risk. Journal of Applied Statistics, 2022, 49: 1598–1610. doi: 10.1080/02664763.2020.1865883
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    Chen X, Fan Y. Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification. Journal of Econometrics, 2006, 135: 125–154. doi: 10.1016/j.jeconom.2005.07.027
    [9]
    Patton A J. Modelling asymmetric exchange rate dependence. International Economic Review, 2006, 47 (2): 527–556. doi: 10.1111/j.1468-2354.2006.00387.x
    [10]
    Giulio G, Ergün A T. Systemic risk measurement: Multivariate GARCH estimation of CoVaR. Journal of Banking and Finance, 2013, 37 (8): 3169–3180. doi: 10.1016/j.jbankfin.2013.02.027
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    Joe H. Multivariate Models and Multivariate Dependence Concepts. London: Chapman & Hall, 1997.
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    [13]
    Beare B K. Copulas and temporal dependence. Econometrica, 2010, 78: 395–410. doi: 10.3982/ECTA8152
    [14]
    Zhao Z, Shi P, Zhang Z. Modeling multivariate time series with copula-linked univariate D-vines. Journal of Business & Economic Statistics, 2021, 40 (2): 690–704. doi: 10.1080/07350015.2020.1859381
    [15]
    Joe H. Families of m-variate distributions with given margins and m(m–1)/2 bivariate dependence parameters. IMS Lecture Notes – Monograph Series, 1996, 28: 120–141. doi: 10.1214/lnms/1215452614
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    [17]
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    Bladt M, McNeil A J. Time series copula models using d-vines and v-transforms. Econometrics and Statistics, 2021, 24: 27–48. doi: 10.1016/j.ecosta.2021.07.004
    [20]
    Brechmann E C, Czado C. COPAR—multivariate time series modeling using the copula autoregressive model. Applied Stochastic Models in Business and Industry, 2015, 31: 495–514. doi: 10.1002/asmb.2043
    [21]
    Smith M S. Copula modelling of dependence in multivariate time series. International Journal of Forecasting, 2015, 31 (3): 815–833. doi: 10.1016/j.ijforecast.2014.04.003
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    Beare B K, Seo J. Vine copula specifications for stationary multivariate Markov chains. Journal of Time Series Analysis, 2015, 36: 228–246. doi: 10.1111/jtsa.12103
    [23]
    Nagler T, Krüger D, Min A. Stationary vine copula models for multivariate time series. Journal of Econometrics, 2022, 277 (2): 305–324. doi: 10.1016/j.jeconom.2021.11.015
    [24]
    Engle R F. Dynamic conditional correlation. Journal of Business and Economic Statistics, 2002, 20: 339–350. doi: 10.1198/073500102288618487
    [25]
    Fioruci J A, Ehlers R S, Andrade M G. Bayesian multivariate GARCH models with dynamic correlations and asymmetric error distributions. Journal of Applied Statistics, 2014, 41 (2): 320–331. doi: 10.1080/02664763.2013.839635
    [26]
    Gong J, Li Y, Peng L, et al. Estimation of extreme quantiles for functions of dependent random variables. Journal of the Royal Statistical Society: Series B, 2015, 77 (5): 1001–1024. doi: 10.1111/rssb.12103
    [27]
    Adrian T, Brunnermeier M K. CoVaR. American Economic Review, 2016, 106: 1705–1741. doi: 10.1257/aer.20120555
    [28]
    Bianchi M L, De Luca G, Rivieccio G. Non-Gaussian models for CoVaR estimation. International Journal of Forecasting, 2023, 39 (1): 391–404. doi: 10.1016/j.ijforecast.2021.12.002
    [29]
    Creal D, Koopman S J, Lucas A. Generalized autoregressive score models with applications. Journal of Applied Econometrics, 2013, 28 (5): 777–795. doi: 10.1002/jae.1279
    [30]
    Joe H. Asymptotic efficiency of the two-stage estimation method for copula-based models. Journal of Multivariate Analysis, 2005, 94 (2): 401–419. doi: 10.1016/j.jmva.2004.06.003
    [31]
    Tsukahara H. Semiparametric estimation in copula model. Canadian Journal of Statistics, 2005, 33 (3): 357–375. doi: 10.1002/cjs.5540330304
    [32]
    Qiang J, Liu B Y, Ying F. Risk dependence of CoVaR and structural change between oil prices and exchange rates: A time-varying copula model. Energy Economics, 2019, 77: 80–92. doi: 10.1016/j.eneco.2018.07.012
    [33]
    Nelson D B. Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 1991, 59 (2): 347–370. doi: 10.2307/2938260
    [34]
    Glosten L R, Jagannathan R, Runkle D E. On the relation between the expected value and the volatility of the nominal excess return on stocks. The Journal of Finance, 1993, 48 (5): 1779–1801. doi: 10.1111/j.1540-6261.1993.tb05128.x
    [35]
    Kupiec P H. Techniques for verifying the accuracy of risk measurement models. The Journal of Derivatives, 1995, 95 (24): 73–84. doi: 10.3905/jod.1995.407942
    [36]
    Christoffersen P F. Evaluating interval forecasts. International Economic Review, 1998, 39 (4): 841–862. doi: 10.2307/2527341
    [37]
    Vaart A W, Wellner J A. Weak Convergence and Empirical Processes. New York: Springer, 1996: 16–28.
    [38]
    Doukhan P, Massart P, Rio E. Invariance principles for absolutely regular empirical processes. Annales de l’Institut Henri Poincar, 1995, 31 (2): 393–427.
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    Figure  1.  The time-varying average correlation among the financial companies and system.

    Figure  2.  The VaR of AMAZ, BA, COCA, GM, IBM, and SP500 at level $ p = 0.05 $, respectively.

    Figure  3.  The CoVaR of SP500 conditional on the financial companies AMAZ, BA, COCA, GM, and IBM being in financial distress.

    Figure  4.  The $ \Delta \text{CoVaR} $ of SP500 conditional on the financial companies AMAZ, BA, COCA, GM, and IBM.

    [1]
    Sklar M. Fonctions de repartition an dimensions et leurs marges. Publications de l’Institut de Statistique de L’Université de Paris, 1959, 8: 229–231.
    [2]
    Nelsen R B. An Introduction to Copulas. New York: Springer, 1999: 414–422.
    [3]
    Whelan N. Sampling from Archimedean copulas. Quantitative Finance, 2004, 4 (3): 339. doi: 10.1088/1469-7688/4/3/009
    [4]
    Bollerslev T. Generalized autoregressive conditional hetero-skedasticity. Journal of Econometrics, 1986, 31: 307–327. doi: 10.1016/0304-4076(86)90063-1
    [5]
    Liu Y, Luger R. Efficient estimation of copula-GARCH models. Computational Statistics & Data Analysis, 2009, 53 (6): 2284–2297. doi: 10.1016/j.csda.2008.01.018
    [6]
    Ghorbel A, Hamma W, Jarboui A. Dependence between oil and commodities markets using time-varying Archimedean copulas and effectiveness of hedging strategies. Journal of Applied Statistics, 2017, 44 (9): 1509–1542. doi: 10.1080/02664763.2016.1155107
    [7]
    Carvalho M D M, Sáfadi T. Risk analysis in the Brazilian stock market: copula-APARCH modeling for value-at-risk. Journal of Applied Statistics, 2022, 49: 1598–1610. doi: 10.1080/02664763.2020.1865883
    [8]
    Chen X, Fan Y. Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification. Journal of Econometrics, 2006, 135: 125–154. doi: 10.1016/j.jeconom.2005.07.027
    [9]
    Patton A J. Modelling asymmetric exchange rate dependence. International Economic Review, 2006, 47 (2): 527–556. doi: 10.1111/j.1468-2354.2006.00387.x
    [10]
    Giulio G, Ergün A T. Systemic risk measurement: Multivariate GARCH estimation of CoVaR. Journal of Banking and Finance, 2013, 37 (8): 3169–3180. doi: 10.1016/j.jbankfin.2013.02.027
    [11]
    Joe H. Multivariate Models and Multivariate Dependence Concepts. London: Chapman & Hall, 1997.
    [12]
    Chen X, Fan Y. Estimation of copula-based semiparametric time series models. Journal of Econometrics, 2006, 130 (2): 307–335. doi: 10.1016/j.jeconom.2005.03.004
    [13]
    Beare B K. Copulas and temporal dependence. Econometrica, 2010, 78: 395–410. doi: 10.3982/ECTA8152
    [14]
    Zhao Z, Shi P, Zhang Z. Modeling multivariate time series with copula-linked univariate D-vines. Journal of Business & Economic Statistics, 2021, 40 (2): 690–704. doi: 10.1080/07350015.2020.1859381
    [15]
    Joe H. Families of m-variate distributions with given margins and m(m–1)/2 bivariate dependence parameters. IMS Lecture Notes – Monograph Series, 1996, 28: 120–141. doi: 10.1214/lnms/1215452614
    [16]
    Bedford T J, Cooke R M. Monte Carlo simulation of vine dependent random variables for applications in uncertainty analysis. In: Proceedings ESREL 2001. Torino: Politecnico di Torino, 2001: 863–870.
    [17]
    Bedford T J, Cooke R M. Probability density decomposition for conditionally dependent random variables modeled by vines. Annals of Mathematics and Artificial Intelligence, 2001, 32: 245–268. doi: 10.1023/A:1016725902970
    [18]
    Aas K, Czado C, Frigessi A, et al. Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics, 2009, 44: 182–198. doi: 10.1016/j.insmatheco.2007.02.001
    [19]
    Bladt M, McNeil A J. Time series copula models using d-vines and v-transforms. Econometrics and Statistics, 2021, 24: 27–48. doi: 10.1016/j.ecosta.2021.07.004
    [20]
    Brechmann E C, Czado C. COPAR—multivariate time series modeling using the copula autoregressive model. Applied Stochastic Models in Business and Industry, 2015, 31: 495–514. doi: 10.1002/asmb.2043
    [21]
    Smith M S. Copula modelling of dependence in multivariate time series. International Journal of Forecasting, 2015, 31 (3): 815–833. doi: 10.1016/j.ijforecast.2014.04.003
    [22]
    Beare B K, Seo J. Vine copula specifications for stationary multivariate Markov chains. Journal of Time Series Analysis, 2015, 36: 228–246. doi: 10.1111/jtsa.12103
    [23]
    Nagler T, Krüger D, Min A. Stationary vine copula models for multivariate time series. Journal of Econometrics, 2022, 277 (2): 305–324. doi: 10.1016/j.jeconom.2021.11.015
    [24]
    Engle R F. Dynamic conditional correlation. Journal of Business and Economic Statistics, 2002, 20: 339–350. doi: 10.1198/073500102288618487
    [25]
    Fioruci J A, Ehlers R S, Andrade M G. Bayesian multivariate GARCH models with dynamic correlations and asymmetric error distributions. Journal of Applied Statistics, 2014, 41 (2): 320–331. doi: 10.1080/02664763.2013.839635
    [26]
    Gong J, Li Y, Peng L, et al. Estimation of extreme quantiles for functions of dependent random variables. Journal of the Royal Statistical Society: Series B, 2015, 77 (5): 1001–1024. doi: 10.1111/rssb.12103
    [27]
    Adrian T, Brunnermeier M K. CoVaR. American Economic Review, 2016, 106: 1705–1741. doi: 10.1257/aer.20120555
    [28]
    Bianchi M L, De Luca G, Rivieccio G. Non-Gaussian models for CoVaR estimation. International Journal of Forecasting, 2023, 39 (1): 391–404. doi: 10.1016/j.ijforecast.2021.12.002
    [29]
    Creal D, Koopman S J, Lucas A. Generalized autoregressive score models with applications. Journal of Applied Econometrics, 2013, 28 (5): 777–795. doi: 10.1002/jae.1279
    [30]
    Joe H. Asymptotic efficiency of the two-stage estimation method for copula-based models. Journal of Multivariate Analysis, 2005, 94 (2): 401–419. doi: 10.1016/j.jmva.2004.06.003
    [31]
    Tsukahara H. Semiparametric estimation in copula model. Canadian Journal of Statistics, 2005, 33 (3): 357–375. doi: 10.1002/cjs.5540330304
    [32]
    Qiang J, Liu B Y, Ying F. Risk dependence of CoVaR and structural change between oil prices and exchange rates: A time-varying copula model. Energy Economics, 2019, 77: 80–92. doi: 10.1016/j.eneco.2018.07.012
    [33]
    Nelson D B. Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 1991, 59 (2): 347–370. doi: 10.2307/2938260
    [34]
    Glosten L R, Jagannathan R, Runkle D E. On the relation between the expected value and the volatility of the nominal excess return on stocks. The Journal of Finance, 1993, 48 (5): 1779–1801. doi: 10.1111/j.1540-6261.1993.tb05128.x
    [35]
    Kupiec P H. Techniques for verifying the accuracy of risk measurement models. The Journal of Derivatives, 1995, 95 (24): 73–84. doi: 10.3905/jod.1995.407942
    [36]
    Christoffersen P F. Evaluating interval forecasts. International Economic Review, 1998, 39 (4): 841–862. doi: 10.2307/2527341
    [37]
    Vaart A W, Wellner J A. Weak Convergence and Empirical Processes. New York: Springer, 1996: 16–28.
    [38]
    Doukhan P, Massart P, Rio E. Invariance principles for absolutely regular empirical processes. Annales de l’Institut Henri Poincar, 1995, 31 (2): 393–427.
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