Skip to main content
SpringerLink
Log in

Temporal Models and Their Applications

  • Chapter
  • First Online:
Hierarchical Archimedean Copulas

Part of the book series: SpringerBriefs in Applied Statistics and Econometrics ((SBASE))

  • 33 Accesses

Abstract

Modeling temporal dependency holds significant importance across diverse domains like finance, economics, and resource allocation. By capturing the intricate relationships between past and future observations, these models not only enhance prediction accuracy but also play a crucial role in effective risk management and optimal resource allocation. In finance, understanding temporal dependencies is essential for making informed investment decisions and quantifying market risks, while in economics, it aids in designing policies that consider the impact of historical factors on future outcomes.

Over time, the HAC has gained attraction in economics and finance, particularly for its efficacy in handling clustered variables and capturing implied correlations. Its prowess in modeling Value-at-Risk across Basel regulatory phases and its role in understanding collateralized debt obligations during the 2008 financial crisis have bolstered its significance. The copula’s reputation, previously challenged by media, has been reinstated through subsequent research. Current financial trends, like high-frequency trading, have opened avenues for accurate joint distribution modeling of daily returns using HAC. These three financial applications are discussed in this chapter.

While running these empirical studies, researchers have noticed that contrary to other models, in HAC not only the parameters but also the structure are changing. This observation led to another series of papers investigating exactly this issue. This chapter discusses in detail two temporal HAC models and several applications of the HAC.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Amato, J., & Gyntelberg, J. (2005). CDS index tranches and the pricing of credit risk correlations. BIS Quarterly Review, 1, 73–87.

    Google Scholar 

  • Andersen, L., & Sidenius, J. (2005). Extensions of the Gaussian copula. Journal of Credit Risk, 1, 29–70.

    Article  Google Scholar 

  • Andersen, T.G., Bollerslev, T., Diebold, F.X., & Ebens, H. (2001a). The distribution of realized stock return volatility. Journal of Financial Economics, 61, 43–76.

    Article  Google Scholar 

  • Andersen, T.G., Bollerslev, T., Diebold, F.X., & Labys, P. (2001b). The distribution of realized exchange rate volatility. Journal of the American Statistical Association, 96, 42–55.

    Article  MathSciNet  Google Scholar 

  • Bickel, P.J., Ritov, Y., & Rydén, T. (1998). Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models. Annals of Statistics, 26(4), 1614–1635.

    Article  MathSciNet  Google Scholar 

  • Breymann, W., Dias, A., & Embrechts, P. (2003). Dependence structures for multivariate high-frequency data in finance. Quantitative Finance, 1, 1–14.

    Article  MathSciNet  Google Scholar 

  • Cappé, O., Moulines, E., & Rydén, T. (2005). Inference in hidden Markov models (Springer series in statistics). Springer.

    Book  Google Scholar 

  • Chen, Y., Härdle, W., & Jeong, S.-O. (2008). Nonparametric risk management with generalized hyperbolic distributions. Journal of the American Statistical Association, 103(483), 910–923.

    Article  MathSciNet  Google Scholar 

  • Choroś-Tomczyk, B., Härdle, W.K., & Okhrin, O. (2013). Valuation of collateralized debt obligations with hierarchical Archimedean copulae. Journal of Empirical Finance, 24, 42–62.

    Article  Google Scholar 

  • Corsi, F. (2009). A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics, 7(2), 174–196.

    Article  Google Scholar 

  • Corsi, F., Audrino, F., & Renò, R. (2012). HAR modeling for realized volatility forecasting. In L. Bauwens, C. Hafner, & S. Laurent (Eds.), Handbook of volatility models and their applications (chapter 15, pp. 363–382). Wiley.

    Google Scholar 

  • Dempster, A.P., Laird, N.M., & Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society: Series B (Methodological), 39(1), 1–22.

    Article  MathSciNet  Google Scholar 

  • Dias, A., & Embrechts, P. (2004). Dynamic copula models for multivariate high-frequency data in finance. Working paper, ETH Zürich.

    Google Scholar 

  • Engle, R.F., & Kroner, K.F. (1995). Multivariate simultaneous generalized ARCH. Econometric Theory, 11(1), 122–150.

    Article  MathSciNet  Google Scholar 

  • Fengler, M.R., & Okhrin, O. (2016). Managing risk with a realized copula parameter. Computational Statistics & Data Analysis, 100, 131–152.

    Article  MathSciNet  Google Scholar 

  • Gao, X., & Song, P.X.-K. (2011). Composite likelihood EM algorithm with applications to multivariate hidden Markov model. Statistica Sinica, 21, 165–185.

    MathSciNet  Google Scholar 

  • Giacomini, E., Härdle, W.K., & Spokoiny, V. (2009). Inhomogeneous dependence modeling with time-varying copulae. Journal of Business and Economic Statistics, 27(2), 224–234.

    Article  MathSciNet  Google Scholar 

  • Glasserman, P., & Suchintabandid, S. (2007). Correlation expansions for cdo pricing. Journal of Banking and Finance, 5, 1375–1398.

    Article  Google Scholar 

  • Górecki, J., Hofert, M., & Okhrin, O. (2021). Outer power transformations of hierarchical Archimedean copulas: Construction, sampling and estimation. Computational Statistics & Data Analysis, 155, 107109.

    Article  MathSciNet  Google Scholar 

  • Gregory, J., & Laurent, J.-P. (2004). In the core of correlation. RISK, 10, 87–91.

    Google Scholar 

  • Härdle, W.K., Okhrin, O., & Okhrin, Y. (2013). Dynamic structured copula models. Statistics and Risk Modeling, 30, 361–388.

    Article  MathSciNet  Google Scholar 

  • Härdle, W.K., Okhrin, O., & Wang, W. (2015). Hidden Markov structures for dynamic copulae. Econometric Theory, 31(5), 1–35.

    Article  MathSciNet  Google Scholar 

  • Hofert, M., & Scherer, M. (2011). CDO pricing with nested Archimedean copulas. Quantitative Finance, 11(5), 775–787.

    Article  MathSciNet  Google Scholar 

  • Jondeau, E., & Rockinger, M. (2006). The copula-GARCH model of conditional dependencies: An international stock market application. Journal of International Money and Finance, 25(5), 827–853.

    Article  Google Scholar 

  • Lamoureux, C.G., & Lastrapes, W.D. (1990). Persistence-in-variance, structural change and the GARCH model. Journal of Business and Economic Statistics, 8, 225–234.

    Article  Google Scholar 

  • Leroux, B.G. (1992). Maximum-likelihood estimation for hidden Markov models. Stochastic Processes and Their Applications, 40, 127–143.

    Article  MathSciNet  Google Scholar 

  • Li, D.X. (1999). Creditmetrics monitor. Riskmetrics.

    Google Scholar 

  • Li, D.X. (2000). On default correlation: A copula function approach. The Journal of Fixed Income, 4, 43–54.

    Article  Google Scholar 

  • Mercurio, D., & Spokoiny, V. (2004). Statistical inference for time-inhomogeneous volatility models. Annals of Statistics, 32(2), 577–602.

    Article  MathSciNet  Google Scholar 

  • Okhrin, O., Okhrin, Y., & Schmid, W. (2013b). On the structure and estimation of hierarchical Archimedean copulas. Journal of Econometrics, 173(2), 189–204.

    Article  MathSciNet  Google Scholar 

  • Okhrin, O., & Tetereva, A. (2017). The realized hierarchical Archimedean copula in risk modelling. Econometrics, 5(2), 26.

    Article  Google Scholar 

  • Okhrin, O., & Xu, Y.F. (2017). A comparison study of pricing credit default swap index tranches with convex combination of copulae. North Americal Journal of Economics and Finance, Vol. 42, 193–217.

    Article  Google Scholar 

  • Patton, A.J. (2004). On the out-of-sample importance of skewness and asymmetric dependence for asset allocation. Journal of Financial Econometrics, 2,130–168.

    Article  Google Scholar 

  • Patton, A.J. (2006). Modeling asymmetric exchange rate dependence. International Economic Review, 47(2), 527–556.

    Article  MathSciNet  Google Scholar 

  • Rodriguez, J.C. (2007). Measuring financial contagion: A copula approach. Journal of Empirical Finance, 14, 401–423.

    Article  Google Scholar 

  • Salmon, F. (2009). Recipe for disaster: The formula that killed wall street. Wired. https://www.wired.com/2009/02/wp-quant/

  • Silvennoinen, A., & Teräsvirta, T. (2009). Multivariate GARCH models. In T.G. Andersen, R.A. Davis, J.-P. Kreiß, & T. Mikosch (Eds.), Handbook of financial time series (pp. 201–233). Springer.

    Google Scholar 

  • Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de statistique de l’Université de Paris, 8, 229–231.

    MathSciNet  Google Scholar 

  • Čížek, P., Härdle, W., & Spokoiny, V. (2009). Adaptive pointwise estimation in time-inhomogeneous conditional heteroscedasticity models. Econometrics Journal, 12(2), 248–271.

    Article  MathSciNet  Google Scholar 

  • Zhu, W., Wang, C.-W., & Tan, K.S. (2016). Structure and estimation of Lévy subordinated hierarchical Archimedean copulas (LSHAC): Theory and empirical tests. Journal of Banking & Finance, 69, 20–36.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Informatics and Mathematics, Silesian University in Opava, Karviná, Czech Republic

    Jan Górecki

  2. Chair of Econometrics and Statistics, esp. Transportation, Institute of Transport and Economics, Technische Universität Dresden, Dresden, Germany

    Ostap Okhrin

Authors
  1. Jan Górecki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ostap Okhrin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Górecki, J., Okhrin, O. (2024). Temporal Models and Their Applications. In: Hierarchical Archimedean Copulas. SpringerBriefs in Applied Statistics and Econometrics. Springer, Cham. https://doi.org/10.1007/978-3-031-56337-9_7

Download citation

Publish with us

Policies and ethics

Search

Navigation