IDEAS home Printed from https://ideas.repec.org/a/wly/emetrp/v89y2021i4p1699-1715.html
   My bibliography  Save this article

A New Parametrization of Correlation Matrices

Author

Listed:
  • Ilya Archakov
  • Peter Reinhard Hansen

Abstract

We introduce a novel parametrization of the correlation matrix. The reparametrization facilitates modeling of correlation and covariance matrices by an unrestricted vector, where positive definiteness is an innate property. This parametrization can be viewed as a generalization of Fisher's Z‐transformation to higher dimensions and has a wide range of potential applications. An algorithm for reconstructing the unique n × n correlation matrix from any vector in Rn(n−1)/2 is provided, and we derive its numerical complexity.

Suggested Citation

  • Ilya Archakov & Peter Reinhard Hansen, 2021. "A New Parametrization of Correlation Matrices," Econometrica, Econometric Society, vol. 89(4), pages 1699-1715, July.
    • Handle: RePEc:wly:emetrp:v:89:y:2021:i:4:p:1699-1715
    DOI: 10.3982/ECTA16910
    as

    Download full text from publisher

    File URL: https://doi.org/10.3982/ECTA16910
    Download Restriction: no
    File URL: https://libkey.io/10.3982/ECTA16910?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Manabu Asai & Chia-Lin Chang & Michael McAleer, 2016. "Realized Matrix-Exponential Stochastic Volatility with Asymmetry, Long Memory and Spillovers," Documentos de Trabajo del ICAE 2016-15, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico.
    2. Bauer, Gregory H. & Vorkink, Keith, 2011. "Forecasting multivariate realized stock market volatility," Journal of Econometrics, Elsevier, vol. 160(1), pages 93-101, January.
    3. Ishihara, Tsunehiro & Omori, Yasuhiro & Asai, Manabu, 2016. "Matrix exponential stochastic volatility with cross leverage," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 331-350.
    4. repec:taf:jnlbes:v:30:y:2012:i:2:p:212-228 is not listed on IDEAS
    5. Peter Reinhard Hansen & Asger Lunde & Valeri Voev, 2014. "Realized Beta Garch: A Multivariate Garch Model With Realized Measures Of Volatility," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 29(5), pages 774-799, August.
    6. BAUWENS, Luc & STORTI, Giuseppe & VIOLANTE, Francesco, 2012. "Dynamic conditional correlation models for realized covariance matrices," LIDAM Discussion Papers CORE 2012060, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. Asai Manabu & So Mike K.P., 2015. "Long Memory and Asymmetry for Matrix-Exponential Dynamic Correlation Processes," Journal of Time Series Econometrics, De Gruyter, vol. 7(1), pages 69-94, January.
    8. Roxana Chiriac & Valeri Voev, 2011. "Modelling and forecasting multivariate realized volatility," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 26(6), pages 922-947, September.
    9. Kawakatsu, Hiroyuki, 2006. "Matrix exponential GARCH," Journal of Econometrics, Elsevier, vol. 134(1), pages 95-128, September.
    10. Linton, Oliver & McCrorie, J. Roderick, 1995. "Differentiation of an Exponential Matrix Function," Econometric Theory, Cambridge University Press, vol. 11(05), pages 1182-1185, October.
    11. Qianqiu Liu, 2009. "On portfolio optimization: How and when do we benefit from high-frequency data?," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 24(4), pages 560-582.
    12. Diaa Noureldin & Neil Shephard & Kevin Sheppard, 2012. "Multivariate high‐frequency‐based volatility (HEAVY) models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 27(6), pages 907-933, September.
    13. Golosnoy, Vasyl & Gribisch, Bastian & Liesenfeld, Roman, 2012. "The conditional autoregressive Wishart model for multivariate stock market volatility," Journal of Econometrics, Elsevier, vol. 167(1), pages 211-223.
    14. Ilya Archakov & Peter Reinhard Hansen & Asger Lunde, 2020. "A Multivariate Realized GARCH Model," Papers 2012.02708, arXiv.org, revised May 2024.
    15. Bollerslev, Tim, 1990. "Modelling the Coherence in Short-run Nominal Exchange Rates: A Multivariate Generalized ARCH Model," The Review of Economics and Statistics, MIT Press, vol. 72(3), pages 498-505, August.
    16. Drew Creal & Siem Jan Koopman & André Lucas, 2013. "Generalized Autoregressive Score Models With Applications," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 28(5), pages 777-795, August.
    17. Gregory Bauer & Keith Vorkink, 2007. "Multivariate Realized Stock Market Volatility," Staff Working Papers 07-20, Bank of Canada.
    18. Roland Weigand, 2014. "Matrix Box-Cox Models for Multivariate Realized Volatility," Working Papers 144, Bavarian Graduate Program in Economics (BGPE).
    19. Tse, Y K & Tsui, Albert K C, 2002. "A Multivariate Generalized Autoregressive Conditional Heteroscedasticity Model with Time-Varying Correlations," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(3), pages 351-362, July.
    20. Engle, Robert, 2002. "Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(3), pages 339-350, July.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. K. B. Gubbels & J. Y. Ypma & C. W. Oosterlee, 2023. "Principal Component Copulas for Capital Modelling and Systemic Risk," Papers 2312.13195, arXiv.org, revised Dec 2024.
    2. Ilya Archakov & Peter Reinhard Hansen & Yiyao Luo, 2024. "A new method for generating random correlation matrices," The Econometrics Journal, Royal Economic Society, vol. 27(2), pages 188-212.
    3. Chen Tong & Peter Reinhard Hansen & Ilya Archakov, 2024. "Cluster GARCH," Papers 2406.06860, arXiv.org.
    4. Ilya Archakov & Peter Reinhard Hansen, 2024. "A Canonical Representation of Block Matrices with Applications to Covariance and Correlation Matrices," The Review of Economics and Statistics, MIT Press, vol. 106(4), pages 1099-1113, July.
    5. Hafner, Christian M. & Wang, Linqi, 2023. "A dynamic conditional score model for the log correlation matrix," Journal of Econometrics, Elsevier, vol. 237(2).
    6. Ayed Alwadain & Rao Faizan Ali & Amgad Muneer, 2023. "Estimating Financial Fraud through Transaction-Level Features and Machine Learning," Mathematics, MDPI, vol. 11(5), pages 1-15, February.
    7. Arias, Jonas E. & Rubio-Ramírez, Juan F. & Shin, Minchul, 2023. "Macroeconomic forecasting and variable ordering in multivariate stochastic volatility models," Journal of Econometrics, Elsevier, vol. 235(2), pages 1054-1086.
    8. Tong, Chen & Hansen, Peter Reinhard, 2023. "Characterizing correlation matrices that admit a clustered factor representation," Economics Letters, Elsevier, vol. 233(C).
    9. Jean-Claude Hessing & Rutger-Jan Lange & Daniel Ralph, 2022. "This article establishes the Poisson optional stopping times (POST) method by Lange et al. (2020) as a near-universal method for solving liquidity-constrained American options, or, equivalently, penal," Tinbergen Institute Discussion Papers 22-007/IV, Tinbergen Institute.
    10. Ilya Archakov & Peter Reinhard Hansen & Asger Lunde, 2020. "A Multivariate Realized GARCH Model," Papers 2012.02708, arXiv.org, revised May 2024.
    11. Joshua Chan, 2023. "BVARs and Stochastic Volatility," Papers 2310.14438, arXiv.org.
    12. D’Innocenzo, Enzo & Lucas, Andre, 2024. "Dynamic partial correlation models," Journal of Econometrics, Elsevier, vol. 241(2).
    13. Dilip B. Madan & King Wang, 2022. "Two sided efficient frontiers at multiple time horizons," Annals of Finance, Springer, vol. 18(3), pages 327-353, September.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ilya Archakov & Peter Reinhard Hansen & Asger Lunde, 2020. "A Multivariate Realized GARCH Model," Papers 2012.02708, arXiv.org, revised May 2024.
    2. Andersen, Torben G. & Bollerslev, Tim & Christoffersen, Peter F. & Diebold, Francis X., 2013. "Financial Risk Measurement for Financial Risk Management," Handbook of the Economics of Finance, in: G.M. Constantinides & M. Harris & R. M. Stulz (ed.), Handbook of the Economics of Finance, volume 2, chapter 0, pages 1127-1220, Elsevier.
    3. Asai, Manabu & McAleer, Michael, 2015. "Forecasting co-volatilities via factor models with asymmetry and long memory in realized covariance," Journal of Econometrics, Elsevier, vol. 189(2), pages 251-262.
    4. Ostap Okhrin & Anastasija Tetereva, 2017. "The Realized Hierarchical Archimedean Copula in Risk Modelling," Econometrics, MDPI, vol. 5(2), pages 1-31, June.
    5. Ishihara, Tsunehiro & Omori, Yasuhiro & Asai, Manabu, 2016. "Matrix exponential stochastic volatility with cross leverage," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 331-350.
    6. BAUWENS Luc, & XU Yongdeng,, 2019. "DCC-HEAVY: A multivariate GARCH model based on realized variances and correlations," LIDAM Discussion Papers CORE 2019025, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. Fengler, Matthias R. & Okhrin, Ostap, 2012. "Realized copula," SFB 649 Discussion Papers 2012-034, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    8. Roxana Halbleib & Valeri Voev, 2011. "Forecasting Covariance Matrices: A Mixed Frequency Approach," CREATES Research Papers 2011-03, Department of Economics and Business Economics, Aarhus University.
    9. Jin, Xin & Maheu, John M., 2016. "Bayesian semiparametric modeling of realized covariance matrices," Journal of Econometrics, Elsevier, vol. 192(1), pages 19-39.
    10. Roland Weigand, 2014. "Matrix Box-Cox Models for Multivariate Realized Volatility," Working Papers 144, Bavarian Graduate Program in Economics (BGPE).
    11. Bauwens, Luc & Braione, Manuela & Storti, Giuseppe, 2017. "A dynamic component model for forecasting high-dimensional realized covariance matrices," Econometrics and Statistics, Elsevier, vol. 1(C), pages 40-61.
    12. repec:hum:wpaper:sfb649dp2012-034 is not listed on IDEAS
    13. Manabu Asai & Chia-Lin Chang & Michael McAleer, 2016. "Realized Matrix-Exponential Stochastic Volatility with Asymmetry, Long Memory and Spillovers," Tinbergen Institute Discussion Papers 16-076/III, Tinbergen Institute.
    14. Asai, Manabu & Chang, Chia-Lin & McAleer, Michael, 2022. "Realized matrix-exponential stochastic volatility with asymmetry, long memory and higher-moment spillovers," Journal of Econometrics, Elsevier, vol. 227(1), pages 285-304.
    15. Dark, Jonathan, 2018. "Multivariate models with long memory dependence in conditional correlation and volatility," Journal of Empirical Finance, Elsevier, vol. 48(C), pages 162-180.
    16. Karmous, Aida & Boubaker, Heni & Belkacem, Lotfi, 2019. "A dynamic factor model with stylized facts to forecast volatility for an optimal portfolio," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 534(C).
    17. Roxana Halbleib & Valeri Voev, 2016. "Forecasting Covariance Matrices: A Mixed Approach," Journal of Financial Econometrics, Oxford University Press, vol. 14(2), pages 383-417.
    18. João F. Caldeira & Guilherme V. Moura & Francisco J. Nogales & André A. P. Santos, 2017. "Combining Multivariate Volatility Forecasts: An Economic-Based Approach," Journal of Financial Econometrics, Oxford University Press, vol. 15(2), pages 247-285.
    19. Fengler, Matthias R. & Okhrin, Ostap, 2016. "Managing risk with a realized copula parameter," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 131-152.
    20. Manabu Asai & Mike K. P. So, 2021. "Quasi‐maximum likelihood estimation of conditional autoregressive Wishart models," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(3), pages 271-294, May.
    21. Yu-Sheng Lai, 2018. "Dynamic hedging with futures: a copula-based GARCH model with high-frequency data," Review of Derivatives Research, Springer, vol. 21(3), pages 307-329, October.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wly:emetrp:v:89:y:2021:i:4:p:1699-1715. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://edirc.repec.org/data/essssea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.