IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v34y2021i4d10.1007_s10959-020-01019-8.html
   My bibliography  Save this article

Tail indices for $$\mathbf{A}\mathbf{X}+\mathbf{B}$$ A X + B Recursion with Triangular Matrices

Author

Listed:
  • Muneya Matsui

    (Nanzan University)

  • Witold Świątkowski

    (University of Wrocław)

Abstract

We study multivariate stochastic recurrence equations (SREs) with triangular matrices. If coefficient matrices of SREs have strictly positive entries, the classical Kesten result says that the stationary solution is regularly varying and the tail indices are the same in all directions. This framework, however, is too restrictive for applications. In order to widen applicability of SREs, we consider SREs with triangular matrices and we prove that their stationary solutions are regularly varying with component-wise different tail exponents. Several applications to GARCH models are suggested.

Suggested Citation

  • Muneya Matsui & Witold Świątkowski, 2021. "Tail indices for $$\mathbf{A}\mathbf{X}+\mathbf{B}$$ A X + B Recursion with Triangular Matrices," Journal of Theoretical Probability, Springer, vol. 34(4), pages 1831-1869, December.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:4:d:10.1007_s10959-020-01019-8
    DOI: 10.1007/s10959-020-01019-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-020-01019-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-020-01019-8?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Jeantheau, Thierry, 1998. "Strong Consistency Of Estimators For Multivariate Arch Models," Econometric Theory, Cambridge University Press, vol. 14(1), pages 70-86, February.
    2. 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.
    3. Fernández, Begoña & Muriel, Nelson, 2009. "Regular variation and related results for the multivariate GARCH(p,q) model with constant conditional correlations," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1538-1550, August.
    4. Sun, Pengfei & Zhou, Chen, 2014. "Diagnosing the distribution of GARCH innovations," Journal of Empirical Finance, Elsevier, vol. 29(C), pages 287-303.
    Full references (including those not matched with items on IDEAS)

    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. Pedersen, Rasmus Søndergaard, 2016. "Targeting Estimation Of Ccc-Garch Models With Infinite Fourth Moments," Econometric Theory, Cambridge University Press, vol. 32(2), pages 498-531, April.
    2. Markus Haas, 2018. "A note on the absolute moments of the bivariate normal distribution," Economics Bulletin, AccessEcon, vol. 38(1), pages 650-656.
    3. Nakatani, Tomoaki & Teräsvirta, Timo, 2008. "Positivity constraints on the conditional variances in the family of conditional correlation GARCH models," Finance Research Letters, Elsevier, vol. 5(2), pages 88-95, June.
    4. Chia-Lin Chang & Yiying Li & Michael McAleer, 2018. "Volatility Spillovers between Energy and Agricultural Markets: A Critical Appraisal of Theory and Practice," Energies, MDPI, vol. 11(6), pages 1-19, June.
    5. Resende, Paulo Angelo Alves & Dorea, Chang Chung Yu, 2016. "Model identification using the Efficient Determination Criterion," Journal of Multivariate Analysis, Elsevier, vol. 150(C), pages 229-244.
    6. Hakim, Abdul & McAleer, Michael, 2009. "Forecasting conditional correlations in stock, bond and foreign exchange markets," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(9), pages 2830-2846.
    7. Luc Bauwens & Sébastien Laurent & Jeroen V. K. Rombouts, 2006. "Multivariate GARCH models: a survey," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 21(1), pages 79-109, January.
    8. Massimiliano Caporin & Michael McAleer, 2010. "Ranking Multivariate GARCH Models by Problem Dimension," CARF F-Series CARF-F-219, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    9. Norbert Henze & María Dolores Jiménez-Gamero, 2019. "A new class of tests for multinormality with i.i.d. and garch data based on the empirical moment generating function," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 499-521, June.
    10. Jirak, Moritz, 2012. "Change-point analysis in increasing dimension," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 136-159.
    11. Paolella, Marc S. & Polak, Paweł, 2015. "ALRIGHT: Asymmetric LaRge-scale (I)GARCH with Hetero-Tails," International Review of Economics & Finance, Elsevier, vol. 40(C), pages 282-297.
    12. Christian M. Hafner & Helmut Herwartz, 2009. "Testing for linear vector autoregressive dynamics under multivariate generalized autoregressive heteroskedasticity," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 63(3), pages 294-323, August.
    13. da Veiga, Bernardo & Chan, Felix & McAleer, Michael, 2008. "Evaluating the impact of market reforms on Value-at-Risk forecasts of Chinese A and B shares," Pacific-Basin Finance Journal, Elsevier, vol. 16(4), pages 453-475, September.
    14. Broto, Carmen & Lamas, Matías, 2020. "Is market liquidity less resilient after the financial crisis? Evidence for US Treasuries," Economic Modelling, Elsevier, vol. 93(C), pages 217-229.
    15. Boubacar Maïnassara, Y. & Kadmiri, O. & Saussereau, B., 2022. "Estimation of multivariate asymmetric power GARCH models," Journal of Multivariate Analysis, Elsevier, vol. 192(C).
    16. Chang, Chia-Lin & McAleer, Michael & Wang, Yanghuiting, 2018. "Testing Co-Volatility spillovers for natural gas spot, futures and ETF spot using dynamic conditional covariances," Energy, Elsevier, vol. 151(C), pages 984-997.
    17. Chia-Lin Chang & Michael McAleer & Jiarong Tian, 2019. "Modeling and Testing Volatility Spillovers in Oil and Financial Markets for the USA, the UK, and China," Energies, MDPI, vol. 12(8), pages 1-24, April.
    18. Abdul Hakim & Michael McAleer, 2009. "VaR Forecasts and Dynamic Conditional Correlations for Spot and Futures Returns on Stocks and Bonds," CARF F-Series CARF-F-178, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    19. Chia-Lin Chang & Michael McAleer & Guangdong Zuo, 2017. "Volatility Spillovers and Causality of Carbon Emissions, Oil and Coal Spot and Futures for the EU and USA," Sustainability, MDPI, vol. 9(10), pages 1-22, October.
    20. Hoti, Suhejla, 2005. "Comparative analysis of risk ratings for the East European region," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 68(5), pages 449-462.

    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:spr:jotpro:v:34:y:2021:i:4:d:10.1007_s10959-020-01019-8. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.