IDEAS home Printed from https://ideas.repec.org/a/eee/econom/v221y2021i2p409-423.html
   My bibliography  Save this article

Testing high-dimensional covariance matrices under the elliptical distribution and beyond

Author

Listed:
  • Yang, Xinxin
  • Zheng, Xinghua
  • Chen, Jiaqi

Abstract

We develop tests for high-dimensional covariance matrices under a generalized elliptical model. Our tests are based on a central limit theorem for linear spectral statistics of the sample covariance matrix based on self-normalized observations. For testing sphericity, our tests neither assume specific parametric distributions nor involve the kurtosis of data. More generally, we can test against any non-negative definite matrix that can even be not invertible. As an interesting application, we illustrate in empirical studies that our tests can be used to test uncorrelatedness among idiosyncratic returns.

Suggested Citation

  • Yang, Xinxin & Zheng, Xinghua & Chen, Jiaqi, 2021. "Testing high-dimensional covariance matrices under the elliptical distribution and beyond," Journal of Econometrics, Elsevier, vol. 221(2), pages 409-423.
    • Handle: RePEc:eee:econom:v:221:y:2021:i:2:p:409-423
    DOI: 10.1016/j.jeconom.2020.05.017
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304407620302384
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jeconom.2020.05.017?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. Fama, Eugene F & French, Kenneth R, 1992. "The Cross-Section of Expected Stock Returns," Journal of Finance, American Finance Association, vol. 47(2), pages 427-465, June.
    2. John Y. Campbell & Martin Lettau & Burton G. Malkiel & Yexiao Xu, 2001. "Have Individual Stocks Become More Volatile? An Empirical Exploration of Idiosyncratic Risk," Journal of Finance, American Finance Association, vol. 56(1), pages 1-43, February.
    3. Jiti Gao & Xiao Han & Guangming Pan & Yanrong Yang, 2017. "High dimensional correlation matrices: the central limit theorem and its applications," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 677-693, June.
    4. Campbell, John Y. & Hentschel, Ludger, 1992. "No news is good news *1: An asymmetric model of changing volatility in stock returns," Journal of Financial Economics, Elsevier, vol. 31(3), pages 281-318, June.
    5. Aït-Sahalia, Yacine & Fan, Jianqing & Li, Yingying, 2013. "The leverage effect puzzle: Disentangling sources of bias at high frequency," Journal of Financial Economics, Elsevier, vol. 109(1), pages 224-249.
    6. Benoit Mandelbrot, 1967. "The Variation of Some Other Speculative Prices," The Journal of Business, University of Chicago Press, vol. 40, pages 393-393.
    7. Christina D. Wang & Per A. Mykland, 2014. "The Estimation of Leverage Effect With High-Frequency Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(505), pages 197-215, March.
    8. Ilze Kalnina & Dacheng Xiu, 2017. "Nonparametric Estimation of the Leverage Effect: A Trade-Off Between Robustness and Efficiency," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(517), pages 384-396, January.
    9. Roll, Richard & Ross, Stephen A, 1980. "An Empirical Investigation of the Arbitrage Pricing Theory," Journal of Finance, American Finance Association, vol. 35(5), pages 1073-1103, December.
    10. Singleton, J. Clay & Wingender, John, 1986. "Skewness Persistence in Common Stock Returns," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 21(3), pages 335-341, September.
    11. Wang, Cheng & Yang, Jing & Miao, Baiqi & Cao, Longbing, 2013. "Identity tests for high dimensional data using RMT," Journal of Multivariate Analysis, Elsevier, vol. 118(C), pages 128-137.
    12. repec:bla:jfinan:v:58:y:2003:i:3:p:975-1008 is not listed on IDEAS
    13. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    14. Weiming Li & Jianfeng Yao, 2018. "On structure testing for component covariance matrices of a high dimensional mixture," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(2), pages 293-318, March.
    15. Amit Goyal & Pedro Santa‐Clara, 2003. "Idiosyncratic Risk Matters!," Journal of Finance, American Finance Association, vol. 58(3), pages 975-1007, June.
    16. Chen, Song Xi & Zhang, Li-Xin & Zhong, Ping-Shou, 2010. "Tests for High-Dimensional Covariance Matrices," Journal of the American Statistical Association, American Statistical Association, vol. 105(490), pages 810-819.
    17. William F. Sharpe, 1964. "Capital Asset Prices: A Theory Of Market Equilibrium Under Conditions Of Risk," Journal of Finance, American Finance Association, vol. 19(3), pages 425-442, September.
    18. Peiro, Amado, 1999. "Skewness in financial returns," Journal of Banking & Finance, Elsevier, vol. 23(6), pages 847-862, June.
    19. Owen, Joel & Rabinovitch, Ramon, 1983. "On the Class of Elliptical Distributions and Their Applications to the Theory of Portfolio Choice," Journal of Finance, American Finance Association, vol. 38(3), pages 745-752, June.
    20. N. H. Bingham & Rudiger Kiesel, 2002. "Semi-parametric modelling in finance: theoretical foundations," Quantitative Finance, Taylor & Francis Journals, vol. 2(4), pages 241-250.
    21. Fan, Jianqing & Fan, Yingying & Lv, Jinchi, 2008. "High dimensional covariance matrix estimation using a factor model," Journal of Econometrics, Elsevier, vol. 147(1), pages 186-197, November.
    22. Birke, Melanie & Dette, Holger, 2005. "A note on testing the covariance matrix for large dimension," Statistics & Probability Letters, Elsevier, vol. 74(3), pages 281-289, October.
    23. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, 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. Seabrook, Isobel & Caccioli, Fabio & Aste, Tomaso, 2022. "Quantifying impact and response in markets using information filtering networks," LSE Research Online Documents on Economics 115308, London School of Economics and Political Science, LSE Library.
    2. Dörnemann, Nina, 2023. "Likelihood ratio tests under model misspecification in high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 193(C).

    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. Choi, Jaewon & Richardson, Matthew, 2016. "The volatility of a firm's assets and the leverage effect," Journal of Financial Economics, Elsevier, vol. 121(2), pages 254-277.
    2. Madhusudan Karmakar, 2007. "Asymmetric Volatility and Risk-return Relationship in the Indian Stock Market," South Asia Economic Journal, Institute of Policy Studies of Sri Lanka, vol. 8(1), pages 99-116, January.
    3. Dimson, Elroy & Mussavian, Massoud, 1999. "Three centuries of asset pricing," Journal of Banking & Finance, Elsevier, vol. 23(12), pages 1745-1769, December.
    4. Haque Mahfuzul & Hassan M. Kabir & Maroney Neal C & Sackley William H, 2004. "An Empirical Examination of Stability, Predictability, and Volatility of Middle Eastern and African Emerging Stock Markets," Review of Middle East Economics and Finance, De Gruyter, vol. 2(1), pages 18-41, April.
    5. Sebastien Valeyre, 2020. "Refined model of the covariance/correlation matrix between securities," Papers 2001.08911, arXiv.org.
    6. Xiafei Li & Chris Brooks & Joëlle Miffre, 2009. "The Value Premium and Time-Varying Volatility," Journal of Business Finance & Accounting, Wiley Blackwell, vol. 36(9-10), pages 1252-1272.
    7. Bodnar, Taras & Reiß, Markus, 2016. "Exact and asymptotic tests on a factor model in low and large dimensions with applications," Journal of Multivariate Analysis, Elsevier, vol. 150(C), pages 125-151.
    8. Xiafei Li & Chris Brooks & Joëlle Miffre, 2009. "The Value Premium and Time‐Varying Volatility," Journal of Business Finance & Accounting, Wiley Blackwell, vol. 36(9‐10), pages 1252-1272, November.
    9. Esther Eiling, 2013. "Industry-Specific Human Capital, Idiosyncratic Risk, and the Cross-Section of Expected Stock Returns," Journal of Finance, American Finance Association, vol. 68(1), pages 43-84, February.
    10. Gilles Boevi Koumou, 2020. "Diversification and portfolio theory: a review," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 34(3), pages 267-312, September.
    11. Campbell, John Y. & Giglio, Stefano & Polk, Christopher & Turley, Robert, 2018. "An intertemporal CAPM with stochastic volatility," Journal of Financial Economics, Elsevier, vol. 128(2), pages 207-233.
    12. Cathy W. S. Chen & Richard H. Gerlach & Ann M. H. Lin, 2011. "Multi-regime nonlinear capital asset pricing models," Quantitative Finance, Taylor & Francis Journals, vol. 11(9), pages 1421-1438, April.
    13. Fong, Wai Mun, 1997. "Robust beta estimation: Some empirical evidence," Review of Financial Economics, Elsevier, vol. 6(2), pages 167-186.
    14. Attiya Yasmeen Javid, 2000. "Alternative Capital Asset Pricing Models: A Review of Theory and Evidence," PIDE Research Report 2000:3, Pakistan Institute of Development Economics.
    15. Zura Kakushadze & Willie Yu, 2016. "Multifactor Risk Models and Heterotic CAPM," Papers 1602.04902, arXiv.org, revised Mar 2016.
    16. Samih Antoine Azar, 2013. "The Spurious Relation between Inflation Uncertainty and Stock Returns: Evidence from the U.S," Review of Economics & Finance, Better Advances Press, Canada, vol. 3, pages 99-109, November.
    17. Sebastien Valeyre & Sofiane Aboura & Denis Grebenkov, 2019. "The Reactive Beta Model," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 42(1), pages 71-113, March.
    18. Bin Liu & Amalia Di Iorio, 2016. "The pricing of idiosyncratic volatility: An Australian study," Australian Journal of Management, Australian School of Business, vol. 41(2), pages 353-375, May.
    19. Chris Brooks & Xiafei Li & Joelle Miffre, 2009. "Time Varying Volatility and the Cross-Section of Equity Returns Â," ICMA Centre Discussion Papers in Finance icma-dp2009-01, Henley Business School, University of Reading.
    20. Phoebe Koundouri & Nikolaos Kourogenis & Nikitas Pittis & Panagiotis Samartzis, 2015. "Factor Models as 'Explanatory Unifiers' versus 'Explanatory Ideals' of Empirical Regularities of Stock Returns," DEOS Working Papers 1507, Athens University of Economics and Business.

    More about this item

    Keywords

    Covariance matrix; High-dimension; Elliptical model; Linear spectral statistics; Central limit theorem;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C55 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Large Data Sets: Modeling and Analysis
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

    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:eee:econom:v:221:y:2021:i:2:p:409-423. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/jeconom .

    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.