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Statistical Inference for the Beta Coefficient

Author

Listed:
  • Taras Bodnar

    (Department of Mathematics, Stockholm University, Roslagsvägen 101, SE-10691 Stockholm, Sweden)

  • Arjun K. Gupta

    (Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403, USA)

  • Valdemar Vitlinskyi

    (Department of Economic and Mathematical Modelling, Kyiv National Economic University, Peremoga Avenue 54/1, 03680 Kyiv, Ukraine)

  • Taras Zabolotskyy

    (Department of Programming, Ivan Franko Lviv National University, Universytetska str. 1, 79000 Lviv, Ukraine)

Abstract

The beta coefficient plays a crucial role in finance as a risk measure of a portfolio in comparison to the benchmark portfolio. In the paper, we investigate statistical properties of the sample estimator for the beta coefficient. Assuming that both the holding portfolio and the benchmark portfolio consist of the same assets whose returns are multivariate normally distributed, we provide the finite sample and the asymptotic distributions of the sample estimator for the beta coefficient. These findings are used to derive a statistical test for the beta coefficient and to construct a confidence interval for the beta coefficient. Moreover, we show that the sample estimator is an unbiased estimator for the beta coefficient. The theoretical results are implemented in an empirical study.

Suggested Citation

  • Taras Bodnar & Arjun K. Gupta & Valdemar Vitlinskyi & Taras Zabolotskyy, 2019. "Statistical Inference for the Beta Coefficient," Risks, MDPI, vol. 7(2), pages 1-14, May.
    • Handle: RePEc:gam:jrisks:v:7:y:2019:i:2:p:56-:d:231435
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    References listed on IDEAS

    as
    1. Frahm, Gabriel & Memmel, Christoph, 2010. "Dominating estimators for minimum-variance portfolios," Journal of Econometrics, Elsevier, vol. 159(2), pages 289-302, December.
    2. Douglas J. Hodgson & Oliver Linton & Keith Vorkink, 2002. "Testing the capital asset pricing model efficiently under elliptical symmetry: a semiparametric approach," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 17(6), pages 617-639, December.
    3. Stephen A. Ross, 2013. "The Arbitrage Theory of Capital Asset Pricing," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 1, pages 11-30, World Scientific Publishing Co. Pte. Ltd..
    4. repec:hal:journl:peer-00741629 is not listed on IDEAS
    5. Gordon J. Alexander & Alexandre M. Baptista, 2004. "A Comparison of VaR and CVaR Constraints on Portfolio Selection with the Mean-Variance Model," Management Science, INFORMS, vol. 50(9), pages 1261-1273, September.
    6. Zhou, Guofu, 1993. "Asset-Pricing Tests under Alternative Distributions," Journal of Finance, American Finance Association, vol. 48(5), pages 1927-1942, December.
    7. Wolfgang Schmid & Taras Zabolotskyy, 2008. "On the existence of unbiased estimators for the portfolio weights obtained by maximizing the Sharpe ratio," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 92(1), pages 29-34, February.
    8. Bodnar, Taras & Parolya, Nestor & Schmid, Wolfgang, 2018. "Estimation of the global minimum variance portfolio in high dimensions," European Journal of Operational Research, Elsevier, vol. 266(1), pages 371-390.
    9. 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.
    10. Berk, Jonathan B., 1997. "Necessary Conditions for the CAPM," Journal of Economic Theory, Elsevier, vol. 73(1), pages 245-257, March.
    11. David Bauder & Rostyslav Bodnar & Taras Bodnar & Wolfgang Schmid, 2019. "Bayesian estimation of the efficient frontier," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 46(3), pages 802-830, September.
    12. Alexander, Gordon J. & Baptista, Alexandre M., 2002. "Economic implications of using a mean-VaR model for portfolio selection: A comparison with mean-variance analysis," Journal of Economic Dynamics and Control, Elsevier, vol. 26(7-8), pages 1159-1193, July.
    13. Gibbons, Michael R & Ross, Stephen A & Shanken, Jay, 1989. "A Test of the Efficiency of a Given Portfolio," Econometrica, Econometric Society, vol. 57(5), pages 1121-1152, September.
    14. Merton, Robert C., 1972. "An Analytic Derivation of the Efficient Portfolio Frontier," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 7(4), pages 1851-1872, September.
    15. Taras Bodnar & Wolfgang Schmid, 2009. "Econometrical analysis of the sample efficient frontier," The European Journal of Finance, Taylor & Francis Journals, vol. 15(3), pages 317-335.
    16. Li, Quan & Reuveny, Rafael, 2003. "Economic Globalization and Democracy: An Empirical Analysis," British Journal of Political Science, Cambridge University Press, vol. 33(1), pages 29-54, January.
    17. Bodnar Taras & Schmid Wolfgang & Zabolotskyy Tara, 2012. "Minimum VaR and minimum CVaR optimal portfolios: Estimators, confidence regions, and tests," Statistics & Risk Modeling, De Gruyter, vol. 29(4), pages 281-314, November.
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    Cited by:

    1. Pankaj Agrrawal, 2023. "The Gibbons, Ross, and Shanken Test for Portfolio Efficiency: A Note Based on Its Trigonometric Properties," Mathematics, MDPI, vol. 11(9), pages 1-19, May.

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