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Testing the Capital Asset Pricing Model Efficiently Under Elliptical Symmetry: A Semiparametric Approach

  • Oliver Linton


  • Douglas J.Hodgson
  • Keith Vorkink

We develop new tests of the capital asset pricing model that take account of and are valid under the assumption that the distribution generating returns is elliptically symmetric; this assumption is neccessary and sufficient for the validity of the CAPM. Our test is based on semi-parametric efficient estimation procedures for a seemingly unrelated regression model where the multvariate error density is elliptically symmetric, but otherwise unrestricted. The elliptical symmetry assumption allows us to avoid the curse of dimensionality problem that typically arises in multivariate semiparametric estimation procedures, because the multivariate elliptically symmetric density function can be written as a function of a scalar transformation of the observed multivariate data. The elliptically symmetric family includes a number of thick-tailed distributions and so is potentially relevant in financial applications. Our estimated betas are lower than the OLS estimates, and our parametric estimates are much less consistent with the CAPM restrictions than the corresponding OLS estimates.

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Paper provided by Financial Markets Group in its series FMG Discussion Papers with number dp382.

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Date of creation: Jun 2001
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Handle: RePEc:fmg:fmgdps:dp382
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  1. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-70, March.
  2. Härdle, W.K., 1992. "Applied Nonparametric Methods," Discussion Paper 1992-6, Tilburg University, Center for Economic Research.
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  4. James Tobin, 1956. "Liquidity Preference as Behavior Towards Risk," Cowles Foundation Discussion Papers 14, Cowles Foundation for Research in Economics, Yale University.
  5. Wesselman, A. M. & Van Praag, B. M. S., 1987. "Elliptical regression operationalized," Economics Letters, Elsevier, vol. 23(3), pages 269-274.
  6. Hodgson, D.J., 1995. "Adaptive Estimation of Error Correlation Models," RCER Working Papers 410, University of Rochester - Center for Economic Research (RCER).
  7. Jeganathan, P., 1995. "Some Aspects of Asymptotic Theory with Applications to Time Series Models," Econometric Theory, Cambridge University Press, vol. 11(05), pages 818-887, October.
  8. Bollerslev, Tim, 1987. "A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return," The Review of Economics and Statistics, MIT Press, vol. 69(3), pages 542-47, August.
  9. Hodgson, Douglas J., 1998. "Adaptive estimation of cointegrating regressions with ARMA errors," Journal of Econometrics, Elsevier, vol. 85(2), pages 231-267, August.
  10. Oliver Linton, 1993. "Second Order Approximation in the Partially Linear Regression Model," Cowles Foundation Discussion Papers 1065, Cowles Foundation for Research in Economics, Yale University.
  11. Fama, Eugene F. & French, Kenneth R., 1993. "Common risk factors in the returns on stocks and bonds," Journal of Financial Economics, Elsevier, vol. 33(1), pages 3-56, February.
  12. Hodgson, Douglas J & Vorkink, Keith P, 2003. "Efficient Estimation of Conditional Asset-Pricing Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 21(2), pages 269-83, April.
  13. 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, 09.
  14. MacKinlay, A. Craig, 1987. "On multivariate tests of the CAPM," Journal of Financial Economics, Elsevier, vol. 18(2), pages 341-371, June.
  15. Gibbons, Michael R., 1982. "Multivariate tests of financial models : A new approach," Journal of Financial Economics, Elsevier, vol. 10(1), pages 3-27, March.
  16. Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
  17. Davidson, Russell & MacKinnon, James G, 1998. "Graphical Methods for Investigating the Size and Power of Hypothesis Tests," The Manchester School of Economic & Social Studies, University of Manchester, vol. 66(1), pages 1-26, January.
  18. 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 1121-52, September.
  19. Oliver Linton, 1993. "Adaptive Estimation in ARCH Models," Cowles Foundation Discussion Papers 1054, Cowles Foundation for Research in Economics, Yale University.
  20. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, 03.
  21. White, Halbert, 1980. "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, Econometric Society, vol. 48(4), pages 817-38, May.
  22. Bollerslev, Tim & Engle, Robert F & Wooldridge, Jeffrey M, 1988. "A Capital Asset Pricing Model with Time-Varying Covariances," Journal of Political Economy, University of Chicago Press, vol. 96(1), pages 116-31, February.
  23. Oliver LINTON, . "Applied nonparametric methods," Statistic und Oekonometrie 9312, Humboldt Universitaet Berlin.
  24. Chamberlain, Gary, 1983. "A characterization of the distributions that imply mean--Variance utility functions," Journal of Economic Theory, Elsevier, vol. 29(1), pages 185-201, February.
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  26. Berk, Jonathan B., 1997. "Necessary Conditions for the CAPM," Journal of Economic Theory, Elsevier, vol. 73(1), pages 245-257, March.
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