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

We develop new tests of the capital asset pricing model (CAPM) that take account of and are valid under the assumption that the distribution generating returns is elliptically symmetric; this assumption is necessary and sufficient for the validity of the CAPM. Our test is based on semiparametric efficient estimation procedures for a seemingly unrelated regression model where the multivariate error density is elliptically symmetric, but otherwise unrestricted. The elliptical symmetry assumption allows us to avert 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 parameter estimates are much less consistent with the CAPM restrictions than the corresponding OLS estimates. Nous développons de nouveaux tests du modèle d'évaluation des actifs financiers (" CAPM ") qui tiennent compte de, et sont valides sous, l'hypothèse que les retours des actifs découlent d'un loi de probabilité elliptiquement symétrique. Cette hypothèse est nécessaire et suffisante pour la validité du CAPM. Notre test utilise un estimateur des paramètres du modèle qui a l'efficacité semiparamétrique quand on a un modèle de régression apparemment sans relation et qui a des erreurs qui suivent une loi elliptiquement symétrique. L'hypothèse de la symétrie elliptique nous permet d'éviter le problème d'estimer non-paramétriquement une fonction de haute dimension parce qu'on peut écrire la densité d'une loi elliptique comme une fonction d'une transformation unidimensionnelle de la variable aléatoire multidimensionnelle. La famille des lois elliptiquement symétriques inclue plusieurs lois leptokurtiques, donc elle est pertinente à des applications financières. Les bêtas obtenus avec notre estimateur sont plus bas que ceux qui sont obtenus en utilisant des moindres carrés, et sont moins compatibles avec le CAPM.

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Paper provided by CREFE, Université du Québec à Montréal in its series Cahiers de recherche CREFE / CREFE Working Papers with number 143.

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Length: 29 pages
Date of creation: Oct 2001
Date of revision:
Handle: RePEc:cre:crefwp:143
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  1. Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
  2. Gibbons, Michael R., 1982. "Multivariate tests of financial models : A new approach," Journal of Financial Economics, Elsevier, vol. 10(1), pages 3-27, March.
  3. Berk, Jonathan B., 1997. "Necessary Conditions for the CAPM," Journal of Economic Theory, Elsevier, vol. 73(1), pages 245-257, March.
  4. 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.
  5. 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.
  6. Hardle, Wolfgang & Linton, Oliver, 1986. "Applied nonparametric methods," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 38, pages 2295-2339 Elsevier.
  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. Hodgson, Douglas J., 1998. "Adaptive Estimation Of Error Correction Models," Econometric Theory, Cambridge University Press, vol. 14(01), pages 44-69, February.
  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. Fan, Yanqin, 1994. "Testing the Goodness of Fit of a Parametric Density Function by Kernel Method," Econometric Theory, Cambridge University Press, vol. 10(02), pages 316-356, June.
  11. 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.
  12. 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.
  13. 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 745-52, June.
  14. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
  15. 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.
  16. Hodgson, D.J., 1997. "Semiparametric Efficient Estimation in Time Series," RCER Working Papers 442, University of Rochester - Center for Economic Research (RCER).
  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, 09.
  18. Linton, Oliver, 1993. "Adaptive Estimation in ARCH Models," Econometric Theory, Cambridge University Press, vol. 9(04), pages 539-569, August.
  19. 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.
  20. 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.
  21. 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.
  22. James Tobin, 1956. "Liquidity Preference as Behavior Towards Risk," Cowles Foundation Discussion Papers 14, Cowles Foundation for Research in Economics, Yale University.
  23. MacKinlay, A. Craig, 1987. "On multivariate tests of the CAPM," Journal of Financial Economics, Elsevier, vol. 18(2), pages 341-371, June.
  24. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, 03.
  25. 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.
  26. repec:cup:etheor:v:11:y:1995:i:5:p:818-87 is not listed on IDEAS
  27. repec:cup:etheor:v:9:y:1993:i:4:p:539-69 is not listed on IDEAS
  28. Amsler, Christine E. & Schmidt, Peter, 1985. "A Monte Carlo investigation of the accuracy of multivariate CAPM tests," Journal of Financial Economics, Elsevier, vol. 14(3), pages 359-375, September.
  29. repec:cup:etheor:v:10:y:1994:i:2:p:316-56 is not listed on IDEAS
  30. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-70, March.
  31. Robinson, Peter M, 1988. "The Stochastic Difference between Econometric Statistics," Econometrica, Econometric Society, vol. 56(3), pages 531-48, May.
  32. Wesselman, A. M. & Van Praag, B. M. S., 1987. "Elliptical regression operationalized," Economics Letters, Elsevier, vol. 23(3), pages 269-274.
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