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Testing Mean-Variance Efficiency in CAPM with Possibly Non-Gaussian Errors: an Exact Simulation-Based Approach

  • Marie-Claude Beaulieu
  • Jean-Marie Dufour
  • Lynda Khalaf

In this paper we propose exact likelihood-based mean-variance efficiency tests of the market portfolio in the context of Capital Asset Pricing Model (CAPM), allowing for a wide class of error distributions which include normality as a special case. These tests are developed in the framework of multivariate linear regressions (MLR). It is well known however that despite their simple statistical structure, standard asymptotically justified MLR-based tests are unreliable. In financial econometrics, exact tests have been proposed for a few specific hypotheses [Jobson and Korkie (Journal of Financial Economics, 1982), MacKinlay (Journal of Financial Economics, 1987), Gibbons, Ross and Shanken (Econometrica, 1989), Zhou (Journal of Finance 1993)], most of which depend on normality. For the gaussian model, our tests correspond to Gibbons, Ross and Shanken's mean-variance efficiency tests. In non-gaussian contexts, we reconsider mean-variance efficiency tests allowing for multivariate Student-t and gaussian mixture errors. Our framework allows to cast more evidence on whether the normality assumption is too restrictive when testing the CAPM. We also propose exact multivariate diagnostic checks (including tests for multivariate GARCH and multivariate generalization of the well known variance ratio tests) and goodness of fit tests as well as a set estimate for the intervening nuisance parameters. Our results [over five-year subperiods] show the following: (i) multivariate normality is rejected in most subperiods, (ii) residual checks reveal no significant departures from the multivariate i.i.d. assumption, and (iii) mean-variance efficiency tests of the market portfolio is not rejected as frequently once it is allowed for the possibility of non-normal errors. Dans cet article, nous proposons des tests exacts, basés sur la vraisemblance de l'efficience du portefeuille de marché dans l'espace moyenne-variance. Ces tests, utilisés ici dans le contexte du modèle du CAPM (Capital Asset Pricing Model), permettent de considérer diverses classes de distributions incluant la loi normale. Les tests sont développés dans le cadre de modèles de régression linéaires multivariés (RLM). Il est, par ailleurs, bien établi que, malgré leur structure simple, les écart-types et tests usuels asymptotiques de ces modèles ne sont pas fiables. En économétrie financière, des tests en échantillons finis ont été proposés pour quelques hypothèses spécifiques, lesquels dépendent pour la plupart de l'hypothèse de normalité [Jobson et Korkie (Journal of Financial Economics, 1982), MacKinlay (Journal of Financial Economics, 1987), Gibbons, Ross et Shanken (Econometrica, 1989), Zhou (Journal of Finance 1993)]. Dans le contexte gaussien, nos tests d'efficience correspondent à ceux de Gibbons, Ross et Shanken. Dans un contexte non-gaussien, nous reconsidérons l'efficience moyenne-variance du portefeuille de marché en permettant des distributions multivariées de Student et des « mélanges de lois normales ». Notre démarche nous permet d'évaluer si l'hypothèse de normalité est trop restrictive lorsque l'on teste le CAPM. Nous proposons aussi des tests diagnostiques multivariés (incluant des tests pour les effets GARCH multivariés et une généralisation multivariée des tests de ratio de variance), des tests de spécification ainsi qu'un estimateur ensembliste pour les paramètres de nuisance pertinents. Nos résultats montrent que i) l'hypothèse de normalité multivariée est rejetée sur la plupart des sous-périodes, ii) les tests diagnostiques appliqués aux résidus de nos estimations ne montrent pas de différences importantes par rapport à l'hypothèse des erreurs i.i.d. multivariées, et iii) les tests d'efficience du portefeuille de marché dans l'espace moyenne-variance ne rejettent aussi fréquemment l'hypothèse d'efficience lorsqu'on s'autorise à considérer des lois non normales sur les erreurs.

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Paper provided by CIRANO in its series CIRANO Working Papers with number 2002s-85.

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Length: 37 pages
Date of creation: 01 Nov 2002
Date of revision:
Handle: RePEc:cir:cirwor:2002s-85
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