Testing Black’s CAPM with possibly non-Gaussian errors: an exact simulation-based approach

In this paper we test the mean variance efficiency of the benchmark portfolio in the absence of a riskfree rate. We propose likelihood-based exact bound mean-variance efficiency tests of the market portfolio, allowing for a wide class of elliptical error distributions. Our tests are developed using multivariate linear regressions (MLR). It is well known that despite their simple statistical structure, standard asymptotically justified MLR-based tests are unreliable. In this context, exact bounds tests have been proposed which depend on normality [Shanken (Journal of Financial Economics, 1986), Zhou (Journal of Finance, 1991), Zhou (Journal of Empirical Finance, 1995) and Velu and Zhou (Journal of Empirical Finance, 1999)]. For the Gaussian model, our tests correspond to Shanken’s bound test. In non-Gaussian contexts, we propose extensions of the latter bound which include an optimal one. Our framework allows to cast more evidence on whether the normality assumption is too restrictive when testing for the mean variance efficiency of the benchmark portfolio when the riskless rate is unobservable. We provide estimates for γ, the zero beta rate, over each time subperiod as well as an exact confidence set estimate for this rate. We show using a small scale simulation study that our exact confidence set for the zero beta rate provides a major improvement over previous measures, with obvious implications for financial applications of the model. We also run exact multivariate diagnostic checks and goodness of fit tests as well as a set estimate for the parameters which define the error distributions. Our results [over five-year subperiods] show the following: (i) multivariate normality is rejected in most subperiods, (ii) multivariate residual checks reveal no significant departures from the i.i.d. assumption, (iii) the exact confidence set for the zero beta rate is very different from the asymptotic one, and (iv) mean variance efficiency of the benchmark portfolio is not rejected as frequently once we allow for the possibility of non-normal errors.