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

In this paper we test the mean variance efficiency of the benchmark portfolio in the absence of a risk-free rate. We propose exact likelihood-based 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)]. We propose extensions of Shanken’s bound which are provably valid with non-Gaussian disturbance distributions. We also suggest a general method which yields tighter bounds in both Gaussian and non-Gaussian cases. Our framework allows one 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 Fieller-type exact confidence intervals 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. The methods proposed are applied to monthly returns on 12 portfolios of New York Stock Exchange (NYSE) firms over the period 1925-1995 divided in five-year subperiods. The results obtained 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 sets for the zero beta rate are 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.