Tests of mean-variance spanning

In this paper, we conduct a comprehensive study of tests for mean-variance spanning. Under the regression framework of Huberman and Kandel (1987), we provide geometric interpretations not only for the popular likelihood ratio test, but also for two new spanning tests based on the Wald and Lagrange multiplier principles. Under normality assumption, we present the exact distributions of the three tests, analyze their power comprehensively. We find that the power is most driven by the difference of the global minimum-variance portfolios of the two minimum-variance frontiers, and it does not always align well with the economic significance. As an alternative, we provide a step-down test to allow better assessment of the power. Under general distributional assumptions, we provide a new spanning test based on the generalized method of moments (GMM), and evaluate its performance along with other GMM tests by simulation.