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 nd that
the power is most driven by the di erence of the global minimum-variance
portfolios of the two minimum-variance frontiers, and it does not always align
well with the economic signi cance. 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.