Spanning tests in return and stochastic discount factor mean-variance frontiers: a unifying approach

We propose new approaches to test for spanning in the return and stochastic discount factor mean-variance frontiers, which assess if either the centred or uncentred mean and cost representing portfolios are shared by the initial and extended sets of assets. We show that our proposed tests are asymptotically equivalent to the existing spanning tests under the null and sequences of local alternatives, and analyse their asymptotic relative e.ciency. We also extend the theory of optimal GMM inference to deal with the singularities that arise in some spanning tests. Finally, we include an empirical application to money markets in Europe.