This study develops and implements a theory and method for analyzing whether introducing new securities or relaxing investment constraints improves the investment opportunity set for risk averse investors. We develop a test procedure for 'stochastic spanning' for two nested polyhedral portfolio sets based on subsampling and Linear Programming. The procedure is statistically consistent and asymptotically exact for a class of weakly dependent processes. Using the stochastic spanning tests, we accept market portfolio efficiency but reject two-fund separation in standard data sets of historical stock market returns. The divergence between the results of the two tests illustrates the role for higher-order moment risk in portfolio choice and challenges representative-investor models of capital market equilibrium.