Encompassing tests when no model is encompassing

This paper considers regression-based tests for encompassing, when none of the models under consideration encompasses all the other models. For both in- and out-of-sample applications, I derive asymptotic distributions and propose feasible procedures to construct confidence intervals and test statistics. Procedures that are asymptotically valid under the null of encompassing (e.g., Davidson and MacKinnon, Econometrica 49 (1981) 781) can have large asymptotic and finite sample distortions. Simulations indicate that the proposed procedures can work well in samples of size typically available, though the divergence between actual and nominal confidence interval coverage sometimes is large.