Before revision:  Encompassing tests generally rely on the existence of strictly consistent loss functions for the functionals under consideration, which do not exist for the ES . However, our encompassing tests are based on recently introduced loss functions and an associated regression framework which considers the ES jointly with the corresponding Value at Risk (VaR). This setup facilitates several testing specifications which allow for both, joint tests for the ES and VaR and stand-alone tests for the ES. We present asymptotic theory for our encompassing tests and verify their finite sample properties through various simulation setups.
After revision:  Encompassing tests generally rely on the existence of strictly consistent loss functions for the functionals under consideration, which do not exist for the ES . However, our encompassing tests are based on recently introduced loss functions and an associated regression framework which considers the ES jointly with the corresponding Value at Risk to set up three ES encompassing test variants. The tests are built on misspecification robust asymptotic theory and we verify the finite sample properties through various simulation setups.