Forecast Encompassing Tests for the Expected Shortfall
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In this paper, we introduce new forecast encompassing tests for the risk measure Expected Shortfall (ES). Forecasting and forecast evaluation techniques for the ES are rapidly gaining attention through the recently introduced Basel III Accords, which stipulate the use of the ES as primary market risk measure for the international banking regulations. 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. In an empirical application, we utilize the encompassing tests in order to demonstrate the superiority of forecast combination methods for the ES for the IBM stock.
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