Encompassing Tests for Value at Risk and Expected Shortfall Multi-Step Forecasts based on Inference on the Boundary [Previous title : "A Regression-based Joint Encompassing Test for Value-at-Risk and Expected Shortfall Forecasts"]

We propose forecast encompassing tests for the Expected Shortfall (ES) jointlywith the Value at Risk (VaR) based on flexible link (or combination) functions.Our setup allows testing encompassing for convex forecast combinations and forlink functions which preclude crossings of the combined VaR and ES forecasts. Asthe tests based on these link functions involve parameters which are on the boundaryof the parameter space under the null hypothesis, we derive and base our tests onnonstandard asymptotic theory on the boundary. Our simulation study shows thatthe encompassing tests based on our new link functions outperform tests based onunrestricted linear link functions for one-step and multi-step forecasts. We furtherillustrate the potential of the proposed tests in a real data analysis for forecastingVaR and ES of the S&P 500 index