Encompassing Tests for Value at Risk and Expected Shortfall Multi-Step Forecasts based on Inference on the Boundary

TL;DR

This paper introduces new encompassing tests for jointly evaluating VaR and ES forecasts using boundary-inference-based methods, demonstrating improved performance over traditional linear link tests in simulations and real data.

Contribution

It develops novel encompassing tests for VaR and ES forecasts that handle boundary parameters, enhancing forecast evaluation accuracy.

Findings

Tests outperform linear link methods in simulations

Boundary-based tests provide more reliable forecast comparisons

Application to S&P 500 data illustrates practical utility

Abstract

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