Regression-Based Expected Shortfall Backtesting

Abstract

This article introduces novel backtests for the risk measure Expected Shortfall (ES) following the testing idea of Mincer and Zarnowitz (1969). Estimating a regression model for the ES stand-alone is infeasible and thus, our tests are based on a joint regression model for the Value at Risk (VaR) and the ES, which allows for different test specifications. These ES backtests are the first which solely backtest the ES in the sense that they only require ES forecasts as input variables. As the tests are potentially subject to model misspecification, we provide asymptotic theory under misspecification for the underlying joint regression. We find that employing a misspecification robust covariance estimator substantially improves the tests’ performance. We compare our backtests to existing joint VaR and ES backtests and find that our tests outperform the existing alternatives throughout all considered simulations. In an empirical illustration, we apply our backtests to ES forecasts for 200 stocks of the S&P 500 index.