Regression-Based Expected Shortfall Backtesting

In this article, we introduce a regression based backtest for the risk measure Expected Shortfall (ES) which is based on a joint regression framework for the quantile and the ES. We also introduce a second variant of this ES backtest which allows for testing one-sided hypotheses by only testing an intercept parameter. These two backtests are the first backtests in the literature which solely backtest the risk measure ES as they only require ES forecasts as input parameters. In contrast, the existing ES backtesting techniques require forecasts for further quantities such as the Value at Risk, the volatility or even the entire (tail) distribution. As the regulatory authorities only receive forecasts for the ES, backtests including further input parameters are not applicable in practice. We compare the empirical performance of our new backtests to existing approaches in terms of their empirical size and power through several different simulation studies. We find that our backtests clearly outperform the existing backtesting procedures in the literature in terms of their size and (size-adjusted) power properties throughout all considered simulation experiments. We provide an R package for these ES backtests which is easily applicable for practitioners.