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 performanceof 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  .
This paper introduces novel backtests for the risk measure Expected Shortfall (ES) following the testing idea of Mincer and Zarnowitz (1969). Estimating a regression framework for the ES stand-alone is infeasible, and thus, our tests are based on a joint regression for the Value at Risk 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 parameters. 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 approaches and find that our tests outperform the competitors throughout all considered simulations. In an empirical illustration, we apply our backtests to ES forecasts for 200 stocks of the S P 500 index .
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