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.

中文翻译：

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基于回归的预期短缺回测

在本文中，我们介绍了针对风险量度预期不足（ES）的基于回归的回测，该测试基于分位数和ES的联合回归框架。我们还介绍了此ES回测的第二种变体，该变体允许仅通过测试拦截参数来测试单方面假设。这两个回测是文献中的第一个回测，它们仅对风险度量ES进行回测，因为它们仅需要ES预测作为输入参数。相反，现有的ES回测技术需要对更多数量进行预测，例如风险价值，波动性甚至整个（尾部）分布。由于监管机构仅收到对ES的预测，因此在实践中不适用包含更多输入参数的回测。我们通过一些不同的模拟研究，将我们的新回测的经验性能与现有方法的经验大小和功效进行了比较。我们发现，在所有考虑的模拟实验中，我们的回测就其尺寸和（尺寸调整后的）功率特性而言，明显优于文献中现有的回测程序。我们为这些ES回测提供R软件包，该软件包很适合从业人员。