We investigate the effect of estimation error on backtests of expected shortfall (ES) forecasts. These backtests are based on first-order conditions of a recently introduced family of jointly consistent loss functions for value-at-risk (VaR) and ES. For both single and multiperiod horizons, we provide explicit expressions for the additional terms in the asymptotic covariance matrix that result from estimation error, and propose robust tests that account for it. Monte Carlo experiments show that the tests that ignore these terms suffer from size distortions, which are more pronounced for higher ratios of out-of-sample to in-sample observations. Robust versions of the backtests perform well with power against common alternatives. We also introduce a novel standardization of the conditional joint test statistic that removes the need to estimate higher-order moments and significantly improves its performance. In an application to VaR and ES forecasts for daily FTSE 100 index returns as generated by (GJR-)GARCH and HEAVY models, we find that estimation error substantially impacts the outcome of the backtests, and is not bound to particular subperiods such as the credit crisis.

中文翻译：

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在存在估计误差的情况下回测风险价值和预期缺口

我们研究了估计误差对预期短缺 (ES) 预测的回测的影响。这些回测基于最近引入的一系列针对风险价值 (VaR) 和 ES 的联合一致损失函数的一阶条件。对于单周期和多周期，我们为渐近协方差矩阵中由估计误差引起的附加项提供了明确的表达式，并提出了对此进行解释的稳健检验。蒙特卡洛实验表明，忽略这些项的检验会出现尺寸失真，这对于样本外与样本内观察值的比率较高时更为明显。强大的回测版本在对抗常见替代方案时表现良好。我们还引入了一种新的条件联合测试统计标准化，它消除了估计高阶矩的需要并显着提高了它的性能。在对 (GJR-)GARCH 和 HEAVY 模型生成的每日 FTSE 100 指数收益的 VaR 和 ES 预测的应用中，我们发现估计误差会显着影响回溯测试的结果，并且不受特定子期的约束，例如信贷危机。