回测 Expected Shortfall：利用二元正交多项式同时考虑持续期与严重程度
Backtesting Expected Shortfall: Accounting for both duration and severity with bivariate orthogonal polynomials
AI总结：
提出结合违规间隔持续期与损失严重程度的二元正交多项式 ES 回测框架及 Wald 检验，可分别诊断模型问题并具有良好有限样本表现。
摘要：
我们提出了一种原创的两部分、持续期—严重程度方法，用于回测 Expected Shortfall（ES）。尽管基于 Probability Integral Transform（PIT）的 ES 回测已经受到欢迎，但它们尚不能分别检验 Value-at-Risk（VaR）违规的频率和严重程度。这是一个关键方面，因为 ES 衡量的是发生此类违规时的平均损失。为克服这一局限，我们引入了一个回测框架，该框架依赖于违规之间的持续期序列以及发生违规时的严重程度序列。通过利用（二元）正交多项式理论，我们推导了这两个序列所满足的正交矩条件。我们的方法包含一个简单、无模型的 Wald 检验，该检验涵盖了针对 VaR 和 ES 的各种无条件和条件覆盖回测。该检验有助于识别银行用于预测 ES 的内部模型中任何设定错误的组成部分。此外，它还可以扩展到分析其他系统性风险度量，例如 Marginal Expected Shortfall。模拟实验表明，在现实样本量下，我们的检验具有良好的有限样本性质。通过对两个股票指数的应用，我们展示了本方法如何能够深入理解检验 ES 有效性时出现拒绝的原因。
英文摘要：
We propose an original two-part, duration-severity approach for backtesting Expected Shortfall (ES). While Probability Integral Transform (PIT) based ES backtests have gained popularity, they have yet to allow for separate testing of the frequency and severity of Value-at-Risk (VaR) violations. This is a crucial aspect, as ES measures the average loss in the event of such violations. To overcome this limitation, we introduce a backtesting framework that relies on the sequence of inter-violation durations and the sequence of severities in case of violations. By leveraging the theory of (bivariate) orthogonal polynomials, we derive orthogonal moment conditions satisfied by these two sequences. Our approach includes a straightforward, model-free Wald test, which encompasses various unconditional and conditional coverage backtests for both VaR and ES. This test aids in identifying any mis-specified components of the internal model used by banks to forecast ES. Moreover, it can be extended to analyze other systemic risk measures such as Marginal Expected Shortfall. Simulation experiments indicate that our test exhibits good finite sample properties for realistic sample sizes. Through application to two stock indices, we demonstrate how our methodology provides insights into the reasons for rejections in testing ES validity.
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