美联储光谱回溯测试无界和折叠

Spectral Backtests Unbounded and Folded

Introduction

Gordy and McNeil (2020) introduced a spectral backtesting framework where the test statistic depends on a spectral transformation of a quantile exceedance indicator function. This framework uses a kernel measure to weight quantile exceedance events according to the validator's priorities for model performance. The present paper extends this framework in two significant ways:

1. Unbounded Kernels: The kernel measure is allowed to be unbounded, subject to an integrability condition. Unbounded kernels provide more powerful tests compared to bounded kernels.
2. Folding Transformation: Data is pre-processed using a folding transformation, which leaves the size of the backtest unchanged but increases its power against common misspecifications of forecast volatility.

Motivation and Context

The research is motivated by recent developments in capital regulation for large banks under the current Basel III rules. Under these rules, banks determine minimum capital requirements based on the 97.5% Expected Shortfall (ES) of their trading books. The regulator continues to validate the bank's models through backtesting, focusing on the realized profit-and-loss (P&L) and its probability integral transform (PIT).

Key Contributions

1. Unbounded Kernels:

   • Powerful Tests: Unbounded kernels yield more powerful tests compared to bounded kernels.
   • Applicability: Continuous kernels that weight every quantile above a threshold (e.g., 0.975) are considered, even some unbounded measures can ensure valid test statistics.

2. Folding Transformation:

   • Tail Focus: The folding transformation maps tail values from the original PIT distribution to the upper tail of the pre-processed distribution.
   • Uniform Distribution Preservation: The transformation ensures that the transformed PIT values remain uniformly distributed under the null hypothesis.

Applications and Implications

• Risk Management: The framework is particularly relevant for validating models throughout the tails of the forecast distribution.
• Regulatory Compliance: It addresses the need for more robust backtesting methods for ES, aligning with the shift from Value-at-Risk (VaR) to ES under Basel III.
• Empirical Evidence: Simulation exercises confirm the increased power of the tests without size distortions.

Conclusion

The spectral backtesting framework, extended to allow unbounded kernels and folding transformations, provides a more flexible and powerful tool for validating forecast models. This approach is especially useful in risk management and regulatory compliance, particularly for expected shortfall (ES) under Basel III regulations.