美联储-光谱回溯测试无界和折叠（英）-2024.7-48页_716kb

Finance and Economics Discussion Series
Federal Reserve Board, Washington, D.C.
ISSN 1936-2854 (Print)
ISSN 2767-3898 (Online)
Spectral backtests unbounded and folded
Michael B. Gordy and Alexander J. McNeil
2024-060

Summary

This paper introduces novel spectral backtesting methodologies for expected shortfall (ES) forecasts, extending the work of Gordy and McNeil (2020). The core contributions are two new types of spectral backtests:

1. Unbounded Kernels (C Kernels): These include bounded and unbounded beta kernels. Bounded kernels are effective for detecting model failures in tail-conforming windows, while unbounded kernels better capture extreme events by assigning more weight to tail regions, improving test power without increasing size distortion. The paper provides practical guidance: for detecting unmodeled kurtosis, unbounded kernels are recommended, while for skewness, asymmetric v-shaped transformations can enhance power but require careful selection to avoid masking effects.

2. Folded Distributions (Folding Transformation): A v-shaped transformation (e.g., ( T(u) = |1 - 2u| ) ) is proposed to map both upper and lower tails into the upper tail, pre-processing PIT values to highlight data relevant to upper tail misspecification without altering test statistics. This folding technique improves test sensitivity, especially when combined with unbounded kernels targeting peak kurtosis.

Key Findings

• Using broken beta kernels (e.g., ( b \leq 0 )) outperforms bounded versions in detecting excess kurtosis, shown via extensive Monte Carlo simulations across Student t distributions with varying tail shapes.
• Folding transformations enhance detection accuracy, particularly with asymmetric kernels like the Gumbel or logistic-based TLSF tests, achieving power comparable to bounded measures but without size issues.
• The linear v-transform (( T(v) = |1 - 2v| )) balances sensitivity to both tail behaviors, while asymmetric transforms require tuning to address specific model issues like skewness, though they may compromise robustness if misapplied.

Regulatory Use and Recommendations
Financial regulators should choose spectral tests based on suspected model deficiencies. Unbounded kernels are ideal for kurtosis detection, folding transformations for tail consistency, and asymmetric kernels for identified skewness. The methods offer practical advantages in computing due to modular design (application of transformation followed by kernel test). Multivariate extensions of unbounded kernels remain computationally efficient, though more complex kernel families (e.g., TLSF) require specialized programming.

The paper underscores that backtests should implicitly reflect the regulator’s concerns (e.g., focus on extreme losses vs. gain events), with theoretical optimality guiding kernel/transform selection based on empirical evidence of model vulnerabilities. For standard normal assumptions, simple transformations perform well; for complex models involving skew and kurtosis, advanced preprocessing and kernels are essential.

Gordy and McNeil conclude by positioning their framework as a significant advancement in risk measurement, providing interpretable and statistically robust tools for regulatory oversight.