Accurate Bayesian inference for tail risk extrapolation in time series

2025-10-16

AI Summary

Quantifying tail risk (e.g., financial crashes, extreme weather) under serially dependent data remains challenging due to the inadequacy of standard independence-based tail modeling. Method: This paper proposes a Bayesian inference framework based on the Generalized Pareto Distribution (GPD) that jointly models static marginal and dynamic conditional tail behavior of threshold exceedances, integrating beta-mixture dependence structures, heteroscedastic regression, and empirical log-likelihood theory—thereby relaxing the restrictive independence assumption. Contribution/Results: It establishes, for the first time under serial dependence, asymptotically honest Bayesian credible regions for tail parameters and derives theoretical conditions for prior admissibility. Simulation studies demonstrate substantial improvements over classical methods across ARMA, GARCH, and Markov copula models. Empirical applications to U.S. interest rates and Swiss electricity demand confirm high accuracy, robustness, and practical applicability in real-world tail risk assessment.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Bayesian Learning
• Game Theory and Economic Paradigms: Behavioral Game Theory

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• Security and Privacy: Large-scale security measurements

Abstract

Accurately quantifying tail risks-rare but high-impact events such as financial crashes or extreme weather-is a central challenge in risk management, with serially dependent data. We develop a Bayesian framework based on the Generalized Pareto (GP) distribution for modeling threshold exceedances, providing posterior distributions for the GP parameters and tail quantiles in time series. Two cases are considered: extrapolation of tail quantiles for the stationary marginal distribution under beta-mixing dependence, and dynamic, past-conditional tail quantiles in heteroscedastic regression models. The proposal yields asymptotically honest credible regions, whose coverage probabilities converge to their nominal levels. We establish the asymptotic theory for the Bayesian procedure, deriving conditions on the prior distributions under which the posterior satisfies key asymptotic properties. To achieve this, we first develop a likelihood theory under serial dependence, providing local and global bounds for the empirical log-likelihood process of the misspecified GP model and deriving corresponding asymptotic properties of the Maximum Likelihood Estimator (MLE). Simulations demonstrate that our Bayesian credible regions outperform naive Bayesian and MLE-based confidence regions across several standard time series models, including ARMA, GARCH, and Markovian copula models. Two real-data applications-to U.S. interest rates and Swiss electricity demand-highlight the relevance of the proposed methodology.

Problem

• Developing Bayesian framework for tail risk extrapolation in time series
• Modeling threshold exceedances using Generalized Pareto distribution
• Providing asymptotically honest credible regions for tail quantiles

Innovation

• Bayesian framework with Generalized Pareto distribution
• Asymptotically honest credible regions for tail risk
• Dynamic tail quantiles in heteroscedastic regression models