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Robust Bayesian estimation in conditionally heteroscedastic time series models

Jeongho Lee, Junmo Song

TL;DR

This work tackles the vulnerability of Bayesian inference for conditionally heteroscedastic time series to outliers by extending the density power divergence (DPD) to a Bayesian setting. The authors construct a gamma-tuned DPD-based posterior, define the EDPE as the posterior mean, and establish a Bernstein–von Mises type result showing asymptotic equivalence between the EDPE and the minimum DPD estimator (MDPDE). Through simulations on GARCH models and an empirical BTC-USD analysis, the method demonstrates robust performance under contamination with only mild efficiency loss under clean data. The approach is implemented with Hamiltonian Monte Carlo and offers practical improvements in volatility forecasting and tail-risk calibration, with guidance on selecting the robustness parameter $\\gamma$ via predictive performance.

Abstract

Outliers can seriously distort statistical inference by inducing excessive sensitivity in the likelihood function, thereby compromising the reliability of Bayesian estimation. To address this issue, we develop a robust Bayesian estimation method for conditionally heteroscedastic time series models by extending the density power divergence (DPD) framework to the Bayesian setting. The resulting DPD-based posterior distribution, controlled by a tuning parameter, achieves a smooth balance between efficiency and robustness. We establish the asymptotic properties of the proposed estimator; specifically, the DPD-based posterior is shown to satisfy a Bernstein-von Mises type theorem, converging to a normal distribution centered at the minimum DPD estimator (MDPDE). Furthermore, the corresponding Bayes estimator, defined as the posterior mean under the DPD-based posterior (EDPE), is asymptotically equivalent to the MDPDE. Monte Carlo simulations based on GARCH(1,1) models confirm that the proposed EDPE performs well under both uncontaminated and contaminated data, maintaining robustness where the ordinary Bayes estimator becomes severely biased. An empirical application to BTC-USD returns further demonstrates the practical advantages of the proposed robust Bayesian framework for financial time series analysis.

Robust Bayesian estimation in conditionally heteroscedastic time series models

TL;DR

This work tackles the vulnerability of Bayesian inference for conditionally heteroscedastic time series to outliers by extending the density power divergence (DPD) to a Bayesian setting. The authors construct a gamma-tuned DPD-based posterior, define the EDPE as the posterior mean, and establish a Bernstein–von Mises type result showing asymptotic equivalence between the EDPE and the minimum DPD estimator (MDPDE). Through simulations on GARCH models and an empirical BTC-USD analysis, the method demonstrates robust performance under contamination with only mild efficiency loss under clean data. The approach is implemented with Hamiltonian Monte Carlo and offers practical improvements in volatility forecasting and tail-risk calibration, with guidance on selecting the robustness parameter via predictive performance.

Abstract

Outliers can seriously distort statistical inference by inducing excessive sensitivity in the likelihood function, thereby compromising the reliability of Bayesian estimation. To address this issue, we develop a robust Bayesian estimation method for conditionally heteroscedastic time series models by extending the density power divergence (DPD) framework to the Bayesian setting. The resulting DPD-based posterior distribution, controlled by a tuning parameter, achieves a smooth balance between efficiency and robustness. We establish the asymptotic properties of the proposed estimator; specifically, the DPD-based posterior is shown to satisfy a Bernstein-von Mises type theorem, converging to a normal distribution centered at the minimum DPD estimator (MDPDE). Furthermore, the corresponding Bayes estimator, defined as the posterior mean under the DPD-based posterior (EDPE), is asymptotically equivalent to the MDPDE. Monte Carlo simulations based on GARCH(1,1) models confirm that the proposed EDPE performs well under both uncontaminated and contaminated data, maintaining robustness where the ordinary Bayes estimator becomes severely biased. An empirical application to BTC-USD returns further demonstrates the practical advantages of the proposed robust Bayesian framework for financial time series analysis.
Paper Structure (9 sections, 10 theorems, 84 equations, 5 figures, 5 tables)

This paper contains 9 sections, 10 theorems, 84 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Density power divergence based Bayesian estimation
  4. Asymptotic Properties
  5. Application to GARCH models
  6. Simulation study
  7. Real data analysis
  8. Conclusion
  9. Appendix

Key Result

Theorem 1

Suppose that conditions C1 -- C4 hold and the parameter space $\Theta$ is compact. Then, the unnormalized posterior densities satisfy the following uniform convergence Additionally, if condition C5 holds, the DPD-based posterior $\pi_\gamma(\theta \mid \mathbf{X}_n)$ converges to the ordinary posterior $\pi(\theta \mid \mathbf{X}_n)$ in total variation, that is,

Figures (5)

  • Figure 1: Total scaled RMSE of posterior mean with respect to $\gamma$ under the uncontaminated setting
  • Figure 2: Total scaled RMSE of posterior mean with respect to $\gamma$ under the 1% contamination setting
  • Figure 3: Daily closing prices (L) and log returns (R) of BTC-USD from September 2021 to December 2024; the vertical dashed line marks the start of the out-of-sample evaluation period
  • Figure 4: RMSE and MAE for conditional variance forecasts (L) and the empirical 95% VaR violation ratio (R)
  • Figure 5: Daily log returns and one-step-ahead 95% VaR forecasts under the EOPE and EDPE for September--December 2024

Theorems & Definitions (13)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 1
  • Proposition 2
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 3 more