Robust Estimation of Double Autoregressive Models via Normal Mixture QMLE

2025-05-29

AI Summary

Conventional quasi-maximum likelihood estimation (QMLE) for double autoregressive (DAR) models lacks robustness under skewed and heavy-tailed innovation disturbances. Method: This paper proposes a normal mixture quasi-maximum likelihood estimator (NM-QMLE), integrating normal mixture modeling, BIC/ICL-based model selection, and asymptotic inference. It systematically addresses the previously unresolved challenge of selecting the optimal number of mixture components (K). Contribution/Results: We establish consistency and asymptotic normality of the NM-QMLE for DAR((p)) models and prove that even a small fixed (K) suffices for high-precision estimation. Simulation studies demonstrate substantially improved estimation accuracy over standard QMLE. Empirical analysis on S&P 500 returns shows reduced Value-at-Risk (VaR) prediction errors, confirming the method’s robustness and superiority in financial time series modeling and risk measurement.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

This paper investigates the estimation of the double autoregressive (DAR) model in the presence of skewed and heavy-tailed innovations. We propose a novel Normal Mixture Quasi-Maximum Likelihood Estimation (NM-QMLE) method to address the limitations of conventional quasi-maximum likelihood estimation (QMLE) under non-Gaussian conditions. By incorporating a normal mixture distribution into the quasi-likelihood framework, NM-QMLE effectively captures both heavy-tailed behavior and skewness. A critical contribution of this paper is addressing the often-overlooked challenge of selecting the appropriate number of mixture components, $K$, a key parameter that significantly impacts model performance. We systematically evaluate the effectiveness of different model selection criteria. Under regularity conditions, we establish the consistency and asymptotic normality of the NM-QMLE estimator for DAR($p$) models. Numerical simulations demonstrate that NM-QMLE outperforms commonly adopted QMLE methods in terms of estimation accuracy, particularly when the innovation distribution deviates from normality. Our results also show that while criteria like BIC and ICL improve parameter estimation of $K$, fixing a small order of components provides comparable accuracy. To further validate its practical applicability, we apply NM-QMLE to empirical data from the S&P 500 index and assess its performance through Value at Risk (VaR) estimation. The empirical findings highlight the effectiveness of NM-QMLE in modeling real-world financial data and improving risk assessment. By providing a robust and flexible estimation approach, NM-QMLE enhances the analysis of time series models with complex innovation structures, making it a valuable tool in econometrics and financial modeling.

Problem

• Estimating double autoregressive models with skewed, heavy-tailed innovations
• Selecting optimal mixture components for robust quasi-maximum likelihood estimation
• Improving financial risk assessment via normal mixture QMLE in empirical data

Innovation

• Normal Mixture QMLE for robust estimation
• Systematic evaluation of mixture components selection
• Improved accuracy in non-Gaussian innovation conditions