An operator-level ARCH Model

2026-03-10

AI Summary

This study addresses a key limitation of existing functional ARCH/GARCH models, which capture only pointwise conditional variances and fail to fully characterize the dynamic evolution of the conditional covariance operator. To overcome this, the paper proposes the first operator-valued ARCH model defined on a general separable Hilbert space, offering a comprehensive framework for modeling the time-varying structure of conditional covariance operators in functional time series. Leveraging tools from functional data analysis and operator theory, the authors develop Yule–Walker-type estimators that yield consistent estimates for infinite-dimensional parameters. The existence of both strictly and weakly stationary solutions for the simplified model is rigorously established. Simulation studies and an empirical application to high-frequency intraday cumulative returns demonstrate the model’s superior performance both theoretically and in practice.

Technology Category

• Cognitive Modeling & Cognitive Systems: Agent Architectures
• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

AutoRegressive Conditional Heteroscedasticity (ARCH) models are standard for modeling time series exhibiting volatility, with a rich literature in univariate and multivariate settings. In recent years, these models have been extended to function spaces. However, functional ARCH and generalized ARCH (GARCH) processes established in the literature have thus far been restricted to model ``pointwise''variances. In this paper, we propose a new ARCH framework for data residing in general separable Hilbert spaces that accounts for the full evolution of the conditional covariance operator. We define a general operator-level ARCH model. For a simplified Constant Conditional Correlation version of the model, we establish conditions under which such models admit strictly and weakly stationary solutions, finite moments, and weak serial dependence. Additionally, we derive consistent Yule--Walker-type estimators of the infinite-dimensional model parameters. The practical relevance of the model is illustrated through simulations and a data application to high-frequency cumulative intraday returns.

Problem

• functional time series
• conditional covariance operator
• ARCH model
• Hilbert space
• volatility modeling

Innovation

• operator-level ARCH
• functional time series
• conditional covariance operator
• Hilbert space
• Yule-Walker estimation