Paper: arXiv 2603.10272

Authors: Alexander Aue, Sebastian Kühnert, Gregory Rice, Jeremy VanderDoes

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 5.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a highly rigorous mathematical extension of ARCH models to general separable Hilbert spaces, overcoming classic identifiability issues related to non-compact operators. While the theoretical framework and Yule-Walker parameter estimation are extremely novel and advanced, the empirical evaluation relies primarily on simulations and a brief intraday return application without released code.

Research Flowchart

  flowchart TD
    A[Research Goal: Model full conditional covariance operators in separable Hilbert spaces] --> B[Framework Development: Propose operator-level ARCH & simplified CCC-ARCH models]
    B --> C[Theoretical Analysis: Prove stationarity, finite moments, and weak serial dependence]
    C --> D[Estimation Method: Derive consistent Yule-Walker-type estimators for parameters]
    D --> E[Empirical Validation: Conduct simulations & evaluate high-frequency cumulative intraday returns]
    E --> F[Key Outcome: Validated functional ARCH model capturing full volatility dynamics]