Alexander Aue, Sebastian Kühnert, Gregory Rice, Jeremy VanderDoes
arXiv 10 Mar 2026 · Statistics — Methodology
arXiv:2603.10272 · PDF · DOI · OpenAlex · Extracted main text
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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Francq, C. and J.-M. Zakoïan (2019) GARCH Models: Structure, Statistical Inference and Financial Applications.\/ (2 ed.) | 1.000 | 8 | 5 | 100% |
| 2 | Cerovecki, C., C. Francq, S. Hörmann, and J.-M. Zakoïan (2019) Functional GARCH models: The quasi-likelihood approach and its applications | 0.941 | 6 | 5 | 83% |
| 3 | Liggett, T (1985) An improved subadditive ergodic theorem | 0.737 | 4 | 3 | 50% |
| 4 | Hörmann, S. and P. Kokoszka (2010) Weakly dependent functional data | 0.644 | 3 | 2 | 67% |
| 5 | Hörmann, S., L. Horváth, and R. Reeder (2013) A functional version of the ARCH model | 0.644 | 2 | 2 | 100% |
| 6 | Hsing, T. and R. Eubank (2015) Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators | 0.644 | 2 | 2 | 100% |
| 7 | Kingman, J. F. C (1973) Subadditive ergodic theory | 0.644 | 2 | 2 | 100% |
| 8 | Kühnert, S (2020) Functional ARCH and GARCH models: A Yule-Walker approach | 0.644 | 2 | 2 | 100% |
| 9 | Hall, P. and A. Meister (2007) A ridge-parameter approach to deconvolution | 0.644 | 2 | 2 | 100% |
| 10 | Kühnert, S., G. Rice, and A. Aue (2026) Estimating invertible processes in Hilbert spaces, with applications to functional ARMA processes | 0.511 | 2 | 2 | 50% |
Showing the top 10 of 42 scored citations.