An operator-level ARCH Model

TL;DR

This paper introduces a novel operator-level ARCH model for Hilbert space-valued data, capturing the full conditional covariance evolution, with theoretical properties and practical applications demonstrated.

Contribution

It proposes the first operator-level ARCH framework in Hilbert spaces, extending volatility modeling beyond pointwise variances with theoretical guarantees and estimation methods.

Findings

Established conditions for stationarity and moments.

Derived consistent Yule--Walker estimators.

Validated through simulations and real data application.

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.