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An operator-level ARCH Model

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.

An operator-level ARCH Model

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.
Paper Structure (21 sections, 12 theorems, 170 equations, 8 figures, 2 tables)

This paper contains 21 sections, 12 theorems, 170 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. General model and assumptions
  3. CCC-op-ARCH model and structure
  4. Strict stationarity
  5. Existence of moments, weak dependence, and weak stationarity
  6. Examples
  7. Estimation in the CCC-op-ARCH model
  8. Finite-dimensional setting
  9. Infinite-dimensional setting
  10. Estimation of the Intercept term
  11. Simulation Study
  12. Implementation Details
  13. Data Generation
  14. Consistency Results
  15. Application to Intra-Day Return Data
  16. ...and 6 more sections

Key Result

Proposition 3.1

The top Lyapunov exponent defined by exists with $\gamma\in[-\infty,\infty)$, where the first limit holds almost surely. Moreover,

Figures (8)

  • Figure 1: S&P500 Data. Visualization of price (left) and overnight cumulative intraday return (right) curves based on 15-minute resolution S&P 500 stock market prices from 2018 to 2020
  • Figure 2: CCC-op-ARCH Examples. (a) Uses $C_\epsilon$ based on Ornstein–Uhlenbeck errors. (b) Uses $C_\epsilon$ based on Brownian motion errors.
  • Figure 3: Relative absolute error $e_{N,\alpha}$ for Tikhonov versus Moore Penrose-based estimators for simulated CCC-op-ARCH$(1)$ data. The left-hand panel considers a low-dimensional setting, and the right-hand panel shows a high-dimensional setting.
  • Figure 4: Estimation Consistency. Relative absolute error for estimation of $\Delta$ and the $\alpha_j$'s of CCC-op-ARCH$(1)$ and CCC-op-ARCH$(5)$ models in the low-dimensional setting.
  • Figure 5: Plots of the curves $\hat{V}_{j,0.05}(\cdot)$ for the CCC-op-ARCH$(p)$ models, $p \in \{1,5\}$, as well as the historical and pw-fARCH$(1)$ model along side the observed OCIDR curves for a particularly volatile period in the S&P 500 index including the COVID-19 Lockdown period in March, 2020.
  • ...and 3 more figures

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 3.1
  • Proposition 3.1
  • Theorem 3.1
  • Proposition 3.2
  • Remark 3.1
  • Example 3.1
  • Proposition 3.3
  • Proposition 3.4
  • Remark 3.2
  • ...and 24 more