[Paper Review] High-frequency sampling of multivariate CARMA processes

Péter Kevei|arXiv (Cornell University)|Sep 11, 2015

Advanced Statistical Process Monitoring14 references3 citations

TL;DR

This paper investigates high-frequency sampling of multivariate continuous-time autoregressive moving average (MCARMA) processes, deriving an asymptotic expansion of the spectral density as the sampling interval Δ approaches zero. It shows that the properly filtered sampled process converges to a vector moving average process, generalizing univariate results to the multivariate case and providing a tractable approximation for high-frequency data analysis.

ABSTRACT

High-frequency sampled multivariate continuous time autoregressive moving average processes are investigated. We obtain asymptotic expansion for the spectral density of the sampled MCARMA process $(Y_{nΔ})_{n \in \mathbb{Z}}$ as $Δ\downarrow 0$, where $(Y_t)_{t \in \mathbb{R}}$ is an MCARMA process. We show that the properly filtered process is a vector moving average process, and determine the asymptotic moving average representation of it, thus generalizing the results by Brockwell et al. in the univariate case to the multivariate model. The determination of the moving average representation of the filtered process, important for the analysis of high-frequency data, is difficult for any fixed positive $Δ$. However, the results established here provide a useful and insightful approximation when $Δ$ is very small.

Motivation & Objective

• To analyze the asymptotic behavior of the spectral density of discretely sampled multivariate CARMA processes as the sampling interval Δ → 0.
• To generalize univariate results on high-frequency sampling of CARMA processes to the multivariate setting.
• To derive the asymptotic moving average representation of the filtered sampled process, enabling practical analysis of high-frequency data.
• To provide a rigorous approximation framework for the moving average representation when Δ is small but positive, overcoming the difficulty of exact computation at fixed Δ.

Proposed method

• Derives the spectral density of the sampled MCARMA process (Y_{nΔ})_{n∈ℤ} using the state space representation and the Lévy-driven stochastic differential equation.
• Applies asymptotic expansion techniques to the spectral density as Δ ↓ 0, focusing on the leading-order term.
• Uses the spectral representation of the MCARMA process in terms of the transfer function g(t) = ∫ e^{i t x} P(ix)^{-1} Q(ix) dx.
• Establishes that the filtered process converges to a vector moving average process by analyzing the asymptotic behavior of the spectral density and the structure of the transfer function.
• Employs Chebyshev polynomials and Eulerian polynomials to characterize the coefficients in the asymptotic expansion.
• Introduces a factorization of the spectral density using roots of generalized Eulerian polynomials and constructs real-valued factorizations even when roots are complex, ensuring real coefficients in the asymptotic representation.

Experimental results

Research questions

• RQ1 What is the asymptotic spectral density of a high-frequency sampled multivariate CARMA process as Δ → 0?
• RQ2 How does the filtered sampled process behave asymptotically, and can it be represented as a vector moving average process?
• RQ3 What is the asymptotic moving average representation of the filtered process, and how does it generalize univariate results to the multivariate case?
• RQ4 Can the leading-order term in the spectral density expansion be factored into a product of terms corresponding to a moving average representation with real coefficients?
• RQ5 Under what conditions are the roots of the relevant polynomials real, and how does this affect the factorization of the spectral density?

Key findings

• The spectral density of the sampled MCARMA process admits an asymptotic expansion as Δ ↓ 0, with the leading-order term proportional to Δ^{2(p−q)−1} (1−cos ω)^{pd} times a trigonometric function.
• The filtered process converges in distribution to a vector moving average process, with the asymptotic moving average representation explicitly derived.
• The leading coefficient in the asymptotic expansion is determined by the product of the eigenvalues of the system matrix A and the Eulerian polynomial coefficients.
• The factorization of the spectral density into terms involving (1−cos ω − ξ_j) allows for a real-valued moving average representation, even when the roots ξ_j are complex, due to conjugate symmetry.
• The method provides a practical approximation for the moving average representation when Δ is small, which is otherwise intractable for fixed Δ > 0.
• The paper conjectures that the roots ξ_{2k−1,j} of the generalized Eulerian polynomials are real and greater than 2, which would ensure real-valued η(ξ_j) and thus real coefficients in the factorization.