[Paper Review] Baxter's inequality for finite predictor coefficients of multivariate long-memory stationary processes

Akihiko Inoue, Yukio Kasahara+1 more|arXiv (Cornell University)|Jul 10, 2015

Financial Risk and Volatility Modeling38 references3 citations

TL;DR

This paper establishes a multivariate extension of Baxter's inequality for finite predictor coefficients in long-memory stationary processes, using novel spectral domain representations via Fourier coefficients of the phase function and an enhanced alternating projection technique. It proves the convergence rate of predictor coefficients and prediction error covariances for q-variate FARIMA processes with common fractional differencing order, providing the first rigorous multivariate Baxter’s inequality for long-memory processes with explicit decay rates.

ABSTRACT

For a multivariate stationary process, we develop explicit representations for the finite predictor coefficient matrices, the finite prediction error covariance matrices and the partial autocorrelation function (PACF) in terms of the Fourier coefficients of its phase function in the spectral domain. The derivation is based on a novel alternating projection technique and the use of the forward and backward innovations corresponding to the predictions based on the infinite past and future, respectively. We show that such representations are ideal for studying the rates of convergence of the finite predictor coefficients, prediction error covariances, and the PACF as well as for proving a multivariate version of Baxter's inequality for a multivariate FARIMA process with a common fractional differencing order for all components of the process.

Motivation & Objective

• To extend Baxter’s inequality to multivariate long-memory stationary processes, where classical proofs relying on bounded spectral density fail.
• To develop explicit representations for finite predictor coefficients, prediction error covariances, and partial autocorrelation functions (PACF) in the spectral domain.
• To establish convergence rates for predictor coefficients and PACF in q-variate FARIMA processes with common fractional differencing order.
• To overcome non-commutativity issues in multivariate settings by introducing matrix-valued phase function Fourier coefficients and tailored alternating projections.
• To prove a multivariate version of Baxter’s inequality under minimal conditions, improving on prior univariate and multivariate results.

Proposed method

• Derives explicit representations of finite predictor coefficients, prediction error covariances, and PACF using the Fourier coefficients of the matrix-valued phase function in the spectral domain.
• Introduces an enhanced alternating projection technique tailored for multivariate processes, overcoming non-commutativity of matrix-valued AR and MA coefficients.
• Uses forward and backward innovation processes based on predictions from the infinite past and future to derive key identities.
• Applies von Neumann’s alternating projection framework to the spectral domain, enabling precise asymptotic analysis.
• Establishes convergence rates by bounding norms of coefficient sequences using properties of the gamma function and spectral conditions.
• Specializes results to q-variate FARIMA processes with common fractional differencing order d ∈ (0, 1/2), ensuring explicit decay rates.

Experimental results

Research questions

• RQ1 Can Baxter’s inequality be extended to multivariate long-memory processes where the spectral density is unbounded?
• RQ2 What are the precise convergence rates of finite predictor coefficients, prediction error covariances, and PACF for multivariate FARIMA processes?
• RQ3 How can the non-commutativity of matrix-valued AR and MA coefficients be handled in deriving predictor coefficient representations?
• RQ4 What spectral domain representation enables the derivation of convergence rates in the multivariate long-memory setting?
• RQ5 Can the finite predictor coefficients of a multivariate FARIMA process be shown to converge to their infinite-horizon counterparts at a rate O(n^{-d}) for d ∈ (0, 1/2)?

Key findings

• For a q-variate FARIMA process with d ∈ (0, 1/2), the finite predictor coefficients converge to the infinite-horizon coefficients at rate O(n^{-d}), as shown in Theorem 6.8.
• The prediction error covariance matrices v_n and ṽ_n decay at rate O(n^{-1-d}) as n → ∞, with explicit constants involving the gamma function and g(1)^{-1}, as stated in Theorem 6.5.
• The partial autocorrelation function α_n decays at rate O(n^{-1-d}) for d ∈ (-1/2, 1/2) ⋯ {0}, establishing precise asymptotic behavior.
• The finite predictor coefficient norm satisfies lim_{n→∞} n^{1+d} ||φ_n|| = ||c_0 g(1)^{-1}|| / |Γ(-d)|, providing a sharp asymptotic expression.
• Baxter’s inequality holds for multivariate FARIMA processes with d ∈ (0, 1/2), with a constant K such that ∑_{j=1}^n ||φ_{n,j} - φ_j|| ≤ K ∑_{j=n+1}^∞ ||φ_j||, as proven in Theorem 6.9.
• A backward Baxter inequality is established via time reversal, showing ∑_{j=1}^n ||φ̃_{n,j} - φ̃_j|| ≤ K̃ ∑_{j=n+1}^∞ ||φ̃_j|| for d ∈ (0, 1/2), confirming symmetry in convergence behavior.