Baxter’s inequality for finite predictor coefficients of multivariate long-memory stationary processes

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 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.2010 Mathematics Subject Classification. Primary 60G25; secondary 62M20, 62M10.

Closed-form expression for finite predictor coefficients of multivariate ARMA processes

“…The proof of the closed-form expression for φ n,j is long. One important ingredient of the proof is the explicit representation of φ n,j (see the proof of Theorem 6 in D below), which was obtained recently in Inoue et al [11], extending the earlier univariate result in Inoue and Kasahara [8]; see also Inoue et al [10] and Inoue and Kasahara [9] for related work. To explain another important ingredient of the proof of the closed-form expression for φ n,j , we recall that, for h : T → C d×d satisfying (1) and (2), there exists h ♯ : T → C d×d that satisfies (2) and…”

“…The representation in [11] holds both for long and short memory processes, and has several applications such as the proof of Baxter's inequality for multivariate long-memory processes in [11]. The representation of φ n,j in [11] is, however, not a closedform expression since it involves infinite series. In this paper, for multivariate ARMA processes, we transform the representation in [11] to a closed-form expression for φ n,j .…”

“…The representation of φ n,j in [11] is, however, not a closedform expression since it involves infinite series. In this paper, for multivariate ARMA processes, we transform the representation in [11] to a closed-form expression for φ n,j . The advantage of the latter is clear from the fact that it can be viewed as a linear-time algorithm to compute φ n,1 , .…”

Matricial Baxter's theorem with a Nehari sequence

“…Since , by von Neumann's alternating projections, converges to the orthogonal projection onto their intersection as . Henceforth, when is rigid, it follows from that In fact, when is rigid, coincides with the space of ‐valued polynomials with degree less than n for , and analogous projection formulas can be used for studying MOPUC, as in the authors' earlier work ; see also Inoue–Kasahara–Pourahmadi for applications to prediction problems.…”

“…In fact, when ℎ ℎ is rigid, 2 ∩ 2 coincides with the space of -valued polynomials with degree less than for = 0, 1, 2 …, and analogous projection formulas can be used for studying MOPUC, as in the authors' earlier work [13]; see also Inoue-Kasahara-Pourahmadi [12] for applications to prediction problems.…”