A prediction perspective on the Wiener-Hopf equations for time series

TL;DR

This paper offers a new perspective on solving Wiener-Hopf equations in time series by combining linear prediction with deconvolution, providing a more interpretable approach and error bounds for spectral density approximations.

Contribution

It introduces an alternative solution to Wiener-Hopf equations that aligns with classical time series methods and derives error bounds for rational spectral density approximations.

Findings

Proposes a solution combining linear prediction and deconvolution.

Provides error bounds for spectral density approximation errors.

Reinterprets Wiener-Hopf equations in a more accessible statistical framework.

Abstract

The Wiener-Hopf equations are a Toeplitz system of linear equations that naturally arise in several applications in time series. These include the update and prediction step of the stationary Kalman filter equations and the prediction of bivariate time series. The celebrated Wiener-Hopf technique is usually used for solving these equations and is based on a comparison of coefficients in a Fourier series expansion. However, a statistical interpretation of both the method and solution is opaque. The purpose of this note is to revisit the (discrete) Wiener-Hopf equations and obtain an alternative solution that is more aligned with classical techniques in time series analysis. Specifically, we propose a solution to the Wiener-Hopf equations that combines linear prediction with deconvolution. The Wiener-Hopf solution requires the spectral factorization of the underlying spectral density function. For ease of evaluation it is often assumed that the spectral density is rational. This allows one to obtain a computationally tractable solution. However, this leads to an approximation error when the underlying spectral density is not a rational function. We use the proposed solution with Baxter's inequality to derive an error bound for the rational spectral density approximation.