Estimation of inverse autocovariance matrices for long memory processes

TL;DR

This paper introduces a method for estimating inverse autocovariance matrices of long memory processes using a modified Cholesky decomposition, demonstrating spectral norm consistency and applications to regression models with long-memory errors.

Contribution

It develops a consistent estimation approach for inverse autocovariance matrices of long memory processes and extends it to regression models with long-memory errors.

Findings

Spectral norm consistency of the estimator is established.

The method performs well in regression models with long-memory errors.

Simulation results support the theoretical properties.

Abstract

This work aims at estimating inverse autocovariance matrices of long memory processes admitting a linear representation. A modified Cholesky decomposition is used in conjunction with an increasing order autoregressive model to achieve this goal. The spectral norm consistency of the proposed estimate is established. We then extend this result to linear regression models with long-memory time series errors. In particular, we show that when the objective is to consistently estimate the inverse autocovariance matrix of the error process, the same approach still works well if the estimated (by least squares) errors are used in place of the unobservable ones. Applications of this result to estimating unknown parameters in the aforementioned regression model are also given. Finally, a simulation study is performed to illustrate our theoretical findings.