Unit roots in moving averages beyond first order

TL;DR

This paper develops a new approach to analyze unit roots in moving average models beyond first order, using joint likelihood with an augmented initial value, and studies the asymptotic behavior of the GLR test.

Contribution

It introduces a novel likelihood-based method for unit root testing in higher-order moving averages, overcoming previous analytical limitations.

Findings

The GLR test performs competitively with existing tests for some alternatives.

The new approach simplifies likelihood computation for higher-order models.

Asymptotic properties of the GLR test are established.

Abstract

The asymptotic theory of various estimators based on Gaussian likelihood has been developed for the unit root and near unit root cases of a first-order moving average model. Previous studies of the MA(1) unit root problem rely on the special autocovariance structure of the MA(1) process, in which case, the eigenvalues and eigenvectors of the covariance matrix of the data vector have known analytical forms. In this paper, we take a different approach to first consider the joint likelihood by including an augmented initial value as a parameter and then recover the exact likelihood by integrating out the initial value. This approach by-passes the difficulty of computing an explicit decomposition of the covariance matrix and can be used to study unit root behavior in moving averages beyond first order. The asymptotics of the generalized likelihood ratio (GLR) statistic for testing unit roots are also studied. The GLR test has operating characteristics that are competitive with the locally best invariant unbiased (LBIU) test of Tanaka for some local alternatives and dominates for all other alternatives.