Open Access
December 2011 Unit roots in moving averages beyond first order
Richard A. Davis, Li Song
Ann. Statist. 39(6): 3062-3091 (December 2011). DOI: 10.1214/11-AOS935


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.


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Richard A. Davis. Li Song. "Unit roots in moving averages beyond first order." Ann. Statist. 39 (6) 3062 - 3091, December 2011.


Published: December 2011
First available in Project Euclid: 24 January 2012

zbMATH: 1246.62183
MathSciNet: MR3012401
Digital Object Identifier: 10.1214/11-AOS935

Primary: 62M10

Keywords: moving average , Unit roots

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 6 • December 2011
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