  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.
