On Estimating an ARMA Model with an MA Unit Root

B.P.M. MCCABE
University of British Columbia
S.J. LEVBOURNE
University of Nottingham
This paper investigates the behavior of the maximum likelihood estimator of a
Gaussian autoregressive moving average model with a unit root in the moving average
polynomial. The results are primarily of interest in testing hypotheses that
involve moving average unit roots as, for example, when testing for stationarity of
a series.

1, INTRODUCTION
It is well known that classical likelihood theory (i.e., T 1/2-consistency and asymptotic
normality) does not apply to autoregressive (AR) models that contain a
unit root. The fact that this is also true of moving average (MA) models with a
unit root seems to be less well known in spite of the work, among others, of Cryer
and Ledolter (1 98 I), Sargan and Bhargava (1983), Tanaka and Satchel1 (1 989),
Potscher (1991), and Shephard (1993). In the MA(1) case when a unit root is
present, the maximum likelihood estimator (MLE) of 0, the MA parameter, is
T-consistent in contrast to the usual T ' I 2 rate when 181< 1. Recently, the asymptotic
distribution of the MLE of 0 has been rigorously derived by Davis, Chen,
and Dunsmuir (1994) and Davis and Dunsmuir (1996) in the Gaussian case. In
addition to very accurate approximations to the distribution they also provide
critical values for the generalized likelihood ratio test and show it has good power
properties for a broad range of values of 8. In the ARMA case the consistency of
the MLE's of 0 and the AR parameters has been established by Potscher in the
presence of an MAunit root even when a Gaussian pseudolikelihood is used. The
object of this paper is to establish the (vector) marginal distribution of the MLE
of the AR parameters in the case where an ARMA(p, 1) model is fitted to observations
from a stationary Gaussian ARMA(p, 1) process and the true value of 0,
the MAparameter, is unity. It turns out that the marginal distribution of the MLE
of the AR parameters, in this situation, is asymptotically the same as the distribution
of parameter estimates (MLE's) in a pure AR(p) model fitted to a stationary pure AR(p) process. In addition, it is shown that the marginal asymptotic
distribution of the MLE of the MA parameter is the same as if a pure MA(1)
model was fitted to an MA(1) process with a unit root. Strictly speaking the
results presented here refer to a consistent root of the score equation and are
termed "local minimizers" by Davis and Dunsmuir (1996). In principle, the results
presented here could be extended along the lines of Potscher but at the
expense of considerable complications in the algebra.