Testing for a Moving Average Unit Root

Testing for a unit root in the moving average model is discussed. First, for the
stationary MA(1) model, we suggest a score type test which is locally best
invariant and unbiased. Performance of the test for finite samples is compared
with the most powerful test. The asymptotic behavior of the test is also
considered by computing the limiting power under a sequence of local
alternatives. We then extend the model to an infinite order MA and suggest a
test for this extended case.
1. INTRODUCTION
Much attention has recently been paid to noninvertible moving average (MA)
models. Estimators and test statistics behave quite differently from those
arising from invertible MA models and various statistical properties have
been explored, for example, in Cryer and Ledolter [5], Sargan and Bhargava
[l 11, Anderson and Takemura [3], and Tanaka and Satchel1 [13].
In this paper we concentrate on testing for a unit root in MA models. The
basic model we shall deal with is the pure MA(1) model
yl=er-ae,+l; ( t = l , . . . ,T ) , (1)
where 1 a 1 S 1 and eo, el ,. . . - NID(0, a2).Then our problem is to test
The distribution of the maximum likelihood estimator (MLE) of CY under Ho
is unknown even asymptotically [13], so we are reluctant to use such tests as
likelihood ratio or Wald tests. The situation remains unchanged if we consider
the nonstationary MA(1) model with eo = 0, although the MLE of a
in the nonstationary case has an asymptotic distribution different from that
in the stationary case [13]. In Section 2 we suggest a score type test which
is locally best invariant and unbiased (LBIU). We can, of course, devise the
corresponding test in the nonstationary case, but that case is much easier to
analyze because of conditioning eo = 0. This is, in part, a reason for concentrating
discussions on the stationary case. We then examine the power property
of the test under finite samples comparing with the most powerful
invariant (MPI) test. The asymptotic properties of our test are also studied
by computing the limiting power as T-, w under a sequence of local alternatives
H, :a = 1 - c/T with c a fixed, positive constant. It is found that
the finite sample power under c = T(l - a )can be well approximated by the
limiting power under the same value of c.