INFERENCE FOR MA(1) PROCESSES WITH A ROOT ON OR NEAR THE
UNIT CIRCLE
Richard A. Davis
Meiching Chen
William T. M. Dunsmuir
Abstract: This paper considers maximum likelihood estimation (MLE) for MA(1) processes
when the moving average parameter is on or near the unit circle. The asymptotic theory to be
presented allows the use of the generalized likelihood ratio test for testing the null hypothesis
of a unit root. The asymptotic distributions of the MLE and the largest local maximizer, the
estimator which yields the local maximum closest to the unit circle, are shown
to be different. The limit distributions of two estimates provide a very accurate
approximation to the finite sample size and power of the tests considered. A comparison is
made of the power of four tests of the null hypothesis that the moving average
parameter is equal to one versus the alternative that it is less than one. The four tests are
based on the MLE, the largest local maximizer, the generalized likelihood ratio
test and Tanaka’s score type test. The use of the generalized likelihood ratio test is
recommended overall since it always dominates the tests based on the MLE and the
largest local maximizer and dominates the score type test for close alternatives
to the null hypothesis. For alternatives very close to the unit circle the score type
test has slightly higher power but this is evident only in the third decimal place.
2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;
Key words and phrases: -