Gaussian likelihood-based inference for non-invertible MA(1) processes with SS noise

A limit theory was developed in the papers of Davis and Dunsmuir (1996) and Davis et al. (1995) for the maximum likelihood estimator, based on a Gaussian likelihood, of the moving average parameter in an MA(1) model when is equal to or close to 1. Using the local parameterization, , where is the sample size, it was shown that the likelihood, as a function of , converged to a stochastic process. From this, the limit distributions of and ( is the maximum likelihood estimator and is the local maximizer of the likelihood closest to 1) were established. As a byproduct of the likelihood convergence, the limit distribution of the likelihood ratio test for testing vs. was also determined. In this paper, we again consider the limit behavior of the local maximizer closest to 1 of the Gaussian likelihood and the corresponding likelihood ratio statistic when the non-invertible MA(1) process is generated by symmetric -stable noise with . Estimates of a similar nature have been studied for causal-invertible ARMA processes generated by infinite variance stable noise. In those situations, the scale normalization improves from the traditional rate obtained in the finite variance case to . In the non-invertible setting of this paper, the rate is the same as in the finite variance case. That is, converges in distribution and the pile-up effect, i.e., , is slightly less than in the finite variance case. It is also of interest to note that the limit distributions of for different values of are remarkably similar.