Gaussian likelihood-based inference for non-invertible MA(1) processes with SαS 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 θ in an MA(1) model when θ is equal to or close to 1. Using the local parameterization, β=T(1−θ), where T is the sample size, it was shown that the likelihood, as a function of β, converged to a stochastic process. From this, the limit distributions of T(θ̂MLE−1) and T(θ̂LM−1) (θ̂MLE is the maximum likelihood estimator and θ̂LM 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 H0: θ=1 vs. θ<1 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 α-stable noise with α∈(0,2). 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 T1/2 rate obtained in the finite variance case to (T/ln T)1/α. In the non-invertible setting of this paper, the rate is the same as in the finite variance case. That is, T(θ̂LM−1) converges in distribution and the pile-up effect, i.e., limT→∞P(θ̂LM=1), is slightly less than in the finite variance case. It is also of interest to note that the limit distributions of T(θ̂LM−1) for different values of α∈(0,2] are remarkably similar.