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

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:77

Article

Davis, RA ; Mikosch, T

关键词: moving average process;    unit roots;    non-invertible moving averages;    maximum likelihood estimation;    stable distribution;    stable integral;   

DOI  :  10.1016/S0304-4149(98)00039-8

来源: Elsevier

【 摘 要 】

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 theta in an MA(1) model when theta is equal to or close to 1. Using the local parameterization, beta = T(1 - theta), where T is the sample size, it was shown that the likelihood, as a function of beta, converged to a stochastic process. From this, the limit distributions of T(<(theta)over cap>(MLE) - 1) and T(<(theta)over cap>(LM) - 1) (<(theta)over cap>(MLE) is the maximum likelihood estimator and <(theta)over cap>(LM) 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 H-0: theta = 1 vs. theta < 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 alpha-stable noise with alpha is an element of (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 T-1/2 rate obtained in the finite variance case to (T/ In T)(1/alpha). In the non-invertible setting of this paper, the rate is the same as in the finite variance case. That is, T(<(theta)over cap>(LM) - 1) converges in distribution and the pile-up effect, i.e., lim(Upsilon-->infinity) P(<(theta)over cap>(LM) = 1), is slightly less than in the finite variance case. It is also of interest to note that the limit distributions of T(<(theta)over cap>(LM) -1) for different values of alpha is an element of (0, 2] are remarkably similar. (C) 1998 Elsevier Science B.V. All rights reserved.

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