Abstract
This paper studies the autoregression models of order one, in a general time series setting that allows for weakly dependent innovations. Let {X t } be a linear process defined by X t = Σ ∞k=0 ψ k ɛ t−k, where {ψ k , k ≥ 0} is a sequence of real numbers and {ɛ k , k = 0, ±1, ±2, …} is a sequence of random variables. Two results are proved in this paper. In the first result, assuming that {ɛ k , k ≥ 1} is a sequence of asymptotically linear negative quadrant dependent (ALNQD) random variables, the authors find the limiting distributions of the least squares estimator and the associated regression t statistic. It is interesting that the limiting distributions are similar to the one found in earlier work under the assumption of i.i.d. innovations. In the second result the authors prove that the least squares estimator is not a strong consistency estimator of the autoregressive parameter α when {ɛ k , k ≥ 1} is a sequence of negatively associated (NA) random variables, and ψ 0 = 1, ψ k = 0, k ≥ 1.
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This research is supported by the National Natural Science Foundation of China under Grant Nos. 10971081 and 11001104, and 985 Project of Jilin University.
This paper was recommended for publication by Editor Guohua ZOU.
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Zhang, Y., Yang, X., Dong, Z. et al. The limit theorem for dependent random variables with applications to autoregression models. J Syst Sci Complex 24, 565–579 (2011). https://doi.org/10.1007/s11424-011-8119-z
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DOI: https://doi.org/10.1007/s11424-011-8119-z