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Maximum likelihood estimation for nearly non‐stationary stable autoregressive processes
Author(s) -
Zhang RongMao,
Chan Ngai Hang
Publication year - 2012
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.2011.00762.x
Subject(s) - autoregressive model , mathematics , unit root , star model , statistics , limit (mathematics) , maximum likelihood , mathematical analysis , autoregressive integrated moving average , time series
The maximum likelihood estimate (MLE) of the autoregressive coefficient of a near‐unit root autoregressive process Y t = ρ n Y t −1 + ɛ t with α ‐stable noise { ɛ t } is studied in this paper. Herein ρ n = 1 − γ / n , γ ≥ 0 is a constant, Y 0 is a fixed random variable and ε t is an α ‐stable random variable with characteristic function φ ( t , θ ) for some parameter θ . It is shown that when 0 < α < 1 or α > 1 and E ɛ 1 = 0, the limit distribution of the MLE of ρ n and θ are mixtures of a stable process and Gaussian processes. On the other hand, when α > 1 and E ɛ 1 ≠ 0, the limit distribution of the MLE of ρ n and θ are normal. A Monte Carlo simulation reveals that the MLE performs better than the usual least squares procedures, particularly for the case when the tail index α is less than 1.