Maximum likelihood estimation for nearly non‐stationary stable autoregressive processes

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.