Likelihood Inference for a Nonstationary Fractional Autoregressive Model

This paper discusses model based inference in an autoregressive model for fractional processes based on the Gaussian likelihood. The model allows for the process to be fractional of order d or d – b; where d ≥ b > 1/2 are parameters to be estimated. We model the data X, …, Xт given the initial values Xº-n, n = 0, 1, …, under the assumption that the errors are i.i.d. Gaussian. We consider the likelihood and its derivatives as stochastic processes in the parameters, and prove that they converge in distribution when the errors are i.i.d. with suitable moment conditions and the initial values are bounded. We use this to prove existence and consistency of the local likelihood estimator, and to find the asymptotic distribution of the estimators and the likelihood ratio test of the associated fractional unit root hypothesis, which contains the fractional Brownian motion of type II.