Likelihood inference for a nonstationary fractional autoregressive model

This paper discusses model-based inference in an autoregressive model for fractional processes which allows the process to be fractional of order d or d􀀀b. Fractional di¤er- encing involves in nitely many past values and because we are interested in nonstation- ary processes we model the data X1; : : : ;XT given the initial values X􀀀n; n = 0; 1; : : :, as is usually done. The initial values are not modeled but assumed to be bounded. This represents a considerable generalization relative to all previous work where it is assumed that initial values are zero. For the statistical analysis we assume the condi- tional Gaussian likelihood and for the probability analysis we also condition on initial values but assume that the errors in the autoregressive model are i.i.d. with suitable moment conditions. We analyze the conditional likelihood and its derivatives as stochastic processes in the parameters, including d and b, and prove that they converge in distribution. We use the results to prove consistency of the maximum likelihood estimator for d; b in a large compact subset of f1=2 < b < d < 1g, and to nd the asymptotic distribution of the estimators and the likelihood ratio test of the associated fractional unit root hypothesis. The limit distributions contain the fractional Brownian motion of type II.