This paper discusses model-based inference in an autoregressive model for fractional processes based on the Gaussian likelihood. The model has a verifiable criterion in terms of the roots of a polynomial for the process to be fractional of order d or d-b. Fractional differencing involves infinitely many past values and because we are interested in nonstationary processes we model the data X_{1},...,X_{T} 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 they are all zero. We consider the Gaussian likelihood and its derivatives as stochastic processes in the parameters which include d and b, and prove that they converge in distribution when the errors are i.i.d. with suitable moment conditions. We use this to prove existence and consistency of the maximum 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.
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Paper provided by Queen's University, Department of Economics in its series Working Papers with number
1172.
Find related papers by JEL classification: C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models
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