Likelihood inference for a nonstationary fractional autoregressive model

Soren Johansen () and Morten Ørregaard Nielsen ()

No 1172, Working Papers from Queen's University, Department of Economics

Abstract: 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 differencing involves in…nitely 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 initial values are zero. For the statistical analysis we assume the conditional 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 {1/2 < b < d < .inf}, and to fi…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.

Keywords: Dickey-Fuller test; fractional unit root; likelihood inference (search for similar items in EconPapers)
JEL-codes: C22 (search for similar items in EconPapers)
New Economics Papers: this item is included in nep-ets and nep-ifn
Date: Written
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http://www.econ.queensu.ca/working_papers/papers/qed_wp_1172.pdf First version 2008 (application/pdf)

Related works:
Working Paper: Likelihood inference for a nonstationary fractional autoregressive model (2007)
Working Paper: Likelihood Inference for a Nonstationary Fractional Autoregressive Model (2007)
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Persistent link: http://EconPapers.repec.org/RePEc:qed:wpaper:1172

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