Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model


  • Søren Johansen,

    1. Dept. of Economics, University of Copenhagen, Øster Farimagsgade 5, building 26, 1353 København K, Denmark and CREATES;
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  • Morten Ørregaard Nielsen

    1. Dept. of Economics, Queen's University, Dunning Hall, room 307, 94 University Avenue, Kingston, Ontario K7L 3N6, Canada and CREATES;
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    • We are very grateful to Jim Stock and five referees for many useful and constructive comments that were above and beyond the call of duty and led to significant improvements to the paper. We are also grateful to Uwe Hassler, James MacKinnon, Ilya Molchanov, and seminar participants at various universities and conferences for comments, and to the Danish Social Sciences Research Council (FSE Grant 275-05-0220), the Social Sciences and Humanities Research Council of Canada (SSHRC Grant 410-2009-0183), and the Center for Research in Econometric Analysis of Time Series (CREATES, funded by the Danish National Research Foundation) for financial support. A previous version of this paper was circulated under the title “Likelihood Inference for a Vector Autoregressive Model Which Allows for Fractional and Cofractional Processes.”


We consider model based inference in a fractionally cointegrated (or cofractional) vector autoregressive model, based on the Gaussian likelihood conditional on initial values. We give conditions on the parameters such that the process Xt is fractional of order d and cofractional of order db; that is, there exist vectors β for which βXt is fractional of order db and no other fractionality order is possible. For b=1, the model nests the I(d−1) vector autoregressive model. We define the statistical model by 0 < bleqslant R: less-than-or-eq, slantd, but conduct inference when the true values satisfy 0leqslant R: less-than-or-eq, slantd0b0<1/2 and b0≠1/2, for which β0Xt is (asymptotically) a stationary process. Our main technical contribution is the proof of consistency of the maximum likelihood estimators. To this end, we prove weak convergence of the conditional likelihood as a continuous stochastic process in the parameters when errors are independent and identically distributed with suitable moment conditions and initial values are bounded. Because the limit is deterministic, this implies uniform convergence in probability of the conditional likelihood function. If the true value b0>1/2, we prove that the limit distribution of inline image is mixed Gaussian, while for the remaining parameters it is Gaussian. The limit distribution of the likelihood ratio test for cointegration rank is a functional of fractional Brownian motion of type II. If b0<1/2, all limit distributions are Gaussian or chi-squared. We derive similar results for the model with d = b, allowing for a constant term.