Likelihood inference for a vector autoregressive model which allows for fractional and cofractional processes ∗ ( Version 20 ) Søren Johansen † University of Copenhagen and CREATES and Morten

This paper discusses model based inference in a vector autoregressive model for cofractional processes based on the Gaussian likelihood. The model allows the process Xt to be fractional of order d and cofractional of order d−b, that is, there exist vectors β for which βXt is fractional of order d − b. The parameters b and d satisfy either d ≥ b ≥ 1/2, d = b ≥ 1/2, or d = d0 ≥ b ≥ 1/2. We model the data X1, . . . , XT given initial values X−n, n = 0, 1, . . ., under the assumption that the errors are i.i.d. Np(0,Ω). We consider the conditional likelihood and its derivatives as stochastic processes in the parameters, and prove that they converge in distribution when errors are i.i.d. with suitable moment conditions and initial values are bounded. We use this to prove existence and consistency of the maximum likelihood estimator, and to find asymptotic distributions of estimators and the likelihood ratio test for cointegrating rank from the asymptotic properties of score and information.