Likelihood inference for a fractionally cointegrated vector autoregressive model

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 X_{t} is fractional of order d and cofractional of order d-b; that is, there exist vectors β for which β′X_{t} is fractional of order d-b, and no other fractionality order is possible. For b=1, the model nests the I(d-1) VAR model. We define the statistical model by 01/2, we prove that the limit distribution of T^{b₀}(β-β₀) is mixed Gaussian and 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.