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 0<b≤d, but conduct inference when the true values satisfy 0≤d₀-b₀<1/2 and b₀≠1/2, for which β₀′X_{t} 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 i.i.d. 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 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.