Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model

Abstract

We consider model based inference in a fractionally cointegrated (or cofractional) vector autoregressive model based on the conditional Gaussian likelihood. The model allows the process Xt to be fractional of order d and cofractional of order d b; that is, there exist vectorsfor which � 0 Xt is fractional of order d b: The parameters d and b satisfy either db � 1=2, d = b � 1=2, or d = d0 � b � 1=2. Our main technical contribution is the proof of consistency of the maximum likelihood estimators on the set 1=2� bdd1 for any d1� d0. To this end, we consider the conditional likelihood as a stochastic process in the parameters, and prove that it converges in distribution when errors are i.i.d. with suitable moment conditions and initial values are bounded. We then prove that the estimator ofis asymptotically mixed Gaussian and estimators of the remaining parameters are asymptotically Gaussian. We also …nd the asymptotic distribution of the likelihood ratio test for cointegration rank, which is a functional of fractional Brownian motion of type II.