45 Pages Posted: 21 May 2010
Date Written: May 18, 2010
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 X(t) to be 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. The parameters d and b satisfy either d≥b≥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≤b≤d≤d₁ 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 of β is asymptotically mixed Gaussian and estimators of the remaining parameters are asymptotically Gaussian. We also find the asymptotic distribution of the likelihood ratio test for cointegration rank, which is a functional of fractional Brownian motion of type II.
Keywords: cofractional processes, cointegration rank, fractional cointegration, likelihood inference, vector autoregressive model
JEL Classification: C32
Suggested Citation: Suggested Citation
Nielsen, Morten Ørregaard and Johansen, Soren, Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model (May 18, 2010). Univ. of Copenhagen Dept. of Economics Discussion Paper No. 10-15. Available at SSRN: https://ssrn.com/abstract=1612765 or http://dx.doi.org/10.2139/ssrn.1612765