We are very grateful to Jim Stock and five referees for many useful and constructive comments that were above and beyond the call of duty and led to significant improvements to the paper. We are also grateful to Uwe Hassler, James MacKinnon, Ilya Molchanov, and seminar participants at various universities and conferences for comments, and to the Danish Social Sciences Research Council (FSE Grant 275-05-0220), the Social Sciences and Humanities Research Council of Canada (SSHRC Grant 410-2009-0183), and the Center for Research in Econometric Analysis of Time Series (CREATES, funded by the Danish National Research Foundation) for financial support. A previous version of this paper was circulated under the title “Likelihood Inference for a Vector Autoregressive Model Which Allows for Fractional and Cofractional Processes.”
Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model
Article first published online: 26 NOV 2012
© 2012 The Econometric Society
Volume 80, Issue 6, pages 2667–2732, November 2012
How to Cite
Johansen, S. and Nielsen, M. Ø. (2012), Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model. Econometrica, 80: 2667–2732. doi: 10.3982/ECTA9299
- Issue published online: 26 NOV 2012
- Article first published online: 26 NOV 2012
- Manuscript received May, 2010; final revision received February, 2012.
- Cofractional processes;
- cointegration rank;
- fractional cointegration;
- likelihood inference;
- 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 Xt is fractional of order d and cofractional of order d−b; that is, there exist vectors β for which β′Xt is fractional of order d−b and no other fractionality order is possible. For b=1, the model nests the I(d−1) vector autoregressive model. We define the statistical model by 0 < bd, but conduct inference when the true values satisfy 0d0−b0<1/2 and b0≠1/2, for which β0′Xt 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 independent and identically distributed 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 is mixed Gaussian, while 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.