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Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model

45 Pages Posted: 21 May 2010  

Morten Ørregaard Nielsen

Queen's University - Department of Economics

Soren Johansen

University of Copenhagen - Department of Economics; Aarhus University - CREATES

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Date Written: May 18, 2010

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 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

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

Morten Orregaard Nielsen (Contact Author)

Queen's University - Department of Economics ( email )

94 University Avenue
Kingston K7L 3N6, Ontario
Canada

HOME PAGE: http://www.econ.queensu.ca/faculty/nielsen/

Soren Johansen

University of Copenhagen - Department of Economics ( email )

Øster Farimagsgade 5
Bygning 26
1353 Copenhagen K.
Denmark

Aarhus University - CREATES ( email )

Nordre Ringgade 1
Aarhus, DK-8000
Denmark

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