Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model

CREATES Research Paper No. 2010-24

46 Pages Posted: 2 Jun 2010

See all articles by Soren Johansen

Soren Johansen

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

Morten Ørregaard Nielsen

Queen's University - Department of Economics

Margit Sommer

School of Economics and Management, University of Aarhus; School of Economics and Management; CREATES Research Papers

Date Written: June 1, 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=d_{0}≥b≥1/2. Our main technical contribution is the proof of consistency of the maximum likelihood estimators on the set 1/2≤b≤d≤d_{1} for any d_{1}≥d_{0}. 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

Johansen, Soren and Nielsen, Morten Orregaard and Sommer, Margit, Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model (June 1, 2010). CREATES Research Paper No. 2010-24, Available at SSRN: https://ssrn.com/abstract=1618653 or http://dx.doi.org/10.2139/ssrn.1618653

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

Morten Orregaard Nielsen

Queen's University - Department of Economics ( email )

94 University Avenue
Kingston K7L 3N6, Ontario
Canada

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

Margit Sommer (Contact Author)

School of Economics and Management, University of Aarhus ( email )

School of Economics and Management
Building 1322, Bartholins Alle 10
DK-8000 Aarhus C
Denmark

School of Economics and Management ( email )

Building 350
DK-8000 Aarhus C
Denmark

CREATES Research Papers ( email )

School of Economics and Management
Building 1322, Bartholins Alle 10
DK-8000 Aarhus C
Denmark

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