An Extension of Cointegration to Fractional Autoregressive Processes

16 Pages Posted: 18 Oct 2010

See all articles by Soren Johansen

Soren Johansen

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

Date Written: October 15, 2010


This paper contains an overview of some recent results on the statistical analysis of cofractional processes, see Johansen and Nielsen (2010b). We first give an brief summary of the analysis of cointegration in the vector autoregressive model and then show how this can be extended to fractional processes. The model allows the process X_{t} to be fractional of order d and cofractional of order d-b≥0; that is, there exist vectors β for which β′X_{t} is fractional of order d-b. We analyse the Gaussian likelihood function to derive estimators and test statistics. The asymptotic properties are derived without the Gaussian assumption, under suitable moment conditions. We assume that the initial values are bounded and show that they do not influence the asymptotic analysis. The estimator of β is asymptotically mixed Gaussian and estimators of the remaining parameters are asymptotically Gaussian. The asymptotic distribution of the likelihood ratio test for cointegration rank is a functional of fractional Brownian motion.

Keywords: cofractional processes, cointegration rank, fractional cointegration, likelihood inference, vector autoregressive model

JEL Classification: C32

Suggested Citation

Johansen, Soren, An Extension of Cointegration to Fractional Autoregressive Processes (October 15, 2010). Univ. of Copenhagen Dept. of Economics Discussion Paper No. 10-28, Available at SSRN: or

Soren Johansen (Contact Author)

University of Copenhagen - Department of Economics ( email )

Øster Farimagsgade 5
Bygning 26
1353 Copenhagen K.

Aarhus University - CREATES ( email )

Nordre Ringgade 1
Aarhus, DK-8000

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