On the Identification of Fractionally Cointegrated VAR Models With the F(d) Condition

ABSTRACT

This article discusses identification problems in the fractionally cointegrated system of Johansen and Johansen and Nielsen. It is shown that several equivalent reparametrizations of the model associated with different fractional integration and cointegration parameters may exist for any choice of the lag-length when the true cointegration rank is known. The properties of these multiple nonidentified models are studied and a necessary and sufficient condition for the identification of the fractional parameters of the system is provided. The condition is named $\mathcal{F}\left(d\right)$. This is a generalization of the well-known I(1) condition to the fractional case. Imposing a proper restriction on the fractional integration parameter, d, is sufficient to guarantee identification of all model parameters and the validity of the $\mathcal{F}\left(d\right)$ condition. The article also illustrates the indeterminacy between the cointegration rank and the lag-length. It is also proved that the model with rank zero and k lags may be an equivalent reparameterization of the model with full rank and k − 1 lags. This precludes the possibility to test for the cointegration rank unless a proper restriction on the fractional integration parameter is imposed.

KEY WORDS:

• Cofractional model
• Fractional cointegration
• Identification
• Lag selection

ACKNOWLEDGMENTS

The authors acknowledge support from CREATES - Center for Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish National Research Foundation. The authors are grateful to Niels Haldrup, Søren Johansen, Katarzyna Lasak, Bent Nielsen and Morten ørregaard Nielsen for their suggestions that improved the quality of this work. The authors are also grateful to an anonymous referee for providing insightful comments. The authors would like to thank also James MacKinnon, Rocco Mosconi, Paolo Paruolo, the participants to the Third Long Memory Symposium (Aarhus 2013), the participants to the CFE’2013 conference (London 2013), and the seminar participants at Queen’s University and at Bologna University for helpful comments. Finally, special thanks to Sølveig Nygaard Sørensen for her careful proofreading of the final version of this article.