On the identification of fractionally cointegrated VAR models with the F(d) condition

This paper discusses identification problems in the fractionally cointegrated system of Johansen (2008) and Johansen and Nielsen (2012). It is shown that several equivalent re-parameterizations of the model associated with different fractional integration and cointegration parameters may exist for any choice of the lag length, also when the true cointegration rank is known. The properties of these multiple non-identified 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 F(d) and it is a generalization to the fractional case of the I(1) condition in the VECM model. The assessment of the F(d) condition in the empirical analysis is relevant for the determination of the fractional parameters as well as the number of lags. The paper also illustrates the indeterminacy between the cointegration rank and lag length. It is proved that, under certain restrictions on the fractional parameters, the model with rank zero and k lags is equivalent to the model with full rank and k - 1 lags. This precludes the possibility to test for the nullity of the cointegration rank.