Some frequency domain properties of fractionally cointegrated processes

Abstract

The paper shows that the multiple squared coherence at the zero frequency for fractionally differenced (fractionally) cointegrated processes is equal to one, while the simple squared coherences assume a value greater than zero but lower than one. In the bivariate case the multiple and simple squared coherence coincide and, therefore, the simple squared coherence at the zero frequency assumes a unitary value. It is also found that processes that are not fractionally cointegrated show, in general, positive, but lower than one, multiple and simple squared coherences at the zero frequency. In the case the dependent and independent variables are driven by different long memory factors, i.e. in the case when the dependent variable is orthogonal at the zero frequency to any of the regressors, the squared multiple coherence will assume a zero value, as any of the squared simple coherences. It is finally shown that all the above results also hold for the series in levels, as the frequency tends to zero.

Notes

 As noted by Granger and Weiss (1983), these results are valid in general for the CI(d, d) case. In the paper we also show that the same results hold for the CI(d, b) case, b > 0, d − b > 0