Spectral Analysis of Fractionally Cointegrated Systems

Cointegration imposes restrictions on the frequency domain behavior of a time series at the zero-frequency. We derive these restrictions for a multivariate fractionally cointegrated system. In particular, we consider a p-vector time series integrated of order d with r cointegrating relations, given by the rows of [I_{r};ß'], where the cointegration errors are integrated of order d-b, d=b>0. We show that, at the zero-frequency, the spectral density matrix of the d'th differenced series has reduced rank (p-r), the coherence and phase measures (multiple and partial) equal unity and zero, respectively, and the gain is the matrix of cointegrating coefficients. Extensions to noncontemporaneous cointegration, seasonal cointegration, and different fractional values of b for each cointegrating relation are considered.