Cointegration in frequency domain

Existence of a cointegration relationship between two time series in the time domain imposes restrictions on the series zero‐frequency behaviour in terms of their squared coherence, phase and gain, in the frequency domain. I derive these restrictions by studying cross‐spectral properties of a cointegrated bivariate system. Specifically, I demonstrate that if two difference stationary series, X(t) and Y(t), are cointegrated with a cointegrating vector [1 b] and thus share a common stochastic trend, then at the zero frequency, the squared coherence of (1 ‐ L)X(t) and (1 ‐ L)Y(t) will equal one, their phase will equal zero, and their gain will equal |b|.