Cointegration in Frequency Domain - Daniel Levy (Journal of Time Series Analysis, 2002)
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 [11emb] 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|.
Keywords: C01; C18; C22; C32; C40; 330; Common Stochastic Trend; Cointegration; Frequency Domain Anlysis; Spectral Analysis; Spectrum; Cross-Spectrum; Zero-Frequency; Short-Run; Long-Run