On the spectral coherence between two periodically correlated processes

We introduce a general class of multivariate periodically correlated processes and their corresponding time-domain and spectral-domain characterizations. A spectral coherence based on the Hilbert-Schmidt inner product of the Fourier transforms is introduced to measure the dependence of two periodically correlated (PC) processes. An estimator for the spectral coherence is introduced and studied. A hypothesis on the presence of significant dependence is formulated and the corresponding testing procedure established. Numerical illustrations on the performance of the spectral coherence and its estimator are given using simulated and real PC time series.