The Time Domain Implications of Phase Angles and Tau

In cross-spectral analysis, the tau statistic is used to infer lead-lag relationships between series in the time domain. Unfortunately, it is frequently inferred that tau is meaningful only when the phase diagram is linear. In this paper it is shown that tau is far more robust than is generally recognized. By appeal to the Cramer Decomposition Theorem, the time-domain interpretation of tau is developed and shown to possess optimum mean square properties. While some of the theoretical background is expository, its inclusion facilitates an intuitive analysis of a number of models. We focus in particular on a model that generates a phase reversal similar to that often found in the spectral analysis of aggregate economic series.