  Distinguishing between cause and effect using time series observational data
is a major challenge in many scientific fields. A new perspective has been
provided based on the principle of Independence of Causal Mechanisms (ICM),
leading to the Spectral Independence Criterion (SIC), postulating that the
power spectral density (PSD) of the cause time series is uncorrelated with the
squared modulus of the frequency response of the filter generating the effect.
Since SIC rests on methods and assumptions in stark contrast with most causal
discovery methods for time series, it raises questions regarding what
theoretical grounds justify its use. In this paper, we provide answers covering
several key aspects. After providing an information theoretic interpretation of
SIC, we present an identifiability result that sheds light on the context for
which this approach is expected to perform well. We further demonstrate the
robustness of SIC to downsampling - an obstacle that can spoil Granger-based
inference. Finally, an invariance perspective allows to explore the limitations
of the spectral independence assumption and how to generalize it. Overall,
these results support the postulate of Spectral Independence is a well grounded
leading principle for causal inference based on empirical time series.
