The definition of causation, discussed in Granger (1980) and elsewhere, has been widely applied in economics and in other disciplines. For this definition, a series y t is said to cause x t +l if it contains information about the forecastability for x t +l contained nowhere else in some large information set, which includes x t − j , j ≥ 0. However, it would be convenient to think of causality being different in extent or direction at seasonal or low frequencies, say, than at other frequencies. The fact that a stationary series is effectively the (uncountably infinite) sum of uncorrelated components, each of which is associated with a single frequency, or a narrow frequency band, introduces the possibility that the full causal relationship can be decomposed by frequency. This is known as the Wiener decomposition or the spectral decomposition of the series, as discussed by Hannan (1970). For any series generated by , where x t , and are both stationary, with finite variances and a ( B ) is a backward filter with B the backward operator, there is a simple, well-known relationship between the spectral decompositions of the two series.
Clive et al. (Thu,) studied this question.