Causality in the Long Run

1. INTRODUCTION
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, is said to cause x,+]if it contains information about the forecastability
for x,+,contained nowhere else in some large information set, which
includes xlPj,j r 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 x;"
generated by x;" = a(B)x,, where x, and x;" 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.