Investigating Causal Relations by Econometric Models and Cross-spectral Methods

There occurs on some occasions a difficulty in deciding the direction of causality between
two related variables and also whether or not feedback is occurring. Testable definitions
of causality and feedback are proposed and illustrated by use of simple two-variable models.
The important problem of apparent instantaneous causality is discussed and it is suggested
that the problem often arises due to slowness in recording information or because a
sufficiently wide class of possible causal variables has not been used. It can be shown that
the cross spectrum between two variables can be decomposed into two parts, each relating
to a single causal arm of a feedback situation. Measures of causal lag and causal strength
can then be constructed. A generalisation of this result with the partial cross spectrum
is suggested.

1. INTRODUCTION
THEOBJECT of this paper is to throw light on the relationships between certain
classes of econometric models involving feedback and the functions arising in
spectral analysis, particularly the cross spectrum and the partial cross spectrum.
Causality and feedback are here defined in an explicit and testable fashion. It is
shown that in the two-variable case the feedback mechanism can be broken down
into two causal relations and that the cross spectrum can be considered as the
sum of two cross spectra, each closely connected with one of the causations.
The next three sections of the paper briefly introduce those aspects of spectral
methods, model building, and causality which are required later. Section 5 presents
the results for. the two-variable case and Section 6 generalises these results for
three variables.