A Linear Theory for Noncausality

A LINEAR THEORY FOR NONCAUSALITY'
Different definitions of noncausality (according to Granger, Sims, Pierce and Haugh),
are analyzed in terms of orthogonality in the Hilbert space of square integrable variables.
Conditions, when necessary, are given for their respective equivalence. Some problems of
testability are mentioned. Finally noncausality is also analyzed in terms of "rational
expectations," extending previous results of Sims.

I . INTRODUCTION
1.1. Presentation of the Paper
FOLLOWINGRANGER'S[9] and Sims' [21] papers, the noncausality concept
has taken on great importance in the econometric literature. This concept is
essentially the same as the concept of transitivity introduced into statistics by
Bahadur [2] and used in sequential analysis (see, e.g., Hall, Wijsman, and Ghosh
[12]). Intuitively, transitivity can be presented in the following way: a sub-process
(z,) of a multivariate stochastic process (x,) is transitive if the past and current
values of z, are sufficient to forecast z,,,. Equivalently, if x, is partitioned
into (z,, y,), we say that the process generating y, does not cause the process generating
z,.
A precise statement of this intuitive definition can be made in different ways.
In some of our previous work, noncausality is couched in terms of sequences of
independence conditions between c-fields (see Florens and Mouchart [6,7] and
Florens, Mouchart, and Rolin [8]). In this paper we propose definitions in terms
of sequences of orthogonality conditions between linear subspaces of a Hilbert
space of random variables. This kind of presentation is implicit in most
econometric papers and was explicitly used by Hosoya [14].