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The relationship between Granger's [8] and Sims' [13] concepts of noncausality is analyzed in terms of conditional independence and Bahadur's [1] concept of transitivity.In particular their equivalence is easily shown to disappear when uncorrelatedness is replaced by independence.IN THIS NOTE the relationship between alternative concepts of noncausality is analyzed using the tool of conditional independence among a-fields. (For the reader who is unfamiliar with this technique, the Appendix sketches the proofs and the basic technical apparatus, along with some basic motivations.) Furthermore, the relationship between the concepts of noncausality and transitivity is made explicit in order to facilitate, in econometric modelling, the use of results already obtained in sequential analysis.2 Consider a stochastic process x,, = (yn, zn) with n E N. Write X,7 for the a-field generated by (x, x,, , ... ., xm), i.e. Xnm = Vn< ?mXj (see Appendix), and similarly for Yn7 and Zn,. Note that for ease of exposition, we identify X,, and xn. Let U stand for the a-field generated by the parameter and/or the initial conditions. Granger's and Sims' concepts of noncausality may be written as follows:
DEFINITION 1 (Granger [8]): y does not cause z in Granger's sense iff
