SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 391

Denoting the covariance matrix of u, and y,_, by W(6) with
columns corresponding to elements of u, and rows corresponding
 to elements of y,_, » and that of w, and u,_, by V(8) (which
is assumed to be independent of ¢), with columns corresponding
 to «#, and rows to u,_, :

(2.4)

W{o)=E (DB) (DV(0i,
8=0

Since, by (R.3), V(8)=o0 for 6>o, this becomes:

(2.5)

W(o) = DV(o).

By (R.2), V(o) is diagonal, hence W(o) is triangular with
zero elements above the principal diagonal. Thus any element
of y, is uncorrelated with all higher-numbered elements of =,
Similarly,

(2.6)

W(1)=S (DB) (DV(0.

Hence all variables which appear in any given equation in (2.1)
save that variable which is to be explained by that equation
are uncorrelated with the disturbance from that equation, as
stated.
We have gone through this demonstration in detail partly
for later purposes and partly to exhibit the way in which each
of the assumptions (R.1)-(R.3) enter. We must now ask
whether those assumptions can be weakened.
In the first place, it is clear that the triangularity of A is
crucial. From (2.5), if A and therefore D is not triangular, then
W (0) will not be triangular either in general, and the elements
of y, cannot be taken as uncorrelated with higher-numbered
disturbances. This is well known, as in this case the system
(2.1) is truly simultaneous. In such a case, ordinary least

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