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 correspond- 
ing to elements of y,_, » and that of w, and u,_, by V(8) (which 
is assumed to be independent of ¢), with columns correspond- 
ing 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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