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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA

By (BR.1), however, D=(I- A)-! is block-triangular, while
V(o) is block diagonal by (BR.2). It follows that their product
is block triangular with the same partitioning. Thus:

(5.3) W(0o)" =o for all I, J=1, …, N and J>I,

but this is equivalent to the proposition in question.
As in the special case of recursive systems, assumption
(BR.3) can be replaced by a somewhat different assumption:

(BR.3*) B is block-triangular with the same partitioning
as A, as is V(6) for all 6>>0. Further, either all B" or all
V(0) (6>0) are zero (I=1, ..., N).

To see that this suffices, observe that in this case every
term in (2.4) will be block-triangular.
Note, however, that whereas (BR.1)-(BR.3) patently suffice
to give W(1)=0 and thus to show that lagged endogenous
variables are uncorrelated with current disturbances, this is
not the case when (BR.3) is replaced by (BR.3*). As in the
similar case for recursive systems, what is implied by (BR.1),
(BR.2), and (BR.3*) in this regard is that W(1) is also blocktriangular
 with zero matrices on the principal diagonal so that
lagged endogenous variables are uncorrelated with the current
disturbances of the same or higher-numbered blocks, but not
necessarily with those of Jower-numbered ones.
If A and B are both block-triangular with the same partitioning,
 then the matrix DB is also block-triangular and the
system of difference equations given by (2.2) is decomposable.
In this case, what occurs in higher-numbered sectors never
influences what occurs in lower-numbered ones, so that there is
in any case no point in using current or lagged endogenous
variables as instruments in lower-numbered sectors. This is
an unlikely circumstance to encounter in an economy-wide
model in any essential way, but it may occur for partitionings
which split off a small group of equations from the rest of the

61 Fisher - pag. 28