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used as instruments (except the variable explained by the equa- 
tion) must be replaced by a linear combination of instruments 
and the dependent variable regressed on such linear combina- 
tions. If the second stage of this procedure is not to involve 
mversion of a singular matrix, then (counting instrumental 
variables appearing in the equation) there must be at least as 
many instruments used as there are parameters to be estimated. 
Further, the linear combinations employed must not be per- 
fectly correlated. Current exogenous variables are simply not 
generally sufficient to meet this requirement in economy-wide 
models. Moreover, they do not cause lagged endogenous 
variables which are likely to be present in a dynamic system. 
Clearly, however, if the system is dynamic it will be pos- 
sible to use lagged exogenous variables as well as current ones. 
Such use may be especially helpful if lagged endogenous va- 
riables are to be treated as endogenous and replaced by linear 
combinations of instruments which can be taken as causing 
them in part. Indeed, if lagged endogenous variables are to 
be taken as endogenous, then exclusive use of current exogenous 
variables as instruments will not satisfy the causal criterion 
for instrumental variables already discussed. Since we have 
already seen that lagged endogenous variables should be used 
as instruments only with caution, it follows that lagged 
exogenous variables may well provide a welcome addition to 
the collection of available instruments. 
Unfortunately, this also is unlikely to suffice. While it is 
true that one can always secure a sufficient number of instru- 
ments by using exogenous variables with larger and larger lags, 
such a procedure runs into several difficulties. In the first 
place, since rather long lags may be required, there may be 
a serious curtailment of available observations at the beginning 
of the time period to be used. Second, exogenous variables in 
the relatively distant past will be relatively indirect causes of 
even the lagged endogenous variables appearing in the equa- 
tion to be estimated: it follows that their use will fail the causal 
61 Fisher - pag. 46