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and if that set includes the instruments explicitly in the equa- 
tion, the latter requirement presents no problem since the nor- 
mal equations of ordinary least squares imply that such cor- 
relations are zero even in the sample (). When different 
instruments are used in the replacement of different variables, 
however, or when the instruments so used do not include those 
explicitly in the equation, the danger of inconsistency from 
this source does arise. 
There are several ways of handling this without sacrificing 
the major benefits of our procedures. One way is simply to 
argue that those procedures are designed to include in the 
regression for any right-hand endogenous variable any instru- 
ment which is correlated with the residuals from that regression 
computed without that instrument. The excluded instruments 
are either those which are known a priori not to be direct or 
indirect causes of the variable to be replaced or those which 
fail to add significantly to the correlation of the regression in 
question. The former instruments are known a priori not to 
appear in equations explaining the variable to be replaced and 
hence cannot be correlated in the probability limit with the 
residual from the regression unless both they and the replaced 
variable are affected by some third variable not included in 
that regression (#). Such a third variable cannot be endogen- 
ous, however, since in that case the excluded instruments in 
question would also be endogenous; moreover, our procedure 
is designed to include explicitly any instrument significantly 
affecting the variable to be replaced. Any such third variable 
must therefore be one omitted from the model and it may not 
be stretching things too far to disregard correlations between 
residuals and excluded instruments stemming from such a 
source. 
(9) This is the case when the reduced form equations are used, for 
example, as in the classic version of two-stage least squares. 
(*) If they were non-negligibly caused by the replaced variable itself 
they would be endogenous. contrarv to assnmption. 
‘61 Fisher - pag. 58