440 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
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there may be some set of » endogenous variables to be replaced
whose regressions together involve less than 7 predetermined
variables. Alternatively, counting the instruments included
in the equation to be estimated there may not be as many
instruments used in the final stage as there are parameters to
be estimated. This can happen, of course, although it is
perhaps relatively unlikely. If it does occur, then it is a sign
that the equation in question is unidentifiable from the sample
available, that the causal information contained in the sample
is insufficient to allow estimation of the equation without re-
laxing the inconsistency requirements. To put it another way,
it can be argued that to rectify this situation by the introduc-
tion in the first-stage regressions of variables failing the causal
test as described is an ad hoc device which adds no causal
information. While such variables may in fact appear in such
regressions with non-zero coefficients in the probability limit,
their use in the sample adds nothing to the quality of the
estimates save the ability to secure numbers and disguise the
problem.
Of course, such an argument is a bit too strong. Whether
a variable adds significantly to correlation is a function of
what one means by significance. The problem is thus a con-
tinuous rather than a discrete one and should be treated as such.
For the criterion of significance used, in some sense, the equa-
tion in question cannot be estimated from the sample in the
circumstance described; it may be estimatable with a less strin-
gent significance criterion. In practice, if the significance
requirements are relaxed, the moment matrix to be inverted
Will pass from singularity to near-singularity and estimated
asymptotic standard errors will be large rather than infinite.
The general point is that if multicollinearity cannot be suffi-
ciently eliminated using causal information, little is to be
gained by eliminating it by introducing more or less irrelevant
variables.
A somewhat related point is that the use of different vari-
6] Fisher - pag. 56
