SEMAINE D'ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
436
but fail to add anything when introduced separately, then the
first one to be tested will not be dropped from the regression,
as omitting it in the presence of the other instrument will signi-
ficantly reduce correlation (622). While it is true that variables
may be dropped because of correlation with variables less pre-
ferred than the T-znd, which are never tested, the exclusion
of the latter variables seems to be a relatively weak reliance on
a priori information.
This brings us to the next point. Clearly, it is possible in
principle that instruments less preferred than the T-2nd would
in fact pass the correlation test described if that test were per-
formed after some lower-numbered instruments were tested and
dropped. Similarly, an instrument dropped at an early stage
might pass the test in the absence of variables later dropped
because of the increased number of degrees of freedom. One
could, of course, repeat the entire procedure in order to test
every previously dropped variable after each decision to omit;
it seems preferable, however, to rely on the a priori preference
ordering in practice and to insist that instruments which come
late in the 3-ordering pass a more stringent empirical test than
those which come early. The rationale behind the 3-ordering
is the belief that it is the earlier instruments in that ordering
which contribute most of the causal information, so that it
seems quite appropriate to calculate the degrees of freedom
for testing a given instrument by subtracting the number of
its place in the ordering from the total number of observations
(and allowing for the constant term) (2b).
Turning to another issue, it may be obje~ted that there is
no guarantee that the suggested procedures will result in a non-
singular moment matrix to be inverted at the last stace. That is
(#) This property was missing in the procedure suggested in the first
draft of this paper in which variables were added in ascending order of
preference and retained if they added significantly to correlation. I am
indebted to ALBERT ANDO for helpful discussions on this point.
(8) Admittedly, this argument loses some of its force when applied to
the modifications of the R-ordering given above
6 |
Fisher - pag. 55
