HISTORY OF RELATIONSHIPS
9
and Watson ( 1973) gave an interesting example inspired by a geophysical
problem.
At about the same time, various other threads were coming together, in-
cluding the use of instrumental variates in estimation. The concepts of iden-
tifiability and over-identifiability (pertinent to many factor analysis models)
and estimation by instrumental variates as proposed by Reiersol ( 1945) were
merged in an interesting paper by Barnett ( 1969). His work was stimulated
by an instrument calibration problem where the aim was to compare the ac-
curacy of new instruments with that of a standard instrument for measuring
a certain lung function.
5. Instrumental variables, calibration, and factor analysis
Reiersol ( 1945) showed that if we observe a variable Z
1
(which may be
measured with or without error, so here we temporarily use the notation to
include a (possibly) error-free
z1
as a particular case), and which is correlated
with the true
x
1
,
y
1
but independent of
u1
,
e1
,
then
P1
may be estimated
by
(6)
b
- mzy
,-
mzx
It is easily shown that b1 converges to
P1

Barnett ( 1969) was interested in a structural relationship problem. He
considered first a two-equation model of the form
(7)
Y=Po,+Pllx,
but, instead of a set of true data points
(x1
,
y
1
,
z1),
each is observed with
an additive independently normally distributed error. Clearly, for estimat-
ing the slope
P11
in (7), the observed Z1 satisfy the conditions for an in-
strumental variable given by Riersol. Barnett showed that, without further
assumptions about variances, the maximum likelihood estimator of
P11
is
given by (6). An interesting feature of this result is that for the paired re-
lationship (7) MLE now works without further assumptions about the error
variances, and Reiersol's more general estimator is equivalent to ML in this
case. MLE works here because the number of parameters to be estimated is
equal to the number of estimating equations. Barnett went on to show that if
we have additional observed instrumental variable
U, V . . .
corresponding
to true unknown
u, v, ... ,
where
u
=
P
03
+ P13x,
etc., then we have an
over-identified model for MLE and also for instrumental variables because
each of Z , U, V, . . . might be used as an instrumental variable. This lat-
ter model is equivalent to a one-factor model in factor analysis, although
reparametrization is needed to put it in familiar factor analysis notation.
It
is now well established that more general factor analysis models can be ex-
pressed as reparametrizations of simultaneous linear structural relationships.
There is a detailed discussion in Anderson ( 1984).
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