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).