10 PETER SPRENT

6. Estimation of the intercept

In most estimation procedures for independent errors, the intercept esti-

mator turns out to be

(8)

so it is customary for algebraic economy to shift the origin to the data mean

and rewrite the relationship to be determined as

y =

[J

1

x.

However, Moran

( 1971) pointed out that for a structural relationship this shift of origin to a

data-determined point is not equivalent to fitting a line through a predeter-

mined origin that is not the data mean. He showed that without assumptions

about error variances MLE now works in the normal case and gives for the

slope estimator b1

=

Y

I

X,

providing

X

-1-

0. In practice this might turn out

to be a very poor estimator, particularly if one or both error variances are

large and the data mean is close to the origin. Furthermore, we note that b1

does not reduce to the usual regression estimator when

x

is error free. The

difficulty seems to be that Moran's estimator requires only first moments and

appears to ignore information about [J

1

contained in the second moments.

This point needs further examination, but it would seem that the method

fails to take into account the fact that we have in reality

n

+

1 data points,

n

of which are determined with error, and one (the origin) determined without

error. This suggests that some sort of weighted least squares approach, or a

constrained optimization approach, may be more appropriate.

7. Better methods of estimation

We have indicated some unsatisfactory features of ML estimation, espe-

cially in matters of consistency. Even in the apparently simple problem of

fitting a line with zero intercept, we saw in Section 6 that there may be

difficulties. Such problems have stimulated a search for other methods of

estimation. These have centered largely on the use of pivotals with distri-

butions independent of the parameters to be estimated. The use of pivotals

has a long history, dating back in a regression context at least to Williams

( 1955) and earlier in other estimation problems. As far as the functional re-

lationship is concerned, this approach culminated in work by Morton ( 1981)

already mentioned, where he introduced the concept of unbiased estimating

equations. Bayesian methods have been proposed by Lindley and El Sayyad

( 1968), and robust or adaptive methods of estimation may be appropriate in

this context; see, e.g., Carroll and Gallo ( 1982) and Bickel and Ritov ( 1987).

8. The Berkson model

A model that is often regarded as something of a curiosity is the so-called

controlled variable model introduced by Berkson ( 1950), which, for estima-

tion purposes, is equivalent to a classical regression model. The type of error

arising in Berkson's model is often a measurement error. The model differs

from a conventional functional or structural relationship model in which

x1

and u1 (although unknown) are fixed once we have chosen our observational