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
Previous Page Next Page