From: thompson@atlas.socsci.umn.edu (T. Scott Thompson)
Subject: Re: multi-colinearity in full lisrel models
Reply-To: thompson@atlas.socsci.umn.edu
Organization: Economics Department, University of Minnesota
Date: Mon, 9 Nov 1992 17:18:21 GMT

mokaba@binah.cc.brandeis.edu writes:

>Could someone help the novice!!
>I am working on estimatic both structural and measurement parameter using
>LISREL. I used 19 indicators of 6 latent varible, three of which are
>eta's (endogenous) and the other three are ksi's (exogenous). 

>Problem :
>In estimating this model I get negative beta coefficients where i expected
>positive ones. Some one suggested that I could be have a mulcolinearity
>problem. How do I test for it in LISREL? How do I resolve it short of 
>eliminating one of the varibales from the model? 
>e-mails appreciated.

(1) Multicolinearity may lead to large standard errors for your
estimates but it should not bias them.  (Extreme multicolinearity can
lead to loss of identification, in which case the bias is not a
very meaningful concept.)

(2) The only way to eliminate multicolinearity is to get better data.
All of the "fix-up" procedures in the literature correspond to
imposing linear restrictions on the parameters.  If these are
justified by a priori considerations then you should be imposing them
anyhow.  If not, then imposing them may seriously distort your
analysis.

Art Goldberger, in his text "A Course in Econometrics", makes the case
that mulicolinearity receives too much attention in the literature
because it has a fancy name, but has the same effects on data analysis
as the problem of not having enough data.  In a very clever parody he
attempts to remedy the imbalance by endowing the latter with the name
"micronumerosity."  Here are some hilights from his parody.  I have
substituted for his mathematical notation where appropriate.

---------------------------------------------------------------------
The following is excerpted (without permission) from "A Course in
Econometrics", Arthur S.  Goldberger, Harvard Press, 1991.

Micronumerosity.

The extreme case, "exact micronumerosity" arises when n=0, in which
case the sample estimate of mu [the mean] is not unique.
(Technically, there is a violation of the rank condition n>0: the
matrix 0 is singular.)  ... "Near micronumerosity" is more subtle, and
yet very serious.  It arises when the rank condition n>0 is barely
satisfied.  Near micronumerosity is very prevalent in empirical
economics.

Consequences of Micronumerosity.  

The consequences ... are serious.  Precision of estimation is reduced.
There are two aspects of this reduction: estimates of [the mean] may
have large errors, and not only that, but [the variance of the
estimate] will be large.  Investigators will sometimes be led to
accept the hypothesis mu=0 because [the t-statistic] is small, even
though the true situation may be not that mu=0 but simply that the
sample data have not enabled us to pick mu up.

The estimate of mu will be very sensitive to sample data, and the
addition of a few more observations can sometimes produce drastic
shifts in the sample mean. ...

Testing for Micronumerosity.

Test for the presence of micronumerosity requires the judicious use of
various fingers.  Some researchers prefer a single finger, others use
their toes, still others let their thumbs rule.

A generally reliable guide is to count the number of observations.
Most of the time in econometric analysis, when n is close to zero ,it
is also far from infinity.  Several test procedures develop critical
values n*, such that micronumerosity is a problem on if n is smaller
than n*.  But those procedures are questionable.

Remedies for Micronumerosity.

If micronumerosity proves serious in the sense that the estimate of mu
has an unsatisfactorily low degree of precision, we are in the
statistical position of not being able to make bricks without straw.
The remedy lies essentially in the acquistion, if possible, of larger
samples from the same population.

But more data are no remedy for micronumerosity if the additional data
are simply "more of the same."  So obtaining lots of small samples
from the same population will not help.
----------------------------------------------------------------------

Goldberger goes on (in a more serious vein) to note that
multicolinearity can actually _improve_ estimates of some linear
combinations of the parameters.
--
T. Scott Thompson              email:  thompson@atlas.socsci.umn.edu
Department of Economics        phone:  (612) 625-0119
University of Minnesota        fax:    (612) 624-0209