THE DIAGNOSIS OF MULTICOLLINEARITY*
| Author | C. L. GILBERT |
| Date | 01 May 1978 |
| DOI | http://doi.org/10.1111/j.1468-0084.1978.mp40002001.x |
| Published date | 01 May 1978 |
OXFORD BULLETIN
of
ECONOMICS and STATISTICS
THE DIAGNOSIS OF MULTICOLLINEARITY*
By C. L. GILBERT
Regression collinearity is not a precise condition and it is therefore not sensible
to look for a precise test for its presence. Certainly a null hypothesis of data
orthogonality is uninteresting in an economic context (Farrar and Glaubman [2],
Haitovsky [3]). The purpose of this paper is to analyse the manner in which the
moment matrix of the data determines the precision of the coefficient estimates
for a given relationship, to use this to provide a characterization of collinearity and
to relate this to a measure of its incidence.
Consider the standard linear model y = Xß + u
with E(u)=O
and E(uu') = a21 (1)
which we take to be written in deviation form. We may gain some intuitive grasp
of the informational content of the sample X by supposing it to have been generated
within a badly designed experiment. We may then ask by how much the precision
of our OLS estimates ß of fi would have been improved by the use of an orthogonal
design. This provides us with a measure of the inefficiency of the design implicit in
the sample X and we may identify this inefficiency with the extent of collinearity
present.
It should be noted that this measure of inefficiency depends both on the sample
of data and on the relationship to be estimated. A particular experimental design
may be efficient relative to one proposed relationship but inefficient relative to a
second.We propose to measure the precision of our estimate ß by the length L(ßß)
of the error vector ß-ß and, in a least squares context, it is natural to define the
length of a vector to be its Euclidean norm. Thus for any vector z we take
L(z)= 11z112__(zz)h12 (2)
* I am indebted to R. W. Bacon, D. K. H. Begg, J. A. C. Brown, A. J. Hughes Hallet and
D. F. Hendry for comment on earlier versions of this paper.
87
Volume 40 May 1978 No. 2
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