Singularity in the Equation Systems of Econometrics Some Aspects of the Problem of Multicollinearity

1. REVIEW
IN ESTIMATING an equation of the form
Yt =f:%Xi$ + ut , t = 1, * * * , T ,
=1
it is well known that linear relationships among the "explanatory"
variables xi, give rise to singularities, indeterminacy, and nonfinite
values sf some parameter estimates.
We define
Least-squares regression estimates of a,have the form
est ai =ai =-I Mi, l ,
I MI
which becomes 010 if xi, variables are linearly related.The simplest case of such linear interrelationships among xi, is perfect
correlation between two xi,'s, say x,, and xjt. If these two variables
were perfectly correlated, the k-th and j-th rows or columns of
M would be proportional. Similarly, these two rows of Mi, would be
proportional since both x,, and xi, would have proportional moments
(about mean values) with y,. If rows of M and of Mi, are proportional,
the respective determinants vanish, and we obtain indeterminate estimates
of a,, .i= 1, * - * , n, in the sense that 010 is indeterminate.