Multicollinearity A Bayesian Interpretation

The probIem of colIinear data sets in the context of the univariate multiple regression model has generated a confusing array of papers,
comments and footnotes. Perusal of this literature does not lead the reader to conclude that the problem has even been rigorously defined.
 The purpose of this paper is to offer several rigorous definitions and to suggest several substantive quantitative summaries of the degree of the problem.

Specifically we address our attention to the regression model y = XP + where p is a k-dimensional parameter vector with p17p2. 7. . pk7where y and are T x 1 vectors where x is a T k matrix. Inferences are to be made about the vector p from observations of y and X. When the columns of x are orthogonal, the design matrix X'X is diagonal. Correlated columns of x imply a nondiagonal design matrix. The collinearity problem has to do with the differences in the inferences that may be drawn in these two situations.