Analysis of the condition number in the raise regression

Abstract

The raise regression was presented as an alternative methodology to estimate models with collinearity, it is to say, when columns of matrix X are nearly linearly dependent. This method consists in transforming matrix X by matrix X that is obtained after raising one of the explanatory variable to move it geometrically away from the span of the remaining variables. In this article, we analyze the behavior of the condition number in the raise regression showing that the effectiveness of the raise method to solve the problem of multicollinearity is limited to the potential reduction of the condition number that presents an asymptote corresponding to the condition number associated to the auxiliary regression of the raised variable. All the contributions are illustrated with an empirical application.

Keywords:

• Multicollinearity
• condition number
• eigenvalue
• raise regression

Notes

1 $\Vert X_i\Vert >0\hspace{1em}\text{and}\hspace{1em}\left|〈X_i,X_j〉\right|\le \Vert X_i\Vert \Vert X_j\Vert \hspace{1em}\text{for}\mathrm{ }\text{each}\mathrm{ }i,j\in \left\{1,2,\dots ,k\right\}.$