The Raise Regression: Justification, properties and application

TL;DR

This paper formalizes the raise regression method, demonstrating its advantages in mitigating multicollinearity without sacrificing inference or fit, and compares it to penalized estimators like ridge regression.

Contribution

It provides a comprehensive formalization of raise regression, including new estimation techniques, theoretical properties, and its relation to other methods like ridge regression.

Findings

Raise regression reduces multicollinearity effects effectively.

It maintains inference and goodness of fit unlike penalized estimators.

Empirical applications demonstrate its practical usefulness.

Abstract

Multicollinearity produces an inflation in the variance of the Ordinary Least Squares estimators due to the correlation between two or more independent variables (including the constant term). A widely applied solution is to estimate with penalized estimators (such as the ridge estimator, the Liu estimator, etc.) which exchange the mean square error by the bias. Although the variance diminishes with these procedures, all seems to indicate that the inference is lost and also the goodness of fit. Alternatively, the raise regression (\cite{Garcia2011} and \cite{Salmeron2017}) allows the mitigation of the problems generated by multicollinearity but without losing the inference and keeping the coefficient of determination. This paper completely formalizes the raise estimator summarizing all the previous contributions: its mean square error, the variance inflation factor, the condition number, the adequate selection of the variable to be raised, the successive raising and the relation between the raise and the ridge estimator. As a novelty, it is also presented the estimation method, the relation between the raise and the residualization, it is analyzed the norm of the estimator and the behaviour of the individual and joint significance test and the behaviour of the mean square error and the coefficient of variation. The usefulness of the raise regression as alternative to mitigate the multicollinearity is illustrated with two empirical applications.