  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.
