Relative Contribution of Regressors

Abstract

The aim of the chapter is to review the approaches suggested in the literature to see the contribution of the explanatory variables to the explained variable. This topic is better known as the relative importance of the regressors. In a regression equation set up, the relative importance may be defined by the relative contribution of each of the regressors to the coefficient of determination, R², considering its effect when combined with the other variables. We have discussed in this chapter various measures of relative importance of predictors, namely, Allocation First, Allocation Last, Hoffman-Pratt Decomposition of R², Shapley Value Decomposition and Relative weights. Shapley Value Decomposition takes care of all possible cases of regression and it seems to be one of the best measures. But it is almost unmanageable for large number of predictors. The procedure of Relative Weight is an alternative measure which give ranking of the predictors close to that of Shapley Value Decomposition. This procedure is quite manageable even for large number of predictors. It has already been seen that the rankings of the predictors are more or less similar whatever measure is taken. To see the contribution of the explanatory variables to the explained variable, we have devised a novel approach—the Set Theoretic Approach. This approach can find out the individual contribution of the regressors as well as the contribution common to the combination of regressors. A new measure of multicollinearity has also been proposed and illustrated with an example.