Gini's multiple regressions: Two approaches and their interaction

Two regressions can be interpreted as based on Gini's Mean Difference (GMD): a semiparametric approach, which relies on weighted average of slopes defined between adjacent observations and a minimization approach, which is based on minimization of the GMD of the residuals. The estimators obtained by the semiparametric approach have representations that resemble the OLS estimators. In addition they are robust with respect to extreme observations and monotonic transformations. The estimators obtained by the minimization approach do not have a closed form. The relationship between the estimators obtained by the two methods is studied in this paper. Combination of the methods provides tools for challenging the specification of the model. In particular it provides tools for assessing the linearity of the model. It can be applied to each explanatory variable individually and to several explanatory variables simultaneously without requiring replications. The semiparametric method and its relationship with the minimization approach are illustrated using consumption data. It is shown that the linearity of the Engel curve, and therefore the 'linear expenditures system' is not supported by the data.