Abstract
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.
Key Words: Gini's Mean Difference, Average Derivative, Linear Expenditure System,
Monotonicity.