7. Gini Simple Regressions

The basic building block in regression is the covariance between the dependent variable and the explanatory variable(s). There are two regression methods that can be interpreted as based on Gini’s Mean Difference (GMD). The first method is based on the fact that one can present the Gini-covariance between the dependent variable and the explanatory variable as a weighted sum of slopes of the regression curve (a semi-parametric approach). The second method is based on the minimization of the GMD of the residuals. The semi-parametric approach is similar in its structure to the Ordinary Least Squares (OLS) method. That is, the regression coefficient in the OLS has an equivalent term in the Gini semi-parametric regression. The equivalent term is constructed by substituting the covariance and the variance in the OLS regression by the Gini-covariance (hereafter co-Gini) and the Gini, respectively. However, unlike the OLS, the Gini regression coefficient and its estimator are not derived by solving a minimization problem. Therefore they do not have optimality properties and cannot be described as “the best,” at least not with respect to a simple target function. On the other hand, the second method, the minimization of the GMD of the residuals implies optimality but it has its drawbacks. Like Mean Absolute Deviation (MAD) and quantile regressions, the regression coefficient does not have an explicit presentation and can be calculated only numerically. The combination of the two methods of Gini regression enables the user to investigate the appropriateness of the assumptions that lie behind the OLS and Gini regressions (e.g., the linearity of the relationship) and therefore can improve the quality of the conclusions that are derived from them. Moreover, when dealing with a multiple regression one can combine the semi-parametric regression method with the OLS regression method. That is, several explanatory variables can be treated as in the OLS, while others are treated using the Gini method. This flexibility enables one to evaluate the effect of the choice of a regression method on the estimated coefficients in a gradual way by substituting the methodology of the estimation for each explanatory variable in a stepwise way rather than in an “all or nothing” way. This issue will be discussed in Chap. 8.