La Regressione Lineare con i Valori Assoluti

The estimation of regression coefficients in the linear model is usually provided by least squares (LS) minimizing the sum of the squares of residuals. An alternative estimator is obtained by minimizing the sum of absolute residuals (MSAE) and was first introduced by Boscovich in 1757 for the straight line. We first provide a short historical background and then we show in detail, from a descriptive point of view, how to obtain the median regression (MSAE) coefficients for the straight line and, for the more general case of the hyperplane, the formulation of the problem as a linear programming problem. Defining the sample quantiles as a solution of a minimization problem, quantile regression, introduced by Koenker and Bassett (1978) provides an extension of this methodology in order to obtain regression coefficients of the hyperplane for a generic quantile of the dependent variable.We introduce quantile regression showing that the use of different loss functions: quadratic, absolute and asymmetric absolute leads respectively to least squares, median and quantile regression. In this thesis we extend these results to the linear regression for quantity quantiles. We first show that quantity quantiles can be defined as the solution to a minimization problem and then we extend the result to the linear regression framework. We finally deal with another use of absolute values in the regression context, in particular we consider the problem of the estimation of the regression coefficients by minimizing the Gini mean difference of the residuals; we show that this apporach fall in the class of R-estimators.