Quantity quantiles linear regression

We show that the definition of the theta th sample quantile as the solution to a minimization problem introduced by Koenker and Bassett (Econometrica 46(1):33-50, 1978) can be easily extended to obtain an analogous definition for the theta th sample quantity quantile widely investigated and applied in the Italian literature. The key point is the use of the first-moment distribution of the variable instead of its distribution function. By means of this definition we introduce a linear regression model for quantity quantiles and analyze some properties of the residuals. In Sect. 4 we show a brief application of the methodology proposed.