We propose a new linear regression model for interval-valued variables. The model uses quantile functions to represent the intervals, thereby considering the distributions within them. In this paper we study the special case where the Uniform distribution is assumed in each observed interval, and we analyze the extension to the Symmetric Triangular distribution. The parameters of the model are obtained solving a constrained quadratic optimization problem that uses the Mallows distance between quantile functions. As in the classical case, a goodness-of-fit measure is deduced. Two applications on up-to-date fields are presented: one predicting duration of unemployment and the other allowing forecasting burned area by forest fires.