Imposing monotonicity on outputs in parametric distance function estimations

The technology set involved in the estimation of a multi-output production frontier theoretically implies monotonicity on outputs. This is because an efficient firm cannot reduce the vector of outputs holding the vector of inputs fixed while it still belongs to the frontier. In empirical studies dealing with the estimation of parametric distance functions, this hypothesis is often violated by observations with far from average characteristics. To overcome this limitation, we propose a new approach for allowing the easy imposition of monotonicity on outputs in this context. This methodology is tested in the educational sector using Spanish student level data from the Programme for International Student Assessment (PISA) database. The results indicate that a nonnegligible 8.33% of the production units break the monotonicity assumption. Furthermore, although there is no statistically significant difference in efficiency distribution by school ownership, our methodology helps to detect a slight worse mathematical performance for students attending public schools.