Direction selection in stochastic directional distance functions

Researchers rely on the distance function to model multiple product production using multiple inputs. A stochastic directional distance function (SDDF) allows for noise in potentially all input and output variables. Yet, when estimated, the direction selected will affect the functional estimates because deviations from the estimated function are minimized in the specified direction. Specifically, the parameters of the parametric SDDF are point identified when the direction is specified; we show that the parameters of the parametric SDDF are set identified when multiple directions are considered. Further, the set of identified parameters can be narrowed via data-driven approaches to restrict the directions considered. We demonstrate a similar narrowing of the identified parameter set for a shape constrained nonparametric method, where the shape constraints impose standard features of a cost function such as monotonicity and convexity.