Single Index Models for Nonparametric Conditional Frontiers

In production theory, a lot of attention has been paid in the literature to the analysis of the effect of environmental variables on the efficiency of firms. The usual and natural way to investigate this issue is to consider conditional frontier models. For nonparametric approaches, this can create serious problems if the number of these potential environmental factors increases, exacerbating the curse of dimensionality characteristic of nonparametric models. In this paper, to address this issue, we investigate whether Single Index Models (SIM) could be used for modeling the effect of these variables on the production process. We propose a test for the SIM hypothesis and analyse the asymptotic properties. If the SIM model is not rejected, we obtain better rates of convergence of the conditional efficiency estimates. The paper investigates, through some Monte-Carlo experiments, the finite sample properties of the proposed test and the properties of the resulting estimates of the SIM when it is not rejected. We illustrate the method with a real data set from the French national postal operator in charge of universal service.