Unobserved heterogeneity and endogeneity in nonparametric frontier estimation

In production theory, firm efficiencies are measured by their distances to a production frontier. In the presence of heterogeneous conditions (like environmental factors) that may influence the shape and the position of the frontier, traditional measures of efficiency obtained in the space of inputs/outputs are difficult to interpret, since they mix managerial inefficiency and shift of the frontier. This can be corrected by using nonparametric conditional efficiencies. In this paper we extend these concepts in the case where the heterogeneity is not observed. We propose a model where the heterogeneity variable is linked to a particular input (or output). It is defined as the part of the input (or the output), independent from some instrumental variable through a nonseparable nonparametric model. We discuss endogeneity issues involved in this model. We show that the model is identified and analyze the asymptotic properties of proposed nonparametric estimators. When using FDH estimators we achieve a limiting Weibull distribution, whereas when using the robust order-m estimators we obtain the asymptotic normality. The method is illustrated with some simulated and real data examples. A Monte-Carlo experiment shows how the procedure works for finite samples.