Approximations and inference for nonparametric production frontiers

Nonparametric methods have been widely used for assessing the performance of organizations in the private and public sector. The most popular ones are based on envelopment estimators, like the FDH or DEA estimators, that estimate the attainable sets and its efficient boundary by enveloping the cloud of observed units in the appropriate input-output space. The statistical properties of these flexible estimators have been established. However these nonparametric techniques do not allow to make sensitivity analysis of the production outputs to some particular inputs, or to infer about marginal products and other coefficients of economic interest. On the contrary, parametric models for production frontiers allow richer and easier economic interpretation but at a cost of restrictive assumptions on the data generating process. In addition, the latter rely mostly on regression methods fitting the center of the cloud of observed points. In this paper we offer a way to avoid these drawbacks and provide approximations of these coefficients of economic interest by "smoothing" the popular nonparametric estimators of the frontiers. Our approach allows to handle fully multivariate cases. We describe the statistical properties for both the full and the partial (robust) frontiers. We consider parametric but also flexible approximations based on local linear tools providing local estimates of all the desired partial derivatives and we show how to deal with environmental factors. An illustration on real data from European Higher Education Institutions (HEI) shows the usefulness of the proposed approach.