Estimation and inference in parametric deterministic frontier models

Abstract

In this paper we consider parametric deterministic frontier models. For example, the production frontier may be linear in the inputs, and the error is purely one-sided, with a known distribution such as exponential or half-normal. The literature contains many negative results for this model. Schmidt (Rev Econ Stat 58:238–239, 1976) showed that the Aigner and Chu (Am Econ Rev 58:826–839, 1968) linear programming estimator was the exponential MLE, but that this was a non-regular problem in which the statistical properties of the MLE were uncertain. Richmond (Int Econ Rev 15:515–521, 1974) and Greene (J Econom 13:27–56, 1980) showed how the model could be estimated by two different versions of corrected OLS, but this did not lead to methods of inference for the inefficiencies. Greene (J Econom 13:27–56, 1980) considered conditions on the distribution of inefficiency that make this a regular estimation problem, but many distributions that would be assumed do not satisfy these conditions. In this paper we show that exact (finite sample) inference is possible when the frontier and the distribution of the one-sided error are known up to the values of some parameters. We give a number of analytical results for the case of intercept only with exponential errors. In other cases that include regressors or error distributions other than exponential, exact inference is still possible but simulation is needed to calculate the critical values. We also discuss the case that the distribution of the error is unknown. In this case asymptotically valid inference is possible using subsampling methods.