Nonparametric, Stochastic Frontier Models with Multiple Inputs and Outputs

Abstract

Stochastic frontier models along the lines of Aigner et al. are widely used to benchmark firms’ performances in terms of efficiency. The models are typically fully parametric, with functional form specifications for the frontier as well as both the noise and the inefficiency processes. Studies such as Kumbhakar et al. have attempted to relax some of the restrictions in parametric models, but so far all such approaches are limited to a univariate response variable. Some (e.g., Simar and Zelenyuk; Kuosmanen and Johnson) have proposed nonparametric estimation of directional distance functions to handle multiple inputs and outputs, raising issues of endogeneity that are either ignored or addressed by imposing restrictive and implausible assumptions. This article extends nonparametric methods developed by Simar et al. and Hafner et al. to allow multiple inputs and outputs in an almost fully nonparametric framework while avoiding endogeneity problems. We discuss properties of the resulting estimators, and examine their finite-sample performance through Monte Carlo experiments. Practical implementation of the method is illustrated using data on U.S. commercial banks.

KEYWORD:

• Efficiency
• Nonparametric
• Stochastic frontier

Supplementary Materials

The Supplementary Appendices give addtional details on the flexibility of the stochastic specification, skewness in the “wrong” direction and alternative approaches for dealing with it, and alternative approaches for estimation. Some extensions are also discussed, including extensions to radial distances, introduction of exogenous environmental variables and stochastic versions of FDH/DEA-type estimators. The Supplementary Appendices also provide Monte Carlo evidence as well as additiona results on the Efficiency of U.S. Banks.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Acknowledgments

The views expressed in this article are those of the authors and do not necessarily reflect the views of the Federal Reserve System, the Board of Governors, or the regional Federal Reserve Banks. Any remaining errors are solely our responsibility.

Additional information

Funding

We are grateful for (i) the Palmetto cluster operated by the Cyber Infrastructure Technology Integration group at Clemson University, and (ii) high-performance computing equipment purchased with funds from U.S. National Science Foundation MRI award no. 1725573.