Optimal bandwidth selection for conditional efficiency measures: A data-driven approach

In productivity analysis an important issue is to detect how external (environmental) factors, exogenous to the production process and not under the control of the producer, might influence the production process and the resulting efficiency of the firms. Most of the traditional approaches proposed in the literature have serious drawbacks. An alternative approach is to describe the production process as being conditioned by a given value of the environmental variables (Cazals, C., Florens, J.P., Simar, L., 2002. Nonparametric Frontier estimation: A robust approach. Journal of Econometrics 106, 1–25; Daraio, C., Simar, L., 2005. Introducing environmental variables in nonparametric Frontier models: A probabilistic approach. Journal of Productivity Analysis 24(1), 93–121). This defines conditional efficiency measures where the production set in the input ×$\times$ output space may depend on the value of the external variables. The statistical properties of nonparametric estimators of these conditional measures are now established (Jeong, S.O., Park, B.U., Simar, L., 2008. Nonparametric conditional efficiency measures: Asymptotic properties. Annals of Operations Research doi: 10.1007/s10479-008-0359-5). These involve the estimation of a nonstandard conditional distribution function which requires the specification of a smoothing parameter (a bandwidth). So far, only the asymptotic optimal order of this bandwidth has been established. This is of little interest for the practitioner. In this paper we fill this gap and we propose a data-driven technique for selecting this parameter in practice. The approach, based on a Least Squares Cross Validation procedure (LSCV), provides an optimal bandwidth that minimizes an appropriate (weighted) integrated Squared Error (ISE). The method is carefully described and exemplified with some simulated data with univariate and multivariate environmental factors. An application on real data (performances of Mutual Funds) illustrates how this new optimal method of bandwidth selection works in practice.