A Bootstrap Approach for Bandwidth Selection in Estimating Conditional Efficiency Measures

Conditional efficiency measures are needed when the production process does not depend only on the inputs and outputs, but may be influenced by external factors and/or environmental variables (Z). They are estimated by means of a nonparametric estimator of the conditional distribution function of the inputs and outputs, conditionally on values of Z. For doing this, smoothing procedures and smoothing parameters, the bandwidths, are involved. So far, Least Squares Cross Validation (LSCV) methods have been used, which have been proven to provide bandwidths with optimal rates for estimating conditional distributions. In efficiency analysis, the main interest is in the estimation of the conditional efficiency score, which typically depends on the boundary of the support of the distribution and not on the full conditional distribution. In this paper, we show indeed that the rate for the bandwidths which is optimal for estimating conditional distributions, may not be optimal for the estimation of the efficiency scores. We propose hence a new approach based on the bootstrap which overcomes these difficulties. We analyze and compare, through Monte Carlo simulations, the performances of LSCV techniques with our bootstrap approach in ï¬ nite samples. As expected, our bootstrap approach shows generally better performances and is more robust to the various Monte Carlo scenarios analyzed. We provide in an Appendix the Matlab code performing our experiments.