Explaining Efficiency in Nonparametric Frontier Models: Recent Developments in Statistical Inference

The explanation of efficiency differentials is an essential step in any frontier analysis study that aims to measure and compare the performance of decision making units. The conditional efficiency measures that have been introduced in recent years (Daraio and Simar, J. Prod. Anal. 24:93–121, 2005) represent an attractive alternative to two-step approaches, to handle external environmental factors, avoiding additional assumptions such as the separability between the input-output space and the space of external factors. Although affected by the curse of dimensionality, nonparametric estimation of conditional measures of efficiency eliminates any potential specification issue associated with parametric approaches. The nonparametric approach requires, however, estimation of a nonstandard conditional distribution function which involves smoothing procedures, and therefore the estimation of a bandwidth parameter. Recently, Bădin et al. (Eur. J. Oper. Res. 201(2):633–640, 2010) proposed a data driven procedure for selecting the optimal bandwidth based on a general result obtained by Hall et al. (J. Am. Stat. Assoc. 99(486):1015–1026, 2004) for estimating conditional probability densities. The method employs least squares cross-validation (LSCV) to determine the optimal bandwidth with respect to a weighted integrated squared error (WISE) criterion.This paper revisits some of the recent advances in the literature on handling external factors in the nonparametric frontier framework. Following the Bădin et al. (Eur. J. Oper. Res. 201(2):633–640, 2010) approach, we provide a detailed description of optimal bandwidth selection in nonparametric conditional efficiency estimation, when mixed continuous and discrete external factors are available. We further propose an heterogeneous bootstrap which allows improving the detection of the impact of the external factors on the production process, by computing pointwise confidence intervals on the ratios of conditional to unconditional efficiency measures.We illustrate these extensions through some simulated data and an empirical application using the sample of U.S. mutual funds previously analyzed in Daraio and Simar (J. Prod. Anal. 24:93–121, 2005; Eur. J. Oper. Res. 175(1):516–542, 2006; Advanced Robust and Nonparametric Methods in Efficiency Analysis: Methodology and Applications, Springer, New York, 2007a).