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Conditional efficiency measures, including conditional FDH, conditional DEA, conditional order-m and conditional order-alpha, have been recently introduced and proved as a useful tool for the investigation on the impact of external-environmental factors on the performance of Decision Making Units in a nonparametric framework (see Daraio and Simar, 2007 for an overview). In this paper we suggest a consistent bootstrap approach to correctly mimicking the DGP and describe how to estimate the bias, the standard deviation and the confidence intervals of the full conditional measures and of the robust conditional measures (order-m and order-alpha). For partial (robust measures) of efficiency, the naive bootstrap can be used consistently, whilst Jeong and Simar (2006) demonstrate that the sub-sampling is consistent for the FDH case. However, the choice of the sub-sampling size in the FDH case may affect the accuracy of the results; hence in this paper we propose a method for selecting the size of the sub-sampling in an effective way. An analytical bias correction is also applied adapting the approach proposed by Badin and Simar (2004) to the framework of conditional FDH case. Further, we illustrate how to compute point-wise confidence intervals on the ratios of conditional on unconditional efficiency measures to improve the detection of the impact of external factors on the production process. Then, a number of simulation exercises, with univariate and multivariate external factors, and with different impact of the external factors, illustrate how to operationalize the statistical inference in this complex framework. |
