This paper proposes a general formulation of a nonparametric frontier model introducing external environmental factors that might influence the production process but are neither inputs nor outputs under the control of the producer. A representation is proposed in terms of a probabilistic model which defines the data generating process. Our approach extends the basic ideas from Cazals, Florens and Simar (2002) to the full multivariate case. We introduce the concepts of conditional efficiency measure and of conditional efficiency measure of order-m. Afterwards we suggest a practical way for computing the nonparametric estimators. Finally, a simple methodology to investigate the influence of these external factors on the production process is proposed. Numerical illustrations through some simulated examples and through a real data set on Mutual Funds show the usefulness of the approach.

The measurement of technical efficiency of decision making units is useful for making comparisons and informing managers and policy makers on existing differentials and potential improvements across a sample of analyzed units. The step further is to relate the obtained efficiency estimates to some external or environmental variables which may influence the production process, affect the performances and explain the efficiency differentials. Conditional efficiency measures (Daraio and Simar, 2005; 2007a), including conditional FDH, conditional DEA, conditional order− and conditional order− , have been recently introduced and became rapidly a useful tool to explore the impact of external-environmental factors on the performance of Decision Making Units in a nonparametric framework. In this paper, we show that analyzing these conditional efficiency scores we can disentangle the impact of these factors on the production process in its components: impact on the attainable set in the input × output space, and/or impact on the distribution of the inefficiency scores. We extend existing methodological tools to investigate these interrelationships, both from an individual and a global perspective. We emphasize the usefulness of regressing the conditional efficiencies on the explaining factors. The analysis of the residuals provides a measure of efficiency whitened from the main effect of the environmental factors. This allows to rank the firms according to their "managerial" efficiency, even when facing heterogeneous environmental conditions. Our approach is illustrated through simulated samples and with a real data set in the Banking industry.

The explanation of productivity differentials is very important to identify the economic conditions that create inefficiency and to improve managerial performance. In literature two main approaches have been developed: one-stage approaches and twostage approaches. Daraio and Simar (2003) propose a full nonparametric methodology based on conditional FDH and conditional order-m frontiers without any convexity assumption on the technology. On the one hand, convexity has always been assumed in mainstream production theory and general equilibrium. On the other hand, in many empirical applications, the convexity assumption can be reasonable and sometimes natural. Leading by these considerations, in this paper we propose a unifying approach to introduce external-environmental variables in nonparametric frontier models for convex and non convex technologies. Developing further the work done in Daraio and Simar (2003) we introduce a conditional DEA estimator, i.e., an estimator of production frontier of DEA type conditioned to some external-environmental variables which are neither inputs nor outputs under the control of the producer. A robust version of this conditional estimator is also proposed. These various measures of efficiency provide also indicators of convexity. Illustrations through simulated and real data (mutual funds) examples are reported.

In this paper, we demonstrate that standard central limit theorem (CLT) results do not hold for means of non-parametric, conditional efficiency estimators, and we provide new CLTs that permit applied researchers to make valid inference about mean conditional efficiency or to compare mean efficiency across groups of producers. The new CLTs are used to develop a test of the restrictive 'separability' condition that is necessary for secondstage regressions of efficiency estimates on environmental variables. We show that if this condition is violated, not only are second-stage regressions difficult to interpret and perhaps meaningless, but also first-stage, unconditional efficiency estimates are misleading. As such, the test developed here is of fundamental importance to applied researchers using nonparametric methods for efficiency estimation. The test is shown to be consistent and its local power is examined. Our simulation results indicate that our tests perform well both in terms of size and power. We provide a real-world empirical example by re-examining the paper by Aly et al. (1990, Review of Economics and Statistics 72, 211-18) and rejecting the separability assumption implicitly assumed by Aly et al., calling into question results that appear in hundreds of papers that have been published in recent years.

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, Florens and Simar, 2002, Daraio andSimar, 2005). This defines conditional efficiency measures where the production set in the input × 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, Park and Simar, 2008). 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 integrated Mean Squared Error (MSE). 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 outperforms former methods.