Conditional nonparametric frontier models for convex and nonconvex technologies: a unifying approach

Abstract

The explanation of productivity differentials is very important to identify the economic conditions that create inefficiency and to improve managerial performance. In the literature two main approaches have been developed: one-stage approaches and two-stage approaches. Daraio and Simar (2005, J Prod Anal 24(1):93–121) propose a fully nonparametric methodology based on conditional FDH and conditional order-m frontiers without any convexity assumption on the technology. However, convexity has always been assumed in mainstream production theory and general equilibrium. In addition, in many empirical applications, the convexity assumption can be reasonable and sometimes natural. Lead by these considerations, in this paper we propose a unifying approach to introduce external-environmental variables in nonparametric frontier models for convex and nonconvex technologies. Extending earlier contributions by Daraio and Simar (2005, J Prod Anal 24(1):93–121) as well as Cazals et al. (2002, J Econometrics 106:1–25), 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 proposed too. These various measures of efficiency provide also indicators of convexity which we illustrate using simulated and real data.

Appendix: Bandwidth selection

DS propose a simple data-driven procedure for choosing the bandwidth, based on a k-nearest neighbor method, based on likelihood cross-validation for the density of Z.

In a first step, a bandwidth h which optimizes the estimation of the density of Z is selected, based on the likelihood cross validation criterion, using a k-NN (Nearest Neighborhood) method (see e.g., Silverman 1986). This allows to obtain bandwidths which are localized, insuring we have always the same number of observations Z _i in the local neighbourhood of the point of interest z when estimating the density of Z.

Hence, for a grid of values of k, we evaluate the leave-one-out kernel density estimate of Z, ${\hat f_k^{(-i)}(Z_i)}$ for i = 1,...,n and find the value of k which maximizes the score function:

$$ CV(k)=n^{-1} \sum_{i=1}^n \log\left(\hat f_k^{(-i)}(Z_i)\right), $$

where

$$ \hat f_k^{(-i)}(Z_i)= \frac{1}{(n-1)h_{Z_i}}\sum_ {j=1, j\ne i}^nK\left(\frac{Z_j-Z_i}{h_{Z_i}}\right), $$

and h _{Z_i} is the local bandwidth chosen such that there exist k points Z _j verifying | Z _j −Z _i | ≤ h _{Z_i} .

In a second step, taking into account for the dimensionality of x and y, and the sparsity of points in larger dimensional spaces, the local bandwidths h _{Z_i} are expanded by a factor 1 + n ^−1/(p+q), increasing with (p + q) but decreasing with n. For more details, see Daraio (2003).

We notice that the calculations of efficiency scores and the evaluation of the influence of external factors is not too sensitive to the choice of the procedure for bandwidth selection. As a matter of fact, we obtained very similar results by applying the global bandwidth obtained with the Sheather and Jones (1991) method for kernel density estimation of Z.