Optimal Bandwidth Selection for Conditional Efficiency Measures: a Data-driven Approach

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 and Simar, 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.Nonparametric efficiency estimation, conditional efficiency measures, environmental factors, conditional distribution function, bandwidth.

Frontier estimation and extreme value theory

In this paper, we investigate the problem of nonparametric monotone frontier estimation from the perspective of extreme value theory. This enables us to revisit the asymptotic theory of the popular free disposal hull estimator in a more general setting, to derive new and asymptotically Gaussian estimators and to provide useful asymptotic confidence bands for the monotone boundary function. The finite-sample behavior of the suggested estimators is explored via Monte Carlo experiments. We also apply our approach to a real data set based on the production activity of the French postal services.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ256 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Frontier Estimation and Extreme Values Theory

In this paper we investigate the problem of nonparametric monotone frontier estimation from an extreme-values theory perspective. This allows to revisit the asymptotic theory of the popular Free Disposal Hull estimator in a general setup, to derive new and asymptotically Gaussian estimators and to provide useful asymptotic confidence bands for the monotone boundary function. The finite sample behavior of the suggested estimators is explored through Monte-Carlo experiments. We also apply our approach to a real data set on the production activity of the French postal services.

Forecasting the Malmquist Productivity Index

The Malmquist Productivity Index (MPI) suggests a convenient way of measuring the productivity change of a given unit between two consequent time periods. Until now, only a static approach for analyzing the MPI was available in the literature. However, this hides a potentially valuable information given by the evolution of productivity over time. In this paper, we introduce a dynamic procedure for forecasting the MPI. We compare several approaches and give credit to a method based on the assumption of circularity. Because the MPI is not circular, we present a new decomposition of the MPI, in which the time-varying indices are circular. Based on that decomposition, a new working dynamic forecasting procedure is proposed and illustrated. To construct prediction intervals of the MPI, we extend the bootstrap method in order to take into account potential serial correlation in the data. We illustrate all the new techniques described above by forecasting the productivityt index of 17 OCDE countries, constructed from their GDP, labor and capital stock.Malmquist Productivity Index, circularity efficiency, smooth bootstrap

Asymptotic distribution of conical-hull estimators of directional edges

Nonparametric data envelopment analysis (DEA) estimators have been widely applied in analysis of productive efficiency. Typically they are defined in terms of convex-hulls of the observed combinations of $\mathrm{inputs}\times\mathrm{outputs}$ in a sample of enterprises. The shape of the convex-hull relies on a hypothesis on the shape of the technology, defined as the boundary of the set of technically attainable points in the $\mathrm{inputs}\times\mathrm{outputs}$ space. So far, only the statistical properties of the smallest convex polyhedron enveloping the data points has been considered which corresponds to a situation where the technology presents variable returns-to-scale (VRS). This paper analyzes the case where the most common constant returns-to-scale (CRS) hypothesis is assumed. Here the DEA is defined as the smallest conical-hull with vertex at the origin enveloping the cloud of observed points. In this paper we determine the asymptotic properties of this estimator, showing that the rate of convergence is better than for the VRS estimator. We derive also its asymptotic sampling distribution with a practical way to simulate it. This allows to define a bias-corrected estimator and to build confidence intervals for the frontier. We compare in a simulated example the bias-corrected estimator with the original conical-hull estimator and show its superiority in terms of median squared error.Comment: Published in at http://dx.doi.org/10.1214/09-AOS746 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotics and Consistent Bootstraps for DEA Estimators in Non-parametric Frontier Models

Non-parametric data envelopment analysis (DEA) estimators based on linear programming methods have been widely applied in analyses of productive efficiency. The distributions of these estimators remain unknown except in the simple case of one input and one output, and previous bootstrap methods proposed for inference have not been proven consistent, making inference doubtful. This paper derives the asymptotic distribution of DEA estimators under variable returns-to-scale. This result is then used to prove that two different bootstrap procedures (one based on sub-sampling, the other based on smoothing) provide consistent inference. The smooth bootstrap requires smoothing the irregularly-bounded density of inputs and outputs as well as smoothing of the DEA frontier estimate. Both bootstrap procedures allow for dependence of the inefficiency process on output levels and the mix of inputs in the case of input-oriented measures, or on inputs levels and the mix of outputs in the case of output-oriented measures.bootstrap, frontier, efficiency, data envelopment analysis, DEA

Regularization of Nonparametric Frontier Estimators

In production theory and efficiency analysis, we are interested in estimating the production frontier which is the locus of the maximal attainable level of an output (the production), given a set of inputs (the production factors). In other setups, we are rather willing to estimate an input (or cost) frontier that is defined as the minimal level of the input (cost) attainable for a given set of outputs (goods or services produced). In both cases the problem can be viewed as estimating a surface under shape constraints (monotonicity, . . . ). In this paper we derive the theory of an estimator of the frontier having an asymptotic normal distribution. The basic tool is the order-m partial frontier where we let the order m to converge to infinity when n ! 1 but at a slow rate. The final estimator is then corrected for its inherent bias. We thus can view our estimator as a regularized frontier estimator which, in addition, is more robust to extreme values and outliers than the usual nonparametric frontier estimators, like FDH. The performances of our estimators are evaluated in finite samples through some Monte-Carlo experiments. We illustrate also how to provide, in an easy way, confidence intervals for the frontier function both with a simulated data set and a real data set.

Efficiency and University Size: Discipline-wise Evidence from European Universities

Strategic management of universities must build the best possible relation between inputs and outputs. One relevant question, in this perspective, is whether the unit is making the best use of existing resources, or whether technical efficiency is in place. Here we address the question of technical efficiency with respect to university’s size. The crucial concept in this analysis is conditional efficiency and the ratio of size-conditional to unconditional efficiency measures. In particular we take use of robust order-m efficiency scores presented in Cazals, Florens and Simar (2002) and generalized in Daraio and Simar (2005a,b) to analyze data from four European countries and four different research fields. Our results are still explorative and mainly show how heterogeneous international datasets could be used to analyze productivity differences.Universities;efficiency;International comparisons