Estimating Production Frontiers Research Articles

Robust nonparametric frontier estimation in two steps

We propose a robust methodology for estimating production frontiers with multi-dimensional input via a two-step nonparametric regression, in which we estimate the level and shape of the frontier before shifting it to an appropriate position. Our main contribution is to derive a novel frontier estimation method under a variety of flexible models which is robust to the presence of outliers and possesses some inherent advantages over traditional frontier estimators. Our approach may be viewed as a simplification, yet a generalization, of those proposed by Martins-Filho and coauthors, who estimate frontier surfaces in three steps. In particular, outliers, as well as commonly seen shape constraints of the frontier surfaces, such as concavity and monotonicity, can be straightforwardly handled by our estimation procedure. We show consistency and asymptotic distributional theory of our resulting estimators under standard assumptions in the multi-dimensional input setting. The competitive finite-sample performances of our estimators are highlighted in both simulation studies and empirical data analysis.

Analytic hierarchy process and data envelopment analysis: A match made in heaven

Analytic hierarchy process (AHP) and data envelopment analysis (DEA) are two popular decision science methods with many business, science, and engineering applications. AHP is a Multi-Attribute Decision-Making (MADM) method for prioritizing alternatives, and DEA is a non-parametric method for estimating production frontiers. Each method has known strengths and weaknesses, and the strengths in one method can overcome the weaknesses in the other. We study several strategies to diminish the weaknesses of DEA with the strengths of pairwise comparisons in AHP. We tackle the low discrimination power inherent to conventional DEA methods. AHP, with its pairwise comparison capability, has been consistently used to increase the discrimination power and accuracy in DEA. We propose and evaluate several new hybrid MADM-DEA models of different computational complexity and consistency, including combinations of the best-worst method (BWM) and its variants with DEA as well as a novel method composed of Measuring Attractiveness by a Categorical Based Evaluation Technique (MACBETH) and DEA. We further develop a new technique for evaluating the similarity among multiple ranking results in MADM. The new simple but powerful technique is called Rank Absolute Deviation (RAD) and is inspired by the mean absolute deviation method. Several numerical examples and a real-world problem are used to demonstrate the applicability and efficacy of the new BWM-DEA, MACBETH-DEA, and RAD methods proposed in this study. We illustrate how less computationally demanding MADM-DEA techniques provide rankings that are highly correlated with the benchmark DEA-AHP and different consensus ranking models.

A new measure of technical efficiency in data envelopment analysis based on the maximization of hypervolumes: Benchmarking, properties and computational aspects

The measurement of technical efficiency attracts considerable interest in the literature. In fact, following on from the seminal works by Koopmans (1951), Debreu (1951), Shephard (1953) and Farrell (1957), a substantial amount of literature has been dedicated to methods for estimating production frontiers and measuring the technical efficiency of production units. Data Envelopment Analysis (DEA) has an important role among current techniques for determining technical efficiency due to its flexibility when it comes to confronting multiple inputs and outputs as well as its non-parametric nature. Many different DEA-based technical efficiency measures have evolved over the last forty years, including radial, hyperbolic, directional, (weighted) additive, slacks-based measures, etc. Nevertheless, there is still room for harnessing new measures that are capable of satisfying interesting properties. For instance, slacks-based measures generally yield solutions with zero values in some slacks, which may mean overloading some of the other dimensions to reach the efficient frontier, thereby involving unbalanced efforts that are unrealistic. Moreover, the projection point determined by standard measures is not unique which is a further weakness from a benchmarking perspective. The purpose of this paper is to propose a novel technical efficiency performance measure in DEA on the premise of maximizing the hypervolume, while dealing with these problems and satisfying additional properties at the same time. In particular, we prove that the new approach yields unique projection points, which is not a usual property of DEA technical efficiency measures. From a computational perspective, second-order cone programming is capable of solving the new measure. Finally, an empirical example extracted from the literature serves to illustrate the new methodology.

Efficiency analysis trees: A new methodology for estimating production frontiers through decision trees

In this paper, we introduce a new methodology based on regression trees for estimating production frontiers satisfying fundamental postulates of microeconomics, such as free disposability. This new approach, baptized as Efficiency Analysis Trees (EAT), shares some similarities with the Free Disposal Hull (FDH) technique. However, and in contrast to FDH, EAT overcomes the problem of overfitting by using cross-validation to prune back the deep tree obtained in the first stage. Finally, the performance of EAT is measured via Monte Carlo simulations, showing that the new approach reduces the mean squared error associated with the estimation of the true frontier by between 13% and 70% in comparison with the standard FDH.

A nonparametric framework to detect outliers in estimating production frontiers

In a large number of organizations, there is an ongoing need to evaluate performance. There is also a need to estimate the production frontier of best performers in such environments. Part of the difficulty in constructing this frontier is that massive amounts of data are generated daily with the result being that the set of best performers is constantly changing. Furthermore, measurement errors can influence datasets as well as the estimation of a production frontier. Outliers can also substantially affect the estimated production frontier. Efforts have been made in the past three decades to deal with datasets that might include such outliers; these methods are mostly semiautomatic or require significant computation time when a large dataset is involved. The few existing research on large datasets also focuses on the computational process of measuring a production frontier without identifying the possible influence of outliers to the estimated frontier. In the current paper, for the first time in the literature of data envelopment analysis (DEA), we develop an automatic framework with the computational capability and accuracy needed when big datasets (with multiple inputs and multiple outputs) are considered. Several examples, simulation experiments, and real-life applications are discussed to demonstrate the power of the proposed framework. A data analysis with illustrative graphs is provided to clearly show the methodology. In terms of estimating the production frontier, the method is robust, user-friendly, and substantially decreases the requirement of user judgment while at the same time allowing for the incorporation of such judgement.

Frontier estimation using kernel smoothing estimators with data transformation

In economics, a production frontier function is a graph that shows the maximum output of production units such as firms, industries, or economies, as a function of their inputs. Practically, estimating production frontiers often requires imposition of constraints such as monotonicity or monotone concavity. However, few constrained estimators of production frontier have been proposed in the literature. They are based on simple envelopment techniques which often suffer from lack of precision and smoothness. Motivated by this observation, we propose a smooth constrained nonparametric frontier estimator respecting constraints by considering kernel smoothing estimators from a transformed data. It is particularly appealing to practitioners who would like to use smooth estimates that, in addition, satisfy theoretical axioms of production. The utility of this method is illustrated through application to one real dataset and simulation evidences are also presented to show its superiority over the most known methods.

Estimating production frontiers and efficiency when output is a discretely distributed economic bad

This article studies the estimation of production frontiers and efficiency scores when the commodity of interest is an economic bad with a discrete distribution. Existing parametric econometric techniques (stochastic frontier methods) assume that output is a continuous random variable but, if output is discretely distributed, then one faces a scenario of model misspecification. Therefore a new class of econometric models has been developed to overcome this problem. The Delaporte subclass of models is studied in detail, and tests of hypotheses are proposed to discriminate among parametric models. In particular, Pearson’s chi-squared test is adapted to construct a new kernel-based consistent Pearson test. A Monte Carlo experiment evaluates the merits of the new model and methods, and these are used to estimate the frontier and efficiency scores of the production of infant deaths in England. Extensions to the model are discussed.

Estimating State‐Contingent Production Frontiers

Chambers and Quiggin (2000) use state‐contingent representations of risky production technologies to establish important theoretical results concerning producer behavior under uncertainty. Unfortunately, perceived problems in the estimation of state‐contingent models have limited the usefulness of the approach in policy formulation. We show that fixed and random effects state‐contingent production frontiers can be conveniently estimated in a finite mixtures framework. An empirical example is provided. Compared to conventional estimation approaches, we find that estimating production frontiers in a state‐contingent framework produces significantly different estimates of elasticities, firm technical efficiencies, and other quantities of economic interest.

Productivity and economic growth in East Asia: innovation, efficiency and accumulation

This paper uses a varying coefficients frontier production function model to examine the sources of growth between 1987 and 1993 in four East Asian economies—Hong Kong, Singapore, Japan and South Korea. Using data for 20 manufacturing sectors at the three-digit SIC level, this study provides the first comprehensive examination of sources of growth that allows one to decompose total factor productivity growth, separating out technical efficiency changes (TECs) from technological progress (TP). We find that while there is ample evidence of the importance of increasing inputs in growth, and there is some support for technical efficiency change, or catching up to the frontier over this period, there is weak or even negative evidence for the role of technological progress, measured as a shift in the estimated production frontier.

Productivity and Economic Growth in East Asia: Innovation, Efficiency and Accumulation

Sources of growth in East Asia have been a controversial subject in the literature. This necessitates more empirical studies to see whether there is more convergence to a particular view of the sources of growth. This study provides a comprehensive examination of sources of growth that allows one to decompose total factor productivity growth, separating out technical efficiency changes from technological progress. This paper applies a varying coefficients frontier production function model to data from 20 manufacturing sectors at the 3-digit SIC level to examine the sources of growth in four East Asian economies. The economies are Hong Kong, Singapore, Japan and South Korea, and the period covered is 1987-93. We find that while there is ample evidence of the importance of increasing inputs in growth, and there is some support for technical efficiency change, or catching up to the frontier over this period, there is weak or even negative evidence for the role of technological progress, measured as a shift in the estimated production frontier.

Nonparametric frontier estimation: a robust approach

Most nonparametric methods for estimating production frontiers (data envelopment analysis and free disposal hall (FDH)) are based on envelopment techniques. Statistical inference based on these estimators is available but, by construction, they are very sensitive to extreme values or outliers. We propose a nonparametric estimator, which is more robust to these extreme values. It is based on a concept of expected minimum input function (or expected maximal output function). We show how this function is related to the efficient frontier itself. The resulting estimator is related to the FDH estimator but it will not envelop all the data. The asymptotic theory is provided. Our approach includes the multiple input and multiple output cases.

Statistical inference in nonparametric frontier models: the state of the art

Efficiency scores of firms are measured by their distance to an estimated production frontier. The economic literature proposes several nonparametric frontier estimators based on the idea of enveloping the data (FDH and DEA-type estimators). Many have claimed that FDH and DEA techniques are non-statistical, as opposed to econometric approaches where particular parametric expressions are posited to model the frontier. We can now define a statistical model allowing determination of the statistical properties of the nonparametric estimators in the multi-output and multi-input case. New results provide the asymptotic sampling distribution of the FDH estimator in a multivariate setting and of the DEA estimator in the bivariate case. Sampling distributions may also be approximated by bootstrap distributions in very general situations. Consequently, statistical inference based on DEA/FDH-type estimators is now possible. These techniques allow correction for the bias of the efficiency estimators and estimation of confidence intervals for the efficiency measures. This paper summarizes the results which are now available, and provides a brief guide to the existing literature. Emphasizing the role of hypotheses and inference, we show how the results can be used or adapted for practical purposes.

A NOTE ON THE CONVERGENCE OF NONPARAMETRIC DEA ESTIMATORS FOR PRODUCTION EFFICIENCY SCORES

Efficiency scores of production units are measured by their distance to an estimated production frontier. Nonparametric data envelopment analysis estimators are based on a finite sample of observed production units, and radial distances are considered. We investigate the consistency and the speed of convergence of these estimated efficiency scores (or of the radial distances) in the very general setup of a multi-output and multi-input case. It is shown that the speed of convergence relies on the smoothness of the unknown frontier and on the number of inputs and outputs. Furthermore, one has to distinguish between the output- and the input-oriented cases.

Sensitivity Analysis of Efficiency Scores: How to Bootstrap in Nonparametric Frontier Models

Efficiency scores of production units are generally measured relative to an estimated production frontier. Nonparametric estimators (DEA, FDH, ⋯) are based on a finite sample of observed production units. The bootstrap is one easy way to analyze the sensitivity of efficiency scores relative to the sampling variations of the estimated frontier. The main point in order to validate the bootstrap is to define a reasonable data-generating process in this complex framework and to propose a reasonable estimator of it. This paper provides a general methodology of bootstrapping in nonparametric frontier models. Some adapted methods are illustrated in analyzing the bootstrap sampling variations of input efficiency measures of electricity plants.

Firm size and technical efficiency: Applications of production frontiers to Indian survey data

This paper is an empirical investigation of the relationship between relative technical efficiency and firm size in four Indian manufacturing industries. A frontier translog production function is used to derive measures of technical inefficiency at the enterprise level. Given the properties of the estimated production frontier these indices of technical efficiency are also indices of total factor productivity (TFP) which is the single most widely accepted measure of productive efficiency. Variations in technical efficiency are correlated with several explanatory variables, including size of firm. Firm size is found to be positively associated with relative productive efficiency in only one of the four industries.