Cross redundancy and sensitivity in DEA models

Data envelopment analysis (DEA) measures the efficiency of each decision making unit (DMU) by maximizing the ratio of virtual output to virtual input with the constraint that the ratio does not exceed one for each DMU. In the case that one output variable has a linear dependence (conic dependence, to be precise) with the other output variables, it can be hypothesized that the addition or deletion of such an output variable would not change the efficiency estimates. This is also the case for input variables. However, in the case that a certain set of input and output variables is linearly dependent, the effect of such a dependency on DEA is not clear. In this paper, we call such a dependency a cross redundancy and examine the effect of a cross redundancy on DEA. We prove that the addition or deletion of a cross-redundant variable does not affect the efficiency estimates yielded by the CCR or BCC models. Furthermore, we present a sensitivity analysis to examine the effect of an imperfect cross redundancy on DEA by using accounting data obtained from United States exchange-listed companies.     

基于超效率DEA模型的我国各省水运业效率分析

应用超效率DEA模型并结合CCR模型和BCC模型对我国各省水运业效率加以分析。通过BCC模型求得各省水运业的纯技术效率和规模效率,分析各省水运业管理现状和所处规模报酬阶段;对比分析了CCR模型和超效率DEA模型得到的效率值差异,从中也反映了应用超效率DEA获得效率值的真实性。实证结果表明,天津、河北、辽宁、上海、海南、贵州、云南和青海8个地区的综合效率、纯技术效率和规模效率都达到有效;CCR模型和超效率DEA模型各省总体均值分别为0.606和1.556,差异很大;最后应用CCR模型松弛变量分析了投入冗余额和产出不足额,给出了各投入产出指标的改进值。     

Efficiency and Global Scale Characteristics on the “No Free Lunch” Assumption Only

In a production technology, the type of returns to scale (RTS) associated with an efficient decision making unit (DMU) is indicative of the direction of marginal rescaling that the DMU should undertake in order to improve its productivity. In this paper a concept of global returns to scale (GRS) is developed as an indicator of the direction in which the most productive scale size (MPSS) of an efficient DMU is achieved. The GRS classes are useful in assisting strategic decisions like those involving mergers of units or splitting into smaller firms. The two characterisations, RTS and GRS, are the same in a convex technology but generally different in a non-convex one. It is shown that, in a non-convex technology, the well-known method of testing RTS proposed by Färe et al. is in fact testing for GRS and not RTS. Further, while there are three types of RTS: constant, decreasing and increasing (CRS, DRS and IRS, respectively), the classification according to GRS includes the fourth type of sub-constant GRS, which describes a DMU able to achieve its MPSS by both reducing and increasing the scale of operations. The notion of GRS is applicable to a wide range of technologies, including the free disposal hull (FDH) and all polyhedral technologies used in data envelopment analysis (DEA).     

Inference by the <Emphasis Type="Italic">m</Emphasis> out of <Emphasis Type="Italic">n</Emphasis> bootstrap in nonparametric frontier models

It is well-known that the naive bootstrap yields inconsistent inference in the context of data envelopment analysis (DEA) or free disposal hull (FDH) estimators in nonparametric frontier models. For inference about efficiency of a single, fixed point, drawing bootstrap pseudo-samples of size m < n provides consistent inference, although coverages are quite sensitive to the choice of subsample size m. We provide a probabilistic framework in which these methods are shown to valid for statistics comprised of functions of DEA or FDH estimators. We examine a simple, data-based rule for selecting m suggested by Politis et al. (Stat Sin 11:1105–1124, 2001), and provide Monte Carlo evidence on the size and power of our tests. Our methods (i) allow for heterogeneity in the inefficiency process, and unlike previous methods, (ii) do not require multivariate kernel smoothing, and (iii) avoid the need for solutions of intermediate linear programs.     

On FDH efficiency analysis: Some methodological issues and applications to retail banking,courts, and urban transit

The methodology of free disposal hull (FDH) measure of productive efficiency is defined and put in perspectivevis-à-vis other nonparametric techniques, in terms of the postulates on which they respectively rest. Computational issues are also considered, in relation to the linear programming techniques used in DEA. The first application bears on a comparison between a private and a public bank, in terms of the relative efficiency of their branches. Important characteristics of the data are revealed by FDH that are not by DEA, due to a better data fit. Next, efficiency estimates of judicial activities are used to evaluate what part of the existing backlog could be reduced by efficiency increases. Finally, with monthly data of an urban transit firm over 12 years, the FDH methodology is extended to a sequential treatment of time series, that supplements efficiency estimation with a measure of technical progress.     

Stochastic FDH/DEA estimators for frontier analysis

In this paper we extend the work of Simar (J Product Ananl 28:183–201, 2007) introducing noise in nonparametric frontier models. We develop an approach that synthesizes the best features of the two main methods in the estimation of production efficiency. Specifically, our approach first allows for statistical noise, similar to Stochastic frontier analysis (even in a more flexible way), and second, it allows modelling multiple-inputs-multiple-outputs technologies without imposing parametric assumptions on production relationship, similar to what is done in non-parametric methods, like Data Envelopment Analysis (DEA), Free Disposal Hull (FDH), etc.... The methodology is based on the theory of local maximum likelihood estimation and extends recent works of Kumbhakar et al. (J Econom 137(1):1–27, 2007) and Park et al. (J Econom 146:185–198, 2008). Our method is suitable for modelling and estimation of the marginal effects onto inefficiency level jointly with estimation of marginal effects of input. The approach is robust to heteroskedastic cases and to various (unknown) distributions of statistical noise and inefficiency, despite assuming simple anchorage models. The method also improves DEA/FDH estimators, by allowing them to be quite robust to statistical noise and especially to outliers, which were the main problems of the original DEA/FDH estimators. The procedure shows great performance for various simulated cases and is also illustrated for some real data sets. Even in the single-output case, our simulated examples show that our stochastic DEA/FDH improves the Kumbhakar et al. (J Econom 137(1):1–27, 2007) method, by making the resulting frontier smoother, monotonic and, if we wish, concave.     

Efficiency characterizations in different DEA models

Relations of efficiency and non-efficiency for the same sets of DMUs (Decision Making Units) are developed for the Charnes, Cooper and Rhodes (CCR) and Barker, Charnes, Cooper (BCC) ratio models, as well as DEA Additive and Multiplicative Models. Surprisingly, additively efficient DMUs are not necessarily multiplicatively efficient. A geometric “stretching” phenomenon is identified for the latter case.     

The effect of foreign ownership on SMEs performance: An efficiency analysis perspective

This paper uses both Data Envelopment Analysis (DEA) and Free Disposal Hull (FDH) models in order to determine different performance levels in a sample of 353 foreign equities operating in the Greek manufacturing sector. Particularly, convex and non-convex models are used alongside with bootstrap techniques in order to determine the effect of foreign ownership on SMEs’ performance. The study illustrates how the recent developments in efficiency analysis and statistical inference can be applied when evaluating performance issues. The analysis among the foreign equities indicates that the levels of foreign ownership have a positive effect on SMEs’ performance.     

How to improve the performances of DEA/FDH estimators in the presence of noise?

In frontier analysis, most nonparametric approaches (DEA, FDH) are based on envelopment ideas which assume that with probability one, all observed units belong to the attainable set. In these “deterministic” frontier models, statistical inference is now possible, by using bootstrap procedures. In the presence of noise, envelopment estimators could behave dramatically since they are very sensitive to extreme observations that might result only from noise. DEA/FDH techniques would provide estimators with an error of the order of the standard deviation of the noise. This paper adapts some recent results on detecting change points [Hall P, Simar L (2002) J Am Stat Assoc 97:523–534] to improve the performances of the classical DEA/FDH estimators in the presence of noise. We show by simulated examples that the procedure works well, and better than the standard DEA/FDH estimators, when the noise is of moderate size in term of signal to noise ratio. It turns out that the procedure is also robust to outliers. The paper can be seen as a first attempt to formalize stochastic DEA/FDH estimators.      

基于数据包络方法的2010年会计师事务所经营效率分析

本文运用数据包络分析方法中基于输入的CCR模型和BCC模型,对2010年会计师事务所的主要投入产出数据进行效率分析,从整体上分析目前我国会计师事务所的运营效率。通过研究发现,一部分会计师事务所已经达到规模效益,但是还有很多会计师事务所在技术和规模上都没有达到理想的水平,在经营效率上还需要很大的改进。     

Axiomatic foundations of efficiency measurement on data-generated technologies

We assess the ability of three well-known technical efficiency indexes, the Debreu-Farrell index, the Färe–Lovell index, and the Zieschang index, to satisfy the Färe–Lovell axioms and continuity axioms (for technologies as well as input quantities) on the class of technologies generated by standard mathematical-programming methods of measuring efficiency: data envelopment analysis (DEA) and free-disposal-hull (FDH) analysis. Our principal conclusions are that (a) restriction to these data-based technologies adds continuity in input quantities to the properties satisfied by the Färe–Lovell and the Zieschang indexes (thus eliminating a salient advantage of the Debreu–Farrell index), but (b) none of the indexes satisfies all Färe–Lovell axioms (nor all continuity axioms) on either DEA or FDH technologies, and hence (c) trade-offs among the indexes remain. These findings provide motivation for the search for an index that does satisfy these axioms on DEA and FDH technologies.     

会计师事务所人力资源投入产出效率相对有效性分析

本文运用数据包络分析(DEA)中的CCR和BCC模型,对40家会计师事务所的人力资源投入产出效率的相对有效性进行评价,使他们看到自己与其他所的差距,从而更清楚地认识自己,分析它们为非DEA有效的原因,并根据DEA的投影定理提出相关的改进建议,为会计师事务所的经营决策提供帮助。     

基于DEA的安徽省物流效率研究

随着我国经济的不断发展,物流行业作为重要的服务业,发展势头十分迅猛。文章运用DEA模型中的两个经典模型:CCR(判断决策单元是否同时为技术有效和规模有效)模型和BCC(判断决策单元是否为技术有效)模型,根据中华人民共和国国家统计局网站和《安徽省统计年鉴》数据,对安徽省2003~2012年的物流投入、产出情况进行实证分析,测算安徽省的物流效率,并对安徽省的物流行业持续健康发展提出建议。     

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.     

Categorical and continuous non-discretionary variables in data envelopment analysis: a comparison of two single-stage models

Non-discretionary or environmental variables are regarded as important in the evaluation of efficiency in Data Envelopment Analysis (DEA), but there is no consensus on the correct treatment of these variables. This paper compares the performance of the standard BCC model as a base case with two single-stage models: the Banker and Morey (1986a) model, which incorporates continuous environmental variables and the Banker and Morey (1986b) model, which incorporates categorical environmental variables. Simulation analyses are conducted using a shifted Cobb-Douglas function, with one output, one non-discretionary input, and two discretionary inputs. The production function is constructed to separate environmental impact from managerial inefficiency, while providing measures of both for comparative purposes. Tests are performed to evaluate the accuracy of each model. The distribution of the inputs, the sample size and the number of categories for the categorical model are varied in the simulations to determine their impact on the performance of each model. The results show that the Banker and Morey models should be used in preference to the standard BCC model when the environmental impact is moderate to high. Both the continuous and categorical models perform equally well but the latter may be better suited to some applications with larger sample sizes. Even when the environmental impact is slight, the use of a simple two-way split of the sample data can produce significantly better results under the Categorical model in comparison to the BCC model.     

The closest strong efficient targets in the FDH technology: an enumeration method

This paper is concerned with the identification of the closest strong efficient target of a Decision Making Unit (DMU) in the Free Disposal Hull (FDH) technology in Data Envelopment Analysis (DEA). The paper uses the geometrical properties of the FDH Production Possibility Set (PPS) to design and test an enumeration algorithm to obtain the minimum distance from a DMU to the strong efficient frontier of the PPS, corresponding to each of the various returns to scale assumptions. The proposed method solves some simple optimization problems whose optimal solutions are obtained by calculating a limited number of ratios. Then, an attempt will be made to mitigate the problem of the lack of unit and translation invariance of the selected distances by considering weighted norms. Finally, the applicability of the presented method is illustrated by a numerical example using real data.

     

The Application of Mathematical Programming Approaches to Estimating Container Port Production Efficiency

Container terminal production is both an important and complicated element in the contemporary global economy. This paper aims to evaluate the efficiency of the world’s most important container ports and terminals using the two alternative techniques of Data Envelopment Analysis (DEA) and the Free Disposal Hull (FDH) model. The results give an insight into the current efficiency ranking of the world’s major container ports and terminals. They also confirm expectations that the available mathematical programming methodologies lead to different results and that appropriate variable definition of input and output factors is a crucial element in meaningful applications of DEA and FDH. It is also concluded that the availability of panel data, rather than cross-sectional data would greatly improve the validity of the efficiency estimates derived from all the mathematical programming techniques applied.JEL Classification:C61, D24, E23, L23, L25, L92     

FDH Directional Distance Functions with an Application to European Commercial Banks

Weextend Free Disposable Hull (FDH) efficiency analysis towardsthe general directional distance function framework. The profitinterpretation of directional distance functions is extendedto the non-convex FDH technologies. In addition, we derive anefficient enumerative algorithm for computing distance measuresin Free Disposable Hull (FDH) technologies, which applies tothe entire (infinitely large) family of directional distancefunctions. A simple numerical example and an application to Europeancommercial banks illustrate the algorithm.     

Stochastic leader–follower DEA models for two-stage systems

Data envelopment analysis (DEA) is a non-parametric approach for measuring the relative efficiencies of peer decision making units (DMUs). In recent years, it has been widely used to evaluate two-stage systems under different organization mechanisms. This study modifies the conventional leader–follower DEA models for two-stage systems by considering the uncertainty of data. The dual deterministic linear models are first constructed from the stochastic CCR models under the assumption that all components of inputs, outputs, and intermediate products are related only with some basic stochastic factors, which follow continuous and symmetric distributions with nonnegative compact supports. The stochastic leader–follower DEA models are then developed for measuring the efficiencies of the two stages. The stochastic efficiency of the whole system can be uniquely decomposed into the product of the efficiencies of the two stages. Relationships between stochastic efficiencies from stochastic CCR and stochastic leader–follower DEA models are also discussed. An example of the commercial banks in China is considered using the proposed models under different risk levels.     

Assessing the influence of external factors on agricultural production in Brazil

This article uses 2017 Brazilian agricultural census data (aggregated at county level) to evaluate the impact of external factors on agricultural efficiency in Brazil. The external factors are defined as access to credit, participation in cooperatives, proportion of literacy, technical assistance, and environmental production practices. All variables are transformed as the log of the municipal (county) rank. The response variable is defined as the free disposal hull (FDH) conditional ratio and a one-inflated beta regression is estimated. The results show that in counties where the FDH ratio is equal to 1, cooperatives have the biggest impact on efficiency, and where the FDH ratio is not equal to 1, environmental practices, which are most often associated with technology, have the largest effect on efficiency. Finally, the regions that are expected to see the largest gains in efficiency with increases in external factors are the Northeastern and the Northern regions of Brazil.