Timo Kuosmanen

Data Envelopment Analysis as Nonparametric Least-Squares Regression

Data envelopment analysis (DEA) is known as a nonparametric mathematical programming approach to productive efficiency analysis. In this paper, we show that DEA can be alternatively interpreted as nonparametric least-squares regression subject to shape constraints on the frontier and sign constraints on residuals. This reinterpretation reveals the classic parametric programming model by Aigner and Chu [Aigner, D., S. Chu. 1968. On estimating the industry production function. Amer. Econom. Rev.58 826--839] as a constrained special case of DEA. Applying these insights, we develop a nonparametric variant of the corrected ordinary least-squares (COLS) method. We show that this new method, referred to as corrected concave nonparametric least squares (C2NLS), is consistent and asymptotically unbiased. The linkages established in this paper contribute to further integration of the econometric and axiomatic approaches to efficiency analysis.

One-stage and two-stage DEA estimation of the effects of contextual variables

Two-stage data envelopment analysis (2-DEA) is commonly used in productive efficiency analysis to estimate the effects of operational conditions and practices on performance. In this method the DEA efficiency estimates are regressed on contextual variables representing the operational conditions. We re-examine the statistical properties of the 2-DEA estimator, and find that it is statistically consistent under more general conditions than earlier studies assume. We further show that the finite sample bias of DEA in the first stage carries over to the second stage regression, causing bias in the estimated coefficients of the contextual variables. This bias is particularly severe when the contextual variables are correlated with inputs. To address this shortcoming, we apply the result that DEA can be formulated as a constrained special case of the convex nonparametric least squares (CNLS) regression. Applying the CNLS formulation, we develop a new semi-nonparametric one-stage estimator for the coefficients of the contextual variables that directly incorporates contextual variables to the standard DEA problem. The proposed method is hence referred to as one-stage DEA (1-DEA). Evidence from Monte Carlo simulations suggests that the new 1-DEA estimator performs systematically better than the conventional 2-DEA estimator both in deterministic and noisy scenarios.

Modelling weak disposability in data envelopment analysis under relaxed convexity assumptions

The treatment of undesirable (bad) outputs in models of efficiency and productivity analysis often requires replacing the assumption of free disposability of outputs by their weak disposability. In a recent publication the authors showed that the Kuosmanen technology is the only correct representation of the fully convex technology exhibiting weak disposability of bad and good outputs. In this paper we relax the assumption of full convexity and consider two further possibilities: the case in which only the output sets are assumed convex and the case in which no convexity is assumed at all. In the first case we show that, although the traditional Shephard technology of nonparametric production analysis satisfies the assumption of convex output sets, it is larger than necessary. Based on the minimum extrapolation principle, we develop a correct model that is based on the assumed axioms. The second case leads to the development of a weakly disposable analogue of the free disposable hull. To complete our study, we give a full axiomatic definition of the Shephard technology.

What Is the Economic Meaning of FDH? A Reply to Thrall

In a recent issue of the Journal of Productivity Analysis, Thrall (1999) called for abandoning the Free Disposable Hull (FDH, Deprins et al. (1984)) approximation of production possibilities as economically meaningless in comparison to the Convex Monotone Hull (CMH; Banker et al. (1984)) approximation. This strong conclusion was solely based on Thralls Principal Theorem, which essentially demonstrates that FDH can give a technically efficient classification to output-input vectors that are inefficient in terms of profit maximisation, i.e. at all non-negative price vectors there exists an alternative output-input vector that yields higher profit. In this short communication, we argue that the economic meaning of the competing empirical production sets cannot be inferred from this theorem. Specifically, we demonstrate that both empirical production sets are economically equally meaningful under the economic conditions that underlie Thralls theorem. In addition, we demonstrate that FDH can be economically more meaningful than CMH under non-trivial alternative economic conditions.

DEA with efficiency classification preserving conditional convexity

Abstract We propose to relax the standard convexity property used in Data Envelopment Analysis (DEA) by imposing additional qualifications for feasibility of convex combinations. We specifically focus on a condition that preserves the Koopmans efficiency classification. This yields an efficiency classification preserving conditional convexity property, which is implied by both monotonicity and convexity, but not conversely. Substituting convexity by conditional convexity, we construct various empirical DEA approximations as the minimal sets that contain all DMUs and are consistent with the imposed production assumptions. Imposing an additional disjunctive constraint to standard convex DEA formulations can enforce conditional convexity. Computation of efficiency measures relative to conditionally convex production set can be performed through Disjunctive Programming (DP).

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.

A More Efficient Algorithm for Convex Nonparametric Least Squares

Convex Nonparametric Least Squares (CNLSs) is a nonparametric regression method that does not require a priori specification of the functional form. The CNLS problem is solved by mathematical programming techniques; however, since the CNLS problem size grows quadratically as a function of the number of observations, standard quadratic programming (QP) and Nonlinear Programming (NLP) algorithms are inadequate for handling large samples, and the computational burdens become significant even for relatively small samples. This study proposes a generic algorithm that improves the computational performance in small samples and is able to solve problems that are currently unattainable. A Monte Carlo simulation is performed to evaluate the performance of six variants of the proposed algorithm. These experimental results indicate that the most effective variant can be identified given the sample size and the dimensionality. The computational benefits of the new algorithm are demonstrated by an empirical application that proved insurmountable for the standard QP and NLP algorithms.

Firm and industry level profit efficiency analysis using absolute and uniform shadow prices

We discuss the nonparametric approach to profit efficiency analysis at the firm and industry levels in the absence of complete price information. Two new insights are developed. First, we measure profit inefficiency in monetary terms using absolute shadow prices. Second, we evaluate all firms using the same input-output prices. This allows us to aggregate firm-level profit inefficiencies to the overall industry inefficiency. Besides the measurement of profit losses, the presented approach enables one to recover absolute price information from quantity data. We conduct a series of Monte Carlo simulations to study the performance of the proposed approach in controlled production environments.

Stochastic Nonparametric Envelopment of Data: Combining Virtues of SFA and DEA in a Unified Framework

The literature of productive efficiency analysis is divided into two main branches: the parametric Stochastic Frontier Analysis (SFA) and nonparametric Data Envelopment Analysis (DEA). This paper attempts to combine the virtues of both approaches in a unified framework. We follow the SFA literature and introduce a stochastic component decomposed into idiosyncratic error and technical inefficiency components imposing the standard SFA assumptions. In contrast to the SFA, we do not make any prior assumptions about the functional form of the deterministic production function. In this respect, we follow the nonparametric route of DEA that only imposes free disposability, convexity, and some specification of returns to scale. From the postulated class of production functions, the proposed method identifies the production function with the best empirical fit to the data. The resulting function will always take a piece-wise linear form analogous to the DEA frontiers. We discuss the practical implementation of the method and illustrate its potential by means empirical examples.

Stochastic Nonparametric Approach to Efficiency Analysis: A Unified Framework

Bridging the gap between axiomatic Data Envelopment Analysis (DEA) and econometric Stochastic Frontier Analysis (SFA) has been one of the most vexing problems in the field of efficiency analysis. Recent developments in multivariate convex regression, particularly Convex Nonparametric Least Squares (CNLS) method, have led to the full integration of DEA and SFA into a unified framework of productivity analysis, referred to as Stochastic Nonparametric Envelopment of Data (StoNED). The unified framework of StoNED offers a general and flexible platform for efficiency analysis and related themes such as frontier estimation and production analysis, allowing one to combine existing tools of efficiency analysis in novel ways across the DEA-SFA spectrum, facilitating new opportunities for further methodological development. This chapter provides an updated and elaborated presentation of the CNLS and StoNED methods. This chapter also extends the scope of the StoNED method in several directions. Most notably, this chapter examines quantile estimation using StoNED and an extension of the StoNED method to the general case of multiple inputs and multiple outputs. This chapter also provides a detailed discussion of how to model heteroscedasticity in the inefficiency and noise terms.