The literature of productive efficiency analysis is dominated by two separate branches: the nonparametric data envelopment analysis (DEA) and the parametric stochastic frontier analysis (SFA). The origins of DEA date back to the seminal paper by FARRELL (1957), but its current popularity is largely due to the influential work by Charnes et al. (1978). The late seventies also saw the birth of SFA in the works of Aigner et al. (1977) and Meeusen and van den Broeck (1977), among others. Today, the literature of DEA and SFA is voluminous and growing rapidly (see e.g. recent surveys by Cherchye and Post, 2003; Murillo-Zamorano, 2004; and Worthington, 2001, 2004).

SFA builds upon the classic econometric regression approaches (e.g. Aigner and Chu, 1968) to production function estimation, which relies heavily on the ex ante specification of the functional form. The main attention has been in the decomposition of the residual into a non-negative inefficiency term and an idiosyncratic error. By contrast, DEA has focused on the nonparametric treatment of the frontier, which does not assume a particular functional form but relies on the general regularity properties such as monotonicity, convexity, and homogeneity. However, DEA attributes all deviations from the frontier to inefficiency, completely ignoring any stochastic noise in the data. In summary, it is generally accepted that the virtues of SFA lie in the stochastic, probabilistic treatment of inefficiency and noise, while the virtues of DEA lie in its general nonparametric frontier (see e.g. Bauer, 1990; and Seiford and Thrall, 1990).

Stochastic Nonparametric Envelopment of Data (StoNED) combines the virtues of both SFA and DEA in a unified framework of frontier analysis by melding the DEA-style nonparametric piece-wise linear frontier with the SFA-style decomposition of residuals into stochastic noise and inefficiency. In this sense, StoNED provides an encompassing framework that includes both SFA and DEA as its constrained special cases.

We estimate the StoNED model by Convex Nonparametric Least Squares regression (CNLS: HILDRETH 1954; Hanson and Pledger, 1976; Groeneboom et al., 2001), which does not require any smoothing parameters. Importantly, CNLS does not assume a priori any particular functional form for the regression function; it identifies the function that best fits the data from the family of continuous, monotonic increasing, concave functions that can be non-differentiable. Although Hildreth (1954) developed CNLS for estimating production functions, already before Farrell's (1957) seminal article, CNLS remained unknown in the field of productive efficiency analysis. Another important overlooked contribution worth recognizing isAFRIAT (1972), who developed the first analytical formulations of the FDH, DEA-VRS, and DEA-CRS frontiers in the single output case, together with the dual formulations, and formal proofs of their minimum extrapolation properties. Afriat's (1972) development of nonparametric frontiers has been ignored in the DEA literature, and his article is usually cited for the statistical inference on efficiency in the parametric stream of literature.

KUOSMANEN (2008) was the first to point out the connection between DEA and CNLS, making use of Afriat's (1972) theorem (see KUOSMANEN AND JOHNSON 2010 for a formal proof). Kuosmanen noted that the single-output DEA model can be regarded as a constrained variant of the CNLS regression. The StoNED method exploits this connection between DEA and CNLS further by introducing a stochastic noise term to a nonparametric regression model that is based on the same global shape constraints as the standard DEA.

StoNED method applies to both cross-sectional and panel data. In panel data, the time-invariant inefficiency components can be estimated in a fully nonparametric fashion by resorting the standard fixed effects or random effects treatments. In the cross-sectional setting, some further distributional assumptions are necessary for identifying inefficiency from noise. Cross-sectional StoNED method consists of two stages. In the first stage, we estimate the shape of the production function by CNLS without making any functional form, distributional or smoothness assumptions. CNLS provides an unbiased, consistent estimator for the shape of the production frontier, but inefficiency and noise terms remain indistinguishable. Therefore, in the second stage we follow Fan et al. (1996) and impose some standard distributional assumptions adopted from the SFA literature, and estimate the conditional expected value of the inefficiency term using the method of moments or pseudolikelihood techniques.

The StoNED method differs from the parametric and semi/nonparametric SFA treatments in that we do not make any functional form or smoothness assumptions, but build upon the global shape constraints (monotonicity, concavity). Compared to DEA, the StoNED method differs in its robustness to outliers and extreme observations and in its probabilistic treatment of inefficiency and noise. While the DEA frontier is typically spanned by a small number of influential observations, StoNED method uses information contained in the entire sample of observations for estimating the frontier. The main appeal of StoNED does not necessarily lie in its attractive properties; the key motivation of this paper is to contribute to better understanding of the connections between SFA and DEA by developing an encompassing framework that contains both methods as its special cases. Such an amalgam framework offers exciting new prospects for cross-fertilization between DEA and SFA paradigms.

Of course, StoNED is not the first attempt to combine features of DEA and SFA, but contributes to a long series of prior studies that have pursued similar aims. For example, Park and Simar (1994), and Park et al. (1998, 2003) have explored semiparametric estimation of SFA models in the context of panel data. On the other hand, the random parameters SFA models (e.g., Tsionas, 2002; Greene, 2005) allow for heterogeneity across firms by introducing firm-specific coefficients that are in common with DEA. On the purely nonparametric side, Cazals et al. (2002) and Aragon et al. (2002) have developed more robust versions of DEA-type estimators, but these approaches still do not allow for rigorous analysis of the stochastic noise. Fan et al., (1996), Kumbhakar et al. (2007), and Henderson and Simar (2005) have explored fully nonparametric SFA estimation by using kernel regression and local maximum likelihood techniques. However, imposing regularity conditions (monotonicity, concavity/convexity) is very difficult in these approaches. The closest relative of StoNED is the constrained maximum likelihood model by Banker and Maindiratta (1992), where the non-parametric DEA-style frontier is combined with the SFA-style noise and inefficiency terms. In practice, however, the constrained maximum likelihood problem of Banker and Maindiratta is extremely difficult to solve: there are no reported empirical applications of the Banker and Maindiratta method.

While the earlier studies come a long way of combining some aspects of DEA and SFA, the conceptual link between the parametric and non-parametric branches has been missing. In this respect, StoNED is the long sought missing link: StoNED is an extension of DEA in the same way as SFA is an extension of COLS. Moreover, both DEA and SFA can be seen as particular special cases of the general StoNED framework, obtained by imposing more restrictive assumptions about the functional form or the noise terms. In essence, StoNED combines DEA and SFA without compromising their attractive features. The main advantage of StoNED to the existing semi- and non-parametric alternatives is its heavy reliance on the established concepts and principles of SFA and DEA without introducing new concepts or tools (such as kernel regression). StoNED builds upon the standard axioms and assumptions that practitioners of SFA and DEA are comfortable with. Thus, readers familiar with classic SFA and DEA approaches will be able to appreciate and apply the proposed approach relatively easily. The conceptual bridges between DEA and SFA are also of methodological value for the further integration of productive efficiency analysis towards a more coherent and unified paradigm.

For more detailed description of the StoNED method, please click to sections "papers" by using the navigation bar on the left.




Afriat, S. (1972): Efficiency Estimation of Production Functions, International Economic Review 13, 568-598. HTTP://WWW.JSTOR.ORG/PSS/2525845.

Aigner, D.J., and S. Chu (1968): On Estimating the Industry Production Function, American Economic Review 58, 826-839.

Aigner, D.J., C.A.K. Lovell, and P. Schmidt (1977): Formulation and Estimation of Stochastic Frontier Models, Journal of Econometrics 6, 21-37.

Aragon, Y., A. Daouia, and C. Thomas-Agnan (2002): Nonparametric Frontier Estimation: A Conditional Quantile-based Approach, Discussion paper, GREMAQ et LSP, UniversitVe de Toulouse.

Bauer, P.W. (1990): Recent Developments in the Econometric Estimation of Frontiers, Journal of Econometrics 46 (1-2), 39-56.

Cazals, C., J.P. Florens, and L. Simar (2002): Nonparametric Frontier Estimation: A Robust Approach, Journal of Econometrics 106, 1-25.

Cherchye, L., and T. Post (2003): Methodological Advances in DEA: A Survey and an Application for the Dutch Electricity Sector, Statistica Neerlandica 57(4), 410-438.

Charnes, A., W.W. Cooper and E. Rhodes (1978): Measuring the Inefficiency of Decision Making Units, European Journal of Operational Research 2(6), 429-444. 44

Fan, Y., Q. Li and A. Weersink (1996): Semiparametric Estimation of Stochastic Production Frontier Models, Journal of Business and Economic Statistics 14(4), 460-468.

Farrell, M.J. (1957): The Measurement of Productive Efficiency, Journal of the Royal Statistical Society Series A. General 120(3): 253-282. HTTP://WWW.JSTOR.ORG/PSS/2343100

Greene, W.H. (2005): Reconsidering Heterogeneity in Panel Data Estimators of the Stochastic Frontier Model, Journal of Econometrics 126, 269-303.

Groeneboom, P., G. Jongbloed, and J.A. Wellner (2001): Estimation of a Convex Function: Characterizations and Asymptotic Theory, Annals of Statistics 29, 1653-1698.

Hanson, D.L., and G. Pledger (1976): Consistency in Concave Regression, Annals of Statistics 4(6), 1038-1050.

Hildreth, C. (1954): Point Estimates of Ordinates of Concave Functions, Journal of the American Statistical Association 49(267), 598-619. HTTP://WWW.JSTOR.ORG/PSS/2281132

Henderson, D.J., and L. Simar (2005): A Fully Nonparametric Stochastic Frontier Model for Panel Data, Discussion Paper 0417, Institut de Statistique, Universite Catholique de Louvain.

Kumbhakar, S.C., B.U. Park, L. Simar, and E.G. Tsionas (2007): Nonparametric Stochastic Frontiers: A Local Maximum Likelihood Approach, Journal of Econometrics 137(1), 1-27.

Kuosmanen, T. (2008): Representation Theorem for Convex Nonparametric Least Squares, Econometrics Journal 11, 308-325.

Kuosmanen, T. and A. Johnson (2010): Data Envelopment Analysis as Nonparametric Least Squares Regression, Operations Research 58(1): 149-160.

Meeusen, W., and J. van den Broeck (1977): Efficiency Estimation from Cobb-Douglas Production Function with Composed Error, International Economic Review 8, 435-444.

Murillo-Zamorano, L.R. (2004): Economic Efficiency and Frontier Techniques, Journal of Economic Surveys 18(1), 33-77.

Park, B., R.C. Sickles, and L. Simar (1998): Stochastic Panel Frontiers: A Semiparametric Approach, Journal of Econometrics 84, 273-301.

Park, B., R.C. Sickles, and L. Simar (2003): Semiparametric Efficient Estimation of AR(1) Panel Data Models, Journal of Econometrics 117, 279-309.

Park, B., and L. Simar (1994): Efficient Semiparametric Estimation in a Stochastic Frontier Model, Journal of the American Statistical Association 89(427), 929-936.

Seiford, L. M., and R.M. Thrall (1990): Recent Developments in DEA: The Mathematical Programming Approach to Frontier Analysis, Journal of Econometrics 46 (1-2), 7-38.

Tsionas, M. (2002): Stochastic frontier models with random coefficients, Journal of Applied Econometrics 17, 127–147.

Worthington, A.C. (2001): An Empirical Survey of Frontier Efficiency Measurement Techniques in Education, Education Economics 9(3), 245-268

Worthington, A.C. (2004): An Empirical Survey of Frontier Efficiency Measurement Techniques in Healthcare Services, Medical Care Research and Review 61(2), 1-36.

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