Stochastic nonsmooth envelopment of data: semiparametric frontier estimation subject to shape constraints
Abstract
The field of productive efficiency analysis is currently divided between two main paradigms: the deterministic, nonparametric Data Envelopment Analysis (DEA) and the parametric Stochastic Frontier Analysis (SFA). This paper examines an encompassing semiparametric frontier model that combines the DEAtype nonparametric frontier, which satisfies monotonicity and concavity, with the SFAstyle stochastic homoskedastic composite error term. To estimate this model, a new twostage method is proposed, referred to as Stochastic Nonsmooth Envelopment of Data (StoNED). The first stage of the StoNED method applies convex nonparametric least squares (CNLS) to estimate the shape of the frontier without any assumptions about its functional form or smoothness. In the second stage, the conditional expectations of inefficiency are estimated based on the CNLS residuals, using the method of moments or pseudolikelihood techniques. Although in a crosssectional setting distinguishing inefficiency from noise in general requires distributional assumptions, we also show how these can be relaxed in our approach if panel data are available. Performance of the StoNED method is examined using Monte Carlo simulations.
Keywords
Data envelopment analysis (DEA) Frontier estimation Nonparametric least squares Productive efficiency analysis Stochastic frontier analysis (SFA)JEL Classification
C14 C51 D241 Introduction
The literature of productive efficiency analysis and frontier estimation is large and growing, consisting of several thousands of studies in the fields of applied economics, econometrics, operations research, and statistics (see e.g., Fried et al. 2008, for an uptodate introduction and literature review). This field is currently dominated by two approaches: the nonparametric data envelopment analysis (DEA: Farrell 1957; Charnes et al. 1978) and the parametric stochastic frontier analysis (SFA: Aigner et al. 1977; Meeusen and van den Broeck 1977). The main appeal of DEA lies in its axiomatic, nonparametric treatment of the frontier, which does not assume a particular functional form but relies on the general regularity properties such as free disposability, convexity, and assumptions concerning the returns to scale. However, the conventional DEA attributes all deviations from the frontier to inefficiency, and ignores any stochastic noise in the data. The key advantage of SFA is its stochastic treatment of these deviations, which are decomposed into a nonnegative inefficiency term and a random disturbance term that accounts for measurement errors and other random noise. However, SFA builds on the parametric regression techniques, which require an ex ante specification of the functional form. Since the economic theory rarely justifies a particular functional form, the flexible functional forms, such as the translog or generalized McFadden are frequently used. In contrast to DEA, the flexible functional forms often violate the monotonicity, concavity/convexity and homogeneity conditions. Further, imposing these conditions can sacrifice the flexibility (see e.g., Sauer 2006). In summary, it is generally accepted that the virtues of DEA lie in its general, nonparametric treatment of the frontier, while the virtues of SFA lie in its stochastic, probabilistic treatment of inefficiency and noise.
Bridging the gap between SFA and DEA has been recognized as one of the most important research objectives in this field, and contributions to this end have accumulated since the early 1990s. The emerging literature on semi/nonparametric stochastic frontier estimation has thus far mainly departed from the SFA side, replacing the parametric frontier function by a nonparametric specification that can be estimated by kernel regression or local maximum likelihood (ML) techniques. Fan et al. (1996) and Kneip and Simar (1996) were among the first to apply kernel regression to frontier estimation in the crosssectional and panel data contexts, respectively. Fan et al. (1996) proposed a twostep method where the shape of the frontier is first estimated by kernel regression, and the conditional expected inefficiency is subsequently estimated based on the residuals, imposing the same distributional assumptions as in standard SFA. Kneip and Simar (1996) similarly use kernel regression for estimating the frontier, but they make use of panel data to avoid the distributional assumptions. Other semi/nonparametric panel data approaches include Park et al. (1998, 2003, 2006) and Henderson and Simar (2005), among others. Recently, Kumbhakar et al. (2007) proposed a more flexible SFA method based on local polynomial ML estimation. While the model is parametrized in a similar way to the standard SFA models, all model parameters are approximated by local polynomials. Simar and Zelenyuk (2008) have further extended the local polynomial ML method to multioutput technologies, building upon results by Hall and Simar (2002) and Simar (2007). Interestingly, Simar and Zelenyuk (2008) also apply DEA to the fitted values of the Kumbhakar et al. (2007) method in order to impose monotonicity and concavity.
Departing from the DEA side, Banker and Maindiratta (1992) were the first to consider ML estimation of the stochastic frontier model subject to the global free disposability and convexity axioms adopted from the DEA literature. While their theoretical model combines the essential features of the classic DEA and SFA models, solving the resulting ML problem has proved extremely difficult, if not impossible in practical applications. We are not aware of any reported empirical applications of the Banker and Maindiratta’s constrained ML method.
While the earlier semi/nonparametric developments come a long way in bridging the gap between DEA and SFA approaches, further elaboration of the interface between these two paradigms is clearly desirable. Since conventional DEA literature emphasizes the fundamental philosophical difference between DEA and the regression techniques (e.g., Cooper et al. 2004), the intimate links between DEA and regression analysis may not have attracted sufficient attention. In this respect, the recent studies Kuosmanen (2008) and Kuosmanen and Johnson (2010) have shown that DEA can be understood as a constrained special case of nonparametric least squares subject to shape constraints. More specifically, Kuosmanen and Johnson (2010) prove formally that the classic outputoriented DEA estimator can be computed in the singleoutput case by solving the convex nonparametric least squares (CNLS) problem (Hildreth 1954; Hanson and Pledger 1976; Groeneboom et al. 2001a,b; Kuosmanen 2008) subject to monotonicity and concavity constraints that characterize the frontier, and a sign constraint on the regression residuals. Thus, DEA can be naturally viewed as a nonparametric counterpart to the parametric programming approach of Aigner and Chu (1968). Building on this analogue, Kuosmanen and Johnson (2010) propose a nonparametric counterpart to the classic COLS method (Greene 1980), which has generally a higher discriminatory power than the conventional DEA in the deterministic setting. However, the deterministic frontier shifting method of Kuosmanen and Johnson (2010) is more sensitive to stochastic noise than the conventional DEA.
Departing from Kuosmanen and Johnson (2010), this paper introduces a stochastic noise term explicitly into the theoretical model to be estimated, and takes it into account in the estimation. In the spirit of Banker and Maindiratta (1992), we examine an encompassing semiparametric frontier model that includes the classic SFA and DEA models as its constrained special cases. More specifically, we assume that the observed data deviates from a nonparametric, DEAstyle piecewise linear frontier production function due to a stochastic SFAstyle composite error term, consisting of homoskedastic noise and inefficiency components. To estimate this theoretical model, we develop a new twostage method, referred to as stochastic nonsmooth envelopment of data (StoNED).^{1} In line with Kuosmanen and Johnson (2010), we first estimate the shape of the frontier by applying the CNLS regression, which does not assume a priori any particular functional form for the regression function. CNLS identifies the function that best fits the data from the family of continuous, monotonic increasing, concave functions that can be nondifferentiable. In the second stage, we estimate the variance parameters of the stochastic inefficiency and noise terms based on the skewness of the CNLS residuals. The noise term is assumed to be symmetric, so the skewness of the regression residuals is attributed to the inefficiency term. Given the parametric distributional assumptions of the inefficiency and the noise terms, we can estimate the variance parameters by using the method of moments (Aigner et al. 1977) or pseudolikelihood (Fan et al. 1996) techniques. The conditional expected value of the inefficiency term can obtained by using the results of Jondrow et al. (1982).
The proposed StoNED method differs from the parametric and semi/nonparametric SFA treatments in that we do not make any assumptions about the functional form or its smoothness, but build upon the global shape constraints (monotonicity, concavity). These shape constraints are equivalent to the free disposability and convexity axioms of DEA. Compared to DEA, the StoNED method differs in its probabilistic treatment of inefficiency and noise. Whereas the DEA frontier is typically spanned by a small number of influential observations, which makes it sensitive to outliers and noise, the StoNED method uses information contained in the entire sample of observations for estimating the frontier, and infers the expected value of inefficiency in a probabilistic fashion.
While this paper focuses on the crosssectional model, we will also briefly suggest how the approach could be extended to the panel data setting. In that case, the timeinvariant inefficiency components can be estimated in a fully nonparametric fashion by resorting the standard fixed effects treatment analogous to Schmidt and Sickles (1984). In the crosssectional setting, imposing some distributional assumptions seems necessary, otherwise inefficiency cannot be distinguished from noise. However, the parametric distributional assumptions should not be taken as the main limitation. While the absolute levels of our frontier and the inefficiency estimates critically depend on the distributional assumptions, the shape of the estimated frontier and the relative rankings of the evaluated units are not affected by these assumptions. In contrast, the classic homoskedastic inefficiency term must be recognized as a more critical assumption. Indeed, even the shape of the frontier and the efficiency rankings tend to be biased if the homoskedasticity assumption is violated (see Sect. 4.5 for a more detailed discussion of this point). Dealing with heteroskedastic inefficiency is left as an interesting and important issue to be addressed in the future research.^{2}
The remainder of the paper is organized as follows. Section 2 introduces the semiparametric model of frontier production function that encompasses the classic DEA and SFA models as its special cases. Section 3 introduces the twostage estimation strategy of the StoNED method: Sect. 3.2 elaborates the first stage consisting of nonparametric estimation of the production function by employing CNLS regression. Based on the CNLS residuals, we estimate the inefficiency and noise terms by means of method of moments and pseudolikelihood techniques, as described in Sect. 3.3. Section 4 discusses some useful extensions to the proposed approach. Section 5 examines how the proposed techniques perform in a controlled environment of Monte Carlo simulations. Finally, Sect. 6 makes concluding remarks. An illustrative example is presented in the “Appendix”. Further supplementary material such as graphical illustrations, example applications, and computational codes are available in the working papers Kuosmanen (2006), Kuosmanen and Kortelainen (2007), and the website: http://www.nomepre.net/stoned/.
2 Encompassing frontier model
This section introduces the theoretical model of frontier production functions to be estimated and the assumptions that will be maintained throughout the paper, except for Sect. 4.1 where a panel data model will be considered. Even in the crosssectional setting we will later introduce more specific assumptions as they become necessary. To maintain direct contact with SFA, we describe the model for the singleoutput multiple input case. The mdimensional input vector is denoted by \( {\mathbf x} \in \Re_{ + }^{m} \) and the scalar output by \( y \in \Re_{ + }^{{}} \). The production technology is represented by the frontier production function \( f:\Re_{ + }^{m} \to \Re_{ + }^{{}} \) that indicates the maximum output that can be produced with the given inputs. Following the classic DEA approach, we assume that function f belongs to the class of continuous, monotonic increasing and globally concave functions that can be nondifferentiable. In what follows, this class of functions will be denoted by F _{2}. In contrast to the traditional SFA literature, no specific functional form for f is assumed a priori; our specification of the production function proceeds along the nonparametric lines of the DEA literature.
In model (1), the deterministic part (i.e., production function f) is defined analogous to DEA, while the stochastic part (i.e., composite error term \( \varepsilon_{i} \)) is defined similar to SFA. As a result, model (1) encompasses the classic SFA and DEA models as its constrained special cases. Specifically, if f is restricted to some specific functional form (instead of the class F _{2}), model (1) boils down to the classic SFA model by Aigner et al. (1977). On the other hand, if we impose the parameter restriction \( \sigma_{v}^{2} = 0 \), we obtain the singleoutput DEA model with an additive outputinefficiency, first considered by Afriat (1972) [see also Banker (1993)]. In this sense, the classic SFA and DEA models can both be seen as constrained special cases of the encompassing model (1).
Although the encompassing frontier model (1) described above is considerably more general than the classic DEA and SFA models, it does impose a number of assumptions that may be viewed as restrictive. From the perspective of DEA, assuming the singleoutput case is clearly restrictive. The multioutput technology could be modeled by using distance functions, but this is left as a topic for future research.^{3} Further, the assumption of global concavity has been subject to debate, but we here restrict to the standard DEA specification.^{4} From the econometric perspective, the additive structure of the composite error term and its components may be restrictive; a more standard multiplicative model will be examined in Sect. 4.3. Finally, assuming homoskedastic inefficiency and noise terms (i.e., \( \sigma_{v}^{2} \) and \( \sigma_{u}^{2} \) are constant across firms) can be very restrictive, as noted in the introduction. Extending the theoretical model to the heteroskedastic setting would be straightforward, but the methods developed in this paper assume the homoskedastic model. We will briefly discuss the possible consequences of the violations of this assumption in Sect. 4.5.
3 Stochastic nonsmooth envelopment of data (StoNED) approach
3.1 Twostage estimation strategy

Stage 1: Estimate the shape of function f by Convex Nonparametric Least Squares (CNLS) regression.

Stage 2: Imposing additional distributional assumptions, estimate the variance parameters \( \sigma_{u}^{2} ,\sigma_{v}^{2} \) based on the skewness of the CNLS residuals obtained in Stage 1, using the method of moments or pseudolikelihood techniques. Given estimates of parameters \( \sigma_{u}^{2} ,\sigma_{v}^{2} \), compute the conditional expected values of inefficiency.
We elaborate the implementation of Steps 1 and 2 in Sects. 3.2 and 3.3, respectively.
Our twostep estimation strategy parallels the modified OLS (MOLS) approach to estimating parametric SFA models, originating from Aigner et al. (1977).^{5} Although SFA models are commonly estimated by maximum likelihood (ML) techniques, MOLS provides a consistent method for estimating the SFA model. While the ML estimators are known to be asymptotically efficient, provided that the distributional assumptions are correct, the MOLS estimators tend to be more robust to violations of the distributional assumptions about inefficiency terms u _{ i } and noise v _{ i }. Note that in MOLS the distributional assumptions about the composite error term do not influence the slope coefficients of f estimated in Step 1. We consider this relative robustness of MOLS with respect to ML as an attractive property, keeping in mind the present semiparametric setting. As mentioned in the introduction, Fan et al. (1996) have earlier explored a parallel twostage approach in the context of kernel estimation.
3.2 Stage 1: CNLS estimation
For estimating the shape of the production function, the coefficients (α_{ i },β _{ i }) have a compelling economic interpretation: vector β _{ i } can be interpreted as the subgradient vector \( \nabla g({\mathbf x}_{i} ) \), and thus it represents the vector of marginal products of inputs at point x _{ i }. Thus, coefficients β _{ i } could be used for nonparametric estimation of substitution and scale elasticities. Note that equation \( y = \alpha_{i} + {\varvec{\upbeta^{\prime}}}_{i} {\mathbf{x}} \) can be interpreted as the tangent hyperplane to the estimated function g at point x _{ i }. Therefore, the coefficients of the QP problem (6) provide a local firstorder Taylor series approximation to any arbitrary function g in the neighborhood of the observed points x _{ i }. In contrast to the flexible functional forms that can be interpreted as secondorder Taylor approximations around a single, unknown expansion point, the CNLS estimator uses all n observations as expansion points for the local linear approximation.
The CNLS problems (4) and (6) are equivalent in the following sense (see Kuosmanen 2008, “Appendix”, for a formal proof).
Theorem 3.1
Denote the optimal solution to the infinite dimensional CNLS problem (4) by \( s_{CNLS}^{2} \) and the optimal solution to the finite quadratic programming problem (6) by \( s_{QP}^{2} \) . Then for any arbitrary data, \( s_{CNLS}^{2} = s_{QP}^{2} \) .
This result shows that the CNLS estimator can be computed in the general multivariate setting. Indeed, it is easy to verify that the univariate CNLS formulation (5) by Hanson and Pledger (1976) is obtained as a special case of (6) when m = 1. We would conjecture that the known statistical properties of the univariate CNLS estimator (consistency, rate of convergence) carry over to the multivariate setting, but this remains to be formally shown. Regarding the rates of convergence, Stone (1980, 1982) has established n ^{−2d/(2d+m)} as the optimal rate of convergence for any arbitrary nonparametric regression estimator, where d equals the degree of differentiability of the true but unknown g. We note that the rate of convergence established by Groeneboom et al. (2001a, b) for the univariate CNLS estimator falls below this optimal rate. Although the rate of convergence for the multivariate CNLS estimator remains unknown, Stone’s general result can be viewed as the theoretical upper bound that the CNLS estimator cannot exceed even under ideal conditions. This is a useful reminder that the CNLS estimator is subject to the “curse of dimensionality”, similar to the conventional DEA estimators (see, e.g., Simar and Wilson 2000, for discussion). In practice, this means that the sample size n needs to be large and the number of inputs m must be sufficiently small for any meaningful estimation. It might be possible to improve the rate of convergence by imposing further restrictions on the third and higher order partial derivatives of g, but it is unclear how the higher derivatives could be utilized in the CNLS estimator. Further, it would be interesting to link the nonsmooth CNLS estimator to the kernel regression and other nonparametric smoothing techniques (see e.g., Mammen and ThomasAgnen 1999; Yatchew 2003). On the other hand, the nonsmooth CNLS estimator is closely related to the classic DEA estimator, which is an appealing property for the purposes of the present paper.
Consider for a moment the deterministic case where \( \sigma_{v}^{2} = 0 \). In this setting, all deviations from the frontier can be attributed to the inefficiency term u. Hence, we could impose an additional signconstraint \( \upsilon_{i} \le 0 \, \forall i = 1, \ldots ,n \) for the composite error terms of the QP problem (6), analogous to the classic parametric programming (PP) approach of Aigner and Chu (1968). Interestingly, Kuosmanen and Johnson (2010) have formally shown that the resulting signconstrained CNLS problem is in fact equivalent to the classic variable returns to scale DEA estimator: the output oriented DEA efficiency estimates are directly obtained from the CNLS residuals. In light of these results, the classic DEA can be interpreted as a signconstrained variant of the CNLS problem (6). Further, the results of Kuosmanen and Johnson (2010) reveal DEA as a nonparametric counterpart to Aigner and Chu’s PP method.
Returning to the stochastic setting, we next elaborate the connection between CNLS and DEA further. Note that Kuosmanen and Johnson (2010) consider the CNLS estimator \( \hat{y}_{i} = \hat{\alpha }_{i} + {\hat{\varvec{\upbeta^{\prime}}}}_{i} {\mathbf{x}}_{i} \) only at the observed input levels x _{ i }, i = 1,…,n. It can be shown that the QP problem (6) always has a unique optimum, and that the fitted values \( \hat{y}_{i} \) are unique. However, estimating the function g at unobserved input levels x proves more complicated.
It is well known in the DEA literature that the input–output weights (shadow prices) of the multiplierside DEA problem are generally not unique. The same is true for the CNLS estimator: the coefficients \( \hat{\alpha }_{i} ,{\hat{\varvec{\upbeta }}}_{i} \) obtained as the optimal solution to (6) need not be unique, even though the fitted values \( \hat{y}_{i} \) are unique for the observed x _{ i }, i = 1,…,n. In general, there are many ways to fit a monotonic and concave function through the finite number of points \( ({\mathbf{x}}_{i} ,\hat{y}_{i} ) \). As Kuosmanen (2008) notes, even the original CNLS problem (4) does not generally have a unique solution: there generally exists a family of alternate optima \( F_{2}^{ * } \).
Interestingly, the lower bound function \( \hat{g}_{\min } \) can be interpreted as the variable returns to scale DEA frontier applied to the predictions \( ({\mathbf{x}}_{i} ,\hat{y}_{i} ) \) of the CNLS estimator. Applying the duality theory of linear programming, we can prove the following^{6}:
Theorem 3.2
In line with the classic DEA, we can resolve the nonuniqueness of the CNLS estimator by resorting to the minimum function \( \hat{g}_{\min } \), which is always unique. Based on Theorems 3.1 and 3.2, we can give function \( \hat{g}_{\min } \) the following minimum extrapolation interpretation (compare with Afriat 1972, and Banker et al. 1984): function \( \hat{g}_{\min } \) is the minimum function that satisfies the axioms of free disposability and concavity and minimizes the sample variance of deviations \( (y_{i}  \hat{g}_{\min } ({\mathbf{x}}_{i} )) \). Recall that the classic DEA estimator has a similar minimum extrapolation property, with the exception that the DEA frontier envelopes all observed data, whereas \( \hat{g}_{\min } \) does not. In the deterministic setting, enveloping all observed data can be desirable. In the stochastic setting, replacing the envelopment axiom by some other axiom seems preferable. Minimization of the sample variance of deviations \( (y_{i}  \hat{g}_{\min } ({\mathbf{x}}_{i} )) \) seems a natural candidate for such an axiom.
Theorem 3.2 is also important for establishing a formal connection between CNLS and DEA estimators for the unobserved input levels x, complementing the results of Kuosmanen and Johnson (2010). Not only do the CNLS and DEA share the same axioms, the DEA estimator has a compelling regression interpretation as a signconstrained variant of CNLS. On the other hand, to interpolate the fitted values of the CNLS regression, the classic DEA estimator provides the tightest lower bound for the family of functions that solve the problem (4).
Despite these compelling links and interpretations, we must recall that the piecewise linear lower bound \( \hat{g}_{\min } ({\mathbf{x}}) \) does not estimate the frontier f(x) but the averagepractice production function g(x). In the present setting, the shape of the averagepractice function g(x) is exactly the same as that of the frontier f(x), because the expected inefficiency μ was assumed to be constant across all firms and thus g(x) = f(x) − μ. In the next section we show how the expected inefficiency μ and the unknown variance parameters \( \sigma_{u}^{2} ,\sigma_{v}^{2} \) can be estimated based on the skewness of the CNLS residuals.
3.3 Efficiency estimation
Given the CNLS residuals \( {\hat{\upsilon }} \equiv (\hat{\upsilon }_{1} , \ldots ,\hat{\upsilon }_{n} ) \), the next challenge is to disentangle inefficiency from noise. At this point, more specific distributional assumptions must be imposed.^{7} We will follow the classic SFA study by Aigner et al. (1977) and assume the halfnormal inefficiency term and a normally distributed noise term: \( u_{i} \mathop \sim \limits_{i.i.d} \left {N(0,\sigma_{u}^{2} )} \right \) and \( v_{i} \mathop \sim \limits_{i.i.d} N(0,\sigma_{v}^{2} ) \). Other distributions such as gamma or exponential are also used for the inefficiency term u _{ i } (e.g., Kumbhakar and Lovell 2000), but in this paper we restrict to the halfnormal specification.
Since the noise term has a symmetric distribution, the negative skewness of the CNLS residuals signals that an asymmetric inefficiency term is present. Of course, the residuals might be skewed in a small sample just by coincidence; it would be advisable to test whether the negative skewness is statistically significant prior to estimation (see, e.g., Kuosmanen and Fosgerau 2009). If skewness is significant, there are at least two possible approaches for estimating the variance parameters \( \sigma_{u}^{2} ,\sigma_{v}^{2} \): the method of moments and pseudolikelihood estimation. We next briefly describe both these approaches and adapt them for our purposes.
3.3.1 Method of moments
3.3.2 Pseudolikelihood estimation
An alternative way to estimate the standard deviations σ_{ u }, σ_{ v } is to apply the pseudolikelihood (PSL) method suggested by Fan et al. (1996). Compared to the MM, PSL is potentially more efficient, but is computationally somewhat more demanding.
3.3.3 Estimation of the inefficiency term
Given a consistent estimator \( \hat{\sigma }_{u}^{{}} \) (obtained by either MM or PSL), the frontier production function f can be consistently estimated as \( \hat{f}({\mathbf{x}}) = \hat{g}_{\min } ({\mathbf{x}}) + \hat{\sigma }_{u} \sqrt {2/\pi } \). In practice, this means that frontier is obtained by shifting the CNLS estimate of the averagepractice production function upwards by the expected value of the inefficiency term, analogous to the MOLS approach.
4 Possible extensions
This section briefly outlines some potential extensions of the proposed method and suggests some interesting avenues for future research. While some extensions are readily implementable, we must emphasize that every topic discussed in this section deserves a more systematic and rigorous examination of its own.
4.1 Panel data model
Panel data enables us to relax the distributional assumptions, and estimate the model in a fully nonparametric fashion. In the following we describe the fixed effects approach to estimating timeinvariant inefficiency. Alternative panel data approaches such as random effects modeling, timevarying inefficiency, and modeling technical progress are left as interesting topics for future research.
4.2 Returns to scale

constant returns to scale (CRS): \( \alpha_{i} = 0 \, \forall i = 1, \ldots ,n \)

nonincreasing returns to scale (NIRS): \( \alpha_{i} \ge 0 \, \forall i = 1, \ldots ,n \)

nondecreasing returns to scale (NDRS): \( \alpha_{i} \le 0 \, \forall i = 1, \ldots ,n \)
Rationale of these constraints is directly analogous to the standard multiplierside DEA formulations where parallel constraints are employed for enforcing RTS assumptions.
While the CNLS regression is easily adapted to alternative RTS assumptions, the implications to the efficiency estimation are somewhat trickier. Specifically, if one estimates the averagepractice technology g subject to CRS, and subsequently shifts the frontier upward by the estimated expected inefficiency, the resulting bestpractice frontier does not generally satisfy CRS. This is due to the mismatch of the additive structure of the inefficiency and noise terms assumed in (1) and the multiplicative nature of the scale properties. If one imposes CRS, NIRS, or NDRS assumptions, it is logically consistent to employ the multiplicative specification of inefficiency and noise, to be discussed next.
4.3 Multiplicative model
Most SFA studies employ a multiplicative error model due to the logtransformations applied to the data (e.g., when the popular CobbDouglas or translog functional forms are used). As noted above, the CRS assumption requires a multiplicative error structure. Moreover, multiplicative error specification might help to alleviate heteroskedasticity from different scale sizes (cf. Caudill and Ford 1993).
Given the composite residuals from model (30) (i.e., \( \upsilon_{i} = \ln y_{i}  \ln \hat{y}_{i} \)), the standard MM or PSL procedures can be applied, as described in Sect. 4. The logtransformation only concerns Step 1, and makes no difference in the estimation of Step 2. However, the interpretation of inefficiency term u _{ i } changes: exp(u _{ i }) provides the Farrell output efficiency measure.
4.4 Cost functions
According to the microeconomic theory, the cost function C is nonnegative and nondecreasing function of both input prices w and the output y. Further, the cost function is known to be continuous, concave and homogenous of degree one in input prices w (Shephard 1953). The known regularity properties of cost functions provide useful shape constraints that can be utilized in the semi and nonparametric estimation.
The interpretation of the inefficiency term also changes from the production function setting: u _{ i } represents (overall) cost inefficiency that captures both technical and allocative aspects of inefficiency. If data of input quantities or cost shares is available, one could disentangle technical inefficiency from allocative inefficiency. Further, one could incorporate the share equations to the CNLS model (35) (see Kumbhakar 1997, for details). Incorporating the share equations, multiple outputs, and variable returns to scale to the CNLS formulation present interesting avenues for future research.
4.5 Heteroskedasticity
We have thus far assumed that standard deviations σ_{ u }, σ_{ v } are the same across all firms. This assumption is referred to as homoskedasticity, and it forms one of the maintained assumptions of the classic SFA model by Aigner et al. (1977). As Caudill and Ford (1993) and Florens and Simar (2005) demonstrate, violation of the homoskedasticity assumption leads to potentially serious problems in the context of parametric frontier estimation. Clearly, similar problems carry over to the present semiparametric setting as well. Thus, a brief discussion about robustness of the proposed method to heteroskedasticity is necessary, although more systematic and rigorous treatment of the topic is left for a separate study.
Firstly, we must distinguish between (1) heteroskedasticity of the noise term (i.e., parameter σ_{ v } varies across firms) and (2) heteroskedasticity of the inefficiency term (i.e., σ_{ u } varies across firms). Let us first consider heteroskedasticity of type (1). Of course, both types of heteroskedasticity may be present at the same time. However, their impacts on the StoNED estimators differ.
Note first that the expected inefficiency \( \mu = \sigma_{u} \sqrt {2/\pi } \) does not depend on σ_{ v }. Therefore, the shape of the averagepractice production function g remains identical to that of the frontier f even if the noise terms are heteroskedastic. Hence, the proposed approach is not particularly sensitive to heteroskedasticity of type (1). Least squares estimators (incl. CNLS) are known to be unbiased and consistent under symmetric heteroskedasticity, even though more efficient estimators are possible if heteroskedasticity is modeled correctly. Given unbiased CNLS residuals, heteroskedastic σ_{ v } will likely increase variance of the parameter estimators \( \hat{\sigma }_{u} ,\hat{\sigma }_{v} \). However, since σ_{ u } is estimated based on the skewness of the residual distribution, and heteroskedasticity in the symmetric noise component does not affect skewness, the estimator \( \hat{\sigma }_{u} \) remains consistent. Thus, frontier f and expected inefficiency μ can be consistently estimated even under heteroskedasticity of type (1). The only problem is that the conditional expected value of inefficiency \( \hat{E}(u_{i} \left {\hat{\varepsilon }_{i} } \right.) \) is a function of heteroskedastic \( \hat{\sigma }_{v} \). Thus, firmspecific efficiency scores and rankings can be affected by heteroskedasticity of type (1).
Heteroskedasticity of type (2) is a much more serious problem because σ_{ u } does directly influence the expected inefficiency E(u _{ i }). When σ_{ u } is heteroskedastic, the expected inefficiency E(u _{ i }) differs across firms, and thus the shape of the averagepractice production function g is no longer identical to that of the frontier f. We stress that this problem arises only in case (2), not in case (1). Since the proposed StoNED method relies on consistent estimation of the averagepractice production g in the step (1), the estimates can be sensitive to the violation of the homoskedasticity assumption for σ_{ u } (see the next section for some evidence from Monte Carlo simulations). Therefore, it is critically important to develop statistical tests of the homoskedasticity assumption and more general estimation methods that can deal with heteroskedastic inefficiency. Fortunately, such tests and methods have been developed for the least squares estimation in the context of the linear regression model (consider, e.g., the generalized least squares (GLS) method). The main challenge is to adapt and extend existing techniques from the linear regression analysis to the CNLS framework. This forms an important topic for future research.
4.6 Statistical inferences
Even though we impose parametric distributional assumption for the inefficiency and noise terms, the conventional methods of statistical inference do not directly apply to the present setting. For example, one might apply the likelihood ratio test for testing significance of two alternative hierarchically nested model variants, but the degrees of freedom are difficult to specify (see Meyer 2003, 2006, for discussion). One could also construct confidence intervals based on the known conditional distribution of the inefficiency term (see Horrace and Schmidt 1996, for details). However, such confidence intervals do not take into account the sampling distribution of the inefficiency estimators, and consequently, have poor coverage properties (Simar and Wilson 2010). In light of these complications, the parametric bootstrap method similar to Simar and Wilson (2010) would appear to be the best suited approach to statistical inference in the present context. Adapting the procedure to the present setting seems straightforward, but it is first important to ensure that the method is consistent and provides valid inferences even in finite samples. We leave this as an interesting research question for future research.
Related to the previous point, we should note that the leastsquares residuals are often skewed in the wrong direction (\( \hat{M}_{3} > 0 \)). In the SFA literature, the usual approach is to set \( \hat{\sigma }_{u} = 0 \), which means that all firms are diagnosed as efficient. It may also occur that the skewness is so great that \( \hat{\sigma }_{u} > \hat{\sigma } \), and thus \( \hat{\sigma }_{v} \) becomes negative. In that case, the typical approach is to set \( \hat{\sigma }_{v} = 0 \) and attribute all observed variation to inefficiency (as in DEA). The “wrong skewness” is conventionally seen as a useful builtin diagnostic, which signals model misspecification or inappropriate data (Greene 2008). Indeed, inspecting the distribution of residuals might reveal some possible sources of model misspecification. However, evidence from several Monte Carlo studies shows that wrongly skewed residuals can arise even in correctly specified frontier models (e.g., Fan et al. 1996; Carree 2002; Simar and Wilson 2010). This is not only a problem for the method of moments, it equally affects the pseudolikelihood method. Interestingly, if Simar and Wilson’s (2010) bootstrap procedure is applicable in the present setting, it could alleviate the wrong skewness problem as well.
5 Monte Carlo simulations
In this section we examine performance of the StoNED method in the controlled environment of Monte Carlo simulations. Our objective is to compare performance of the StoNED method with the standard DEA and SFA under alternative conditions where the distributional assumptions of the StoNED model are violated.^{8} The data generating processes used in the simulations has been adopted from Simar and Zelenyuk (2008). Systematic performance comparisons with other semi and nonparametric methods is left as a topic for future research.^{9}
For DEA, the standard outputoriented variable returns to scale (VRS) specification is used. Given the DEA efficiency score \( \theta = \hat{f}^{DEA} ({\mathbf{x}}_{i} )/y_{i} \), the DEA inefficiency estimator is obtained as \( \hat{u}_{i}^{DEA} = (\theta  1)y_{i} \). For SFA, we use the CobbDouglas production function with the halfnormal inefficiency term. The MOLS estimator is used to ensure comparability with the StoNED method. For the StoNED method, we assume the multiplicative specification (27) and the halfnormal inefficiency distribution. Since the MC simulations are computationally intensive, we restrict to the simpler method of moment (MM) estimator in this section. In the MM estimation of SFA and StoNED models, we have dealt with the wrong skewness problem as follows. If \( \hat{M}_{3} \) is nonnegative, we set \( \hat{M}_{3} \) = −0.0001. On the other hand, if \( \hat{\sigma }_{v} \) is negative, we set \( \hat{\sigma }_{v} \) = 0.0001. These settings ensure that the algorithm runs smoothly even in those scenarios where the DGP is inconsistent with the model assumptions (e.g., there are outliers or no inefficiency). Of course, the wrong skewness can be a signal of model misspecification (e.g., in scenarios involving outliers), but in these MC simulations we disregard this potentially useful information and force the postulated skewness to the estimated distributions of the composite error term.
5.1 Univariate CobbDouglas frontier
Performance in estimating frontier f; univariate CD frontier
Scenario  Description  MSE_{ DEA }  MSE_{ SFA }  MSE_{ StoNED } 

a)  n = 100, ρ _{ nts } = 0  0.0002  0.0060  0.0052 
b)  n = 103, 3 outliers  0.0999  0.0068  0.0064 
c)  n = 100, ρ _{ nts } = 1  0.0398  0.0070  0.0067 
d)  n = 200, ρ _{ nts } = 1  0.0640  0.0068  0.0067 
e)  n = 500, ρ _{ nts } = 1  0.0966  0.0058  0.0057 
f)  n = 500, ρ _{ nts } = 2  0.7053  0.0077  0.0075 
Performance in estimating inefficiency term u; univariate CD frontier
Scenario  Description  MSE_{ DEA }  MSE_{ SFA }  MSE_{ StoNED } 

a)  n = 100, ρ _{ nts } = 0  0.0161  0.0109  0.0097 
b)  n = 103, 3 outliers  0.0854  0.0322  0.0317 
c)  n = 100, ρ _{ nts } = 1  0.0424  0.0294  0.0282 
d)  n = 200, ρ _{ nts } = 1  0.0600  0.0301  0.0288 
e)  n = 500, ρ _{ nts } = 1  0.0829  0.0265  0.0258 
f)  n = 500, ρ _{ nts } = 2  0.6236  0.0377  0.0362 
5.2 Trivariate CobbDouglas frontier
Performance in estimating frontier f; trivariate CD frontier
Scenario  Description  MSE_{ DEA }  MSE_{ SFA }  MSE_{ StoNED } 

a)  n = 100, ρ _{ nts } = 0  0.0014  0.0028  0.0020 
b)  n = 100, ρ _{ nts } = 0.5  0.0013  0.0028  0.0021 
c)  n = 100, ρ _{ nts } = 1  0.0063  0.0028  0.0029 
d)  n = 200, ρ _{ nts } = 1  0.0084  0.0037  0.0036 
e)  n = 300, ρ _{ nts } = 1  0.0137  0.0031  0.0028 
f)  n = 300, ρ _{ nts } = 2  0.1583  0.0073  0.0080 
Performance in estimating inefficiency term u; trivariate CD frontier
Scenario  Description  MSE_{ DEA }  MSE_{ SFA }  MSE_{ StoNED } 

a)  n = 100, ρ _{ nts } = 0  0.0334  0.0011  0.0010 
b)  n = 100, ρ _{ nts } = 0.5  0.0295  0.0163  0.0135 
c)  n = 100, ρ _{ nts } = 1  0.0283  0.0267  0.0250 
d)  n = 200, ρ _{ nts } = 1  0.0268  0.0309  0.0297 
e)  n = 300, ρ _{ nts } = 1  0.0284  0.0265  0.0262 
f)  n = 300, ρ _{ nts } = 2  0.1288  0.0512  0.0511 
5.3 Trivariate CobbDouglas frontier with heteroskedastic inefficiency
We next adapt the DGP of the previous section by introducing heteroskedasticity in the inefficiency term u. Following Simar and Zelenyuk (2008) Section 3.1.4, we draw inefficiency terms from the halfnormal distribution as \( u_{i} \left {\mathbf x}_{i} \right.\sim \left {N(0,(\sigma_{u} (x_{1,i} + x_{2,i} )/2)^{2} )} \right \), where σ_{ u } = 0.3. Note that variance of inefficiency distribution depends on inputs 1 and 2, which results as heteroskedasticity. The noise term is homoskedastic normal, \( v_{i} \sim N(0,\sigma_{v}^{2} ) \), where \( \sigma_{v} = \rho_{nts} \cdot \sigma_{u} \cdot \sqrt {(\pi  2)/\pi } \). Parameter \( \rho_{nts} \) can be interpreted as the average noise to signal ratio, and it is varied across scenarios.
Performance in estimating frontier f; trivariate CD frontier with heteroskedastic inefficiency
Scenario  Description  MSE_{ DEA }  MSE_{ SFA }  MSE_{ StoNED } 

a)  n = 100, ρ _{ nts } = 0  0.0036  0.0016  0.0042 
b)  n = 100, ρ _{ nts } = 0.5  0.0024  0.0015  0.0038 
c)  n = 100, ρ _{ nts } = 1  0.0051  0.0030  0.0051 
d)  n = 200, ρ _{ nts } = 1  0.0071  0.0017  0.0038 
e)  n = 300, ρ _{ nts } = 1  0.0067  0.0011  0.0023 
f)  n = 300, ρ _{ nts } = 2  0.0895  0.0036  0.0041 
Performance in estimating inefficiency term u; trivariate CD frontier with heteroskedastic inefficiency
Scenario  Description  MSE_{ DEA }  MSE_{ SFA }  MSE_{ StoNED } 

a)  n = 100, ρ _{ nts } = 0  0.0574  0.0108  0.0192 
b)  n = 100, ρ _{ nts } = 0.5  0.0498  0.0191  0.0210 
c)  n = 100, ρ _{ nts } = 1  0.0439  0.0401  0.0363 
d)  n = 200, ρ _{ nts } = 1  0.0371  0.0370  0.0377 
e)  n = 300, ρ _{ nts } = 1  0.0358  0.0346  0.0335 
f)  n = 300, ρ _{ nts } = 2  0.0651  0.0629  0.0613 
In conclusion, the proposed StoNED estimator proved a competitive alternative to the conventional DEA and SFA estimators in the simulations adopted from Simar and Zelenuyk (2008). We should note that the distributional assumptions for the inefficiency term were incorrect in all scenarios that were considered. Despite this specification error, the StoNED estimator performed better than the distributionfree DEA estimator in many of the scenarios considered. This suggests it may often be preferable to model noise even at the risk of making a specification error in the distributional assumptions than assume away noise completely. The StoNED estimator also achieved a lower MSE than the corresponding SFA estimator in a majority of scenarios, even though the functional form of the frontier was correctly specified for the SFA estimator (the inefficiency term was wrongly specified, exactly the same way as for the StoNED estimator). It appears that the better empirical fit in the estimation of the frontier can also partly offset the possible specification errors in the estimation of the inefficiency distribution. Of course, evidence from any Monte Carlo study is limited, and the present comparison is restricted to the most basic variants of DEA and SFA. We recognize the need to compare the performance of the proposed method with other recently developed semiparametric and nonparametric approaches that were briefly reviewed in the Introduction, but we also realize that designing and implementing a comparison of many computationally intensive methods in a fair and objective manner is a daunting task that deserves a thorough investigation of its own.
6 Conclusions and discussion
We have developed a new encompassing framework for productive efficiency analysis, referred to as stochastic nonsmooth envelopment of data (StoNED). One of our main objectives was to show how the StoNED method can be used to estimate a semiparametric frontier model that combines a nonparametric DEAlike frontier with a stochastic SFAlike inefficiency and noise terms. We also demonstrated that both classic DEA and SFA can be viewed as special cases of this encompassing model, obtainable by imposing some more restrictive assumptions to the model.
In our approach, we employed a twostage estimation strategy that is commonly used in many areas of econometrics. In the first stage, the shape of the frontier is consistently estimated by using convex nonparametric least squares (CNLS), which does not assume any smoothing parameters, building upon the same shape constraints as DEA. In the second stage, we apply method of moments or pseudolikelihood techniques, adopted from the SFA literature, to disentangle the inefficiency and noise components from the CNLS residuals. Although this stepwise estimation strategy may not be as efficient as the constrained maximum likelihood, it has some important advantages, including the relative robustness of the CNLS estimator to distributional assumptions of inefficiency and noise terms, and substantially lower computational barriers (i.e., the constrained ML estimators are often computationally infeasible in the present setting).
This study has established further connections between CNLS regression and DEA, complementing the prior work of Kuosmanen (2008) and Kuosmanen and Johnson (2010). We find that DEA can be formulated as a constrained special case of the CNLS regression, and that CNLS has a minimum extrapolation interpretation analogous to that of the conventional DEA. While we mainly focused on the estimation of production functions under variable returns to scale, we also demonstrated how the method can be extended to the estimation of cost functions and to allow one to postulate for alternative specifications of returns to scale. Moreover, the performance of the approach was examined in the controlled environment of Monte Carlo simulations. The evidence from the simulations suggests the proposed method is a competitive alternative to standard DEA and SFA methods even when the distribution of the inefficiency term is wrongly specified.
The proposed StoNED approach shares many common features with SFA and DEA, being an amalgam of the two. Thus, many of the existing tools and techniques for SFA and DEA can be incorporated into the proposed framework. The hybrid nature of StoNED also implies that there are many important differences to both SFA and DEA, which should be kept in mind. For example, the interpretation of the StoNED input coefficients differs considerably from those of SFA coefficients. Moreover, in contrast to DEA, all observations influence the shape of the frontier. While the StoNED approach combines the appealing features of DEA and SFA, it also shares many of their limitations. Similar to DEA, the nonparametric orientation of StoNED can make it vulnerable to the curse of dimensionality, which means that the sample size needs to be very large when the number of input variables is high. On the other hand, the composite error term assumptions of SFA are rather restrictive, and might often be inappropriate. In this respect, we again emphasize that the focus of this paper has been on the development of an operational estimation strategy for an encompassing model that includes the classic DEA and SFA models as its special cases. Improving upon DEA and/or SFA aspects of the model is another challenge, which falls beyond the scope of the present study.
 1.
Adapting the known econometric and statistical methods for dealing with heteroskedasticity, endogeneity, sample selection, and other potential sources of bias, to the context of CNLS and StoNED estimators.
 2.
Extending the proposed approach to a multiple output setting.
 3.
Extending the proposed approach to account for relaxed concavity assumptions (e.g., quasiconcavity).
 4.
Developing more efficient computational algorithms or heuristics for solving the CNLS problem.
 5.
Examining the statistical properties of the CNLS estimator, especially in the multivariate case.
 6.
Investigating the axiomatic foundation of the CNLS and StoNED estimators.
 7.
Implementing alternative distributional assumptions and estimating the distribution of the inefficiency term by semi or nonparametric methods in the crosssectional setting.
 8.
Distinguishing timeinvariant inefficiency from heterogeneity across firms, and identifying intertemporal frontier shifts and catching up in panel data models.
 9.
Extending the proposed approach to the estimation of cost, revenue, and profit functions as well as to distance functions.
 10.
Developing a consistent bootstrap algorithm and/or other statistical inference methods.
 11.
Conducting further Monte Carlo simulations to examine the performance of the proposed estimators under a wider range of conditions, and comparing the performance with other semi and nonparametric frontier estimators.
 12.
Applying the proposed method to empirical data, and adapting the method to better serve the needs of specific empirical applications.
These twelve points could be seen as limitations of the proposed approach, but also as an outline of a research program to address these challenges. We hope this study could inspire other researchers to join us in further theoretical and empirical work along the lines sketched above, and to expand our list of research questions further. Finally, we hope that this study could contribute to further crossfertilization and unification of the parametric and nonparametric streams of productive efficiency analysis.
Footnotes
 1.
In earlier working papers Kuosmanen (2006) and Kuosmanen and Kortelainen (2007) the term “stochastic nonparametric envelopment of data” was used. However, as the Associate Editor and two anonymous reviewers of this journal correctly noted, the proposed method is actually semiparametric due the parametric distributional assumptions imposed on the inefficiency and noise terms.
 2.
In the SFA literature, the problem of heteroskedasticity was recognized in the early 1990s (Caudill and Ford 1993; see also Florens and Simar 2005). The econometric literature provides many useful tools for dealing with heteroskedasticity, but suitability of these tools to the present setting deserves a thorough examination that falls beyond the scope of the present study.
 3.
Simar (2007) presents a formal description of a data generation process for a stochastic multioutput frontier model, which could be a useful starting point for multioutput extensions (see also Simar and Zelenyuk 2008). The working paper Kuosmanen (2006) suggests how the CNLS problem could be formulated in terms of the directional distance function.
 4.
 5.
MOLS should not be confused with the deterministic COLS (= corrected OLS) approach (Greene, 1980), where the frontier is shifted upward according to the largest OLS residual so as to envelop all observations.
 6.
The proof involves straightforward mechanical calculations and it is hence omitted. Details of the proof are available from the authors by request.
 7.
 8.
 9.
Since we replicate some of the simulations conducted by Simar and Zelenyuk (2008), an interested reader may compare our results with those reported by Simar and Zelenyuk for their local maximum likelihood estimator. However, it is worth noting that the synthetic data sets used in the different simulations are not exactly identical, but each random draw from the DGP yields unique data, which may have effect on the performance of estimators. The results reported here are averages over 50 replications of each scenario, whereas Simar and Zelenyuk (2008) report results of a single simulation run for each scenario.
Notes
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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