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The econometric analysis of cost functions is based on the analysis of the conditional distribution of the cost Y given the level of the outputs X ∈ R+p and given a set of environmental variables Z ∈ Rd. The model basically describes the conditional distribution of Y given X ≥ x and Z = z. In many applications, the dimension of Z is naturally large and a fully nonparametric specification of the model is limited by the curse of the dimensionality. Most of the approaches so far are based on two-stage estimations when the frontier level does not depend on the value of Z. But even in the case of separability of the frontier, the estimation procedure suffers from several problems, mainly due to the inherent bias of the estimated efficiency scores and the poor rates of convergence of the frontier estimates. In this paper we suggest an alternative semi-parametric model which avoids the drawbacks of the two-stage methods. It is based on a class of model called the Proportional Incremental Cost Functions (PICF), adapted to our setup from the Cox proportional hazard models extensively used in survival analysis for durations models. We define the PICF model, then we examine its properties and propose a semi-parametric estimation. By this way of modeling, we avoid the first stage nonparametric estimation of the frontier and avoid the curse of dimensionality keeping the parametric √n rates of convergence for the parameters of interest. We are also able to derive √n-consistent estimator of the conditional order-m

We use mollification to regularize the problem of deconvolution of random variables. This regularization method offers a unifying and generalizing framework in order to compare the benefits of various filter-type techniques like deconvolution kernels, Tikhonov, or spectral cutoff methods. In particular, the mollifier approach allows to relax some restrictive assumptions required for the deconvolution kernels, and has better stabilizing properties compared with spectral cutoff or Tikhonov. We show that this approach achieves optimal rates of convergence for both finitely and infinitely smoothing convolution operators under Besov and Sobolev smoothness assumptions on the unknown probability density. The qualification can be arbitrarily high depending on the choice of the mollifier function. We propose an adaptive choice of the regularization parameter using the Lepskiĭ method, and we provide simulations to compare the finite sample properties of our estimator with respect to the well-known regularization methods.

Nonparametric methods are widely used for assessing the performance of firms and other organizations in the private and public sectors. Typically, FDH or DEA estimators that estimate the attainable sets and its efficient boundary by enveloping the cloud of observed units in the appropriate input-output space are used. The statistical properties of these estimators have been established and inference is available using appropriate nonparametric techniques. In particular, hypotheses on model structure and the production process can be tested using using recent theoretical results including Central Limit Theorems on limiting distribution of means of efficiency scores. This chapter shows how these results can be used for testing the equality of means of efficiency, convexity of production sets and separability with respect to environmental factors are considered, and in addition for analyzing the dynamics of the production process over time. An empirical illustration is provided by using the various results and tests to examine the performance of municipal governments in the U.S. in providing local public goods.

The Malmquist productivity index (MPI) has gained popularity among studies on the dynamic change of productivity of decision-making units (DMUs). In practice, this index is frequently reported at aggregate levels (e.g., public and private firms) in the form of simple, equally weighted arithmetic or geometric means of individual MPIs. A number of studies emphasize that it is necessary to account for the relative importance of individual DMUs in the aggregations of indices in general and of the MPI in particular. Whereas more suitable aggregations of MPIs have been introduced in the literature, their statistical properties have not been revealed yet, preventing applied researchers from making essential statistical inferences, such as confidence intervals and hypothesis testing. In this paper, we fill this gap by developing a full asymptotic theory for an appealing aggregation of MPIs. On the basis of this, meaningful statistical inferences are proposed, their finite-sample performances are verified via extensive Monte Carlo experiments, and the importance of the proposed theoretical developments is illustrated with an empirical application to real data.

A simple yet easy to implement method is proposed to further improve the finite sample approximation of the recently developed central limit theorems for aggregates of envelopment estimators. Focusing on the simple mean efficiency, we propose using the bias-corrected individual efficiency estimate to improve the variance estimator. The extensive Monte-Carlo experiments confirm that, for relatively small sample sizes (≤ 100), with both low dimensions and especially for high dimensions, our new method combined with the data sharpening method generally provides better ‘coverage’ (of the true values by the estimated confidence intervals) than the previously developed approaches.

The Malmquist index gives a measure of productivity in dynamic settings and has been widely applied in empirical work. The index is typically estimated using envelopment estimators, particularly data envelopment analysis (DEA) estimators. Until now, inference about productivity change measured by Malmquist indices has been problematic, including both inference regarding productivity change experienced by particular firms as well as mean productivity change. This paper establishes properties of a DEA-type estimator of distance to the conical hull of a variable returns to scale production frontier. In addition, properties of DEA estimators of Malmquist indices for individual producers are derived as well as properties of geometric means of these estimators. The latter requires new central limit theorem results, extending the work of Kneip, Simar, and Wilson (2015, Econometric Theory 31, 394–422). Simulation results are provided to give applied researchers an idea of how well inference may work in practice in finite samples. Our results extend easily to other productivity indices, including the Luenberger and Hicks–Moorsteen indices.

Several tests of model structure developed by Kneip et al. (J Bus Econ Stat 34:435–456, 2016) and Daraio et al. (Econ J 21:170–191, 2018) rely on comparing sample means of two different efficiency estimators, one appropriate under the conditions of the null hypothesis and the other appropriate under the conditions of the alternative hypothesis. These tests rely on central limit theorems developed by Kneip et al. (Econ Theory 31:394–422, 2015) and Daraio et al. (Econ J 21:170–191, 2018), but require that the original sample be split randomly into two independent subsamples. This introduces some ambiguity surrounding the sample-split, which may be determined by choice of a seed for a random number generator. We develop a method that eliminates much of this ambiguity by repeating the random splits a large number of times. We use a bootstrap algorithm to exploit the information from the multiple sample-splits. Our simulation results show that in many cases, eliminating this ambiguity results in tests with better size and power than tests that employ a single sample-split.

Stochastic frontier models along the lines of Aigner et al. are widely used to benchmark firms’ performances in terms of efficiency. The models are typically fully parametric, with functional form specifications for the frontier as well as both the noise and the inefficiency processes. Studies such as Kumbhakar et al. have attempted to relax some of the restrictions in parametric models, but so far all such approaches are limited to a univariate response variable. Some (e.g., Simar and Zelenyuk; Kuosmanen and Johnson) have proposed nonparametric estimation of directional distance functions to handle multiple inputs and outputs, raising issues of endogeneity that are either ignored or addressed by imposing restrictive and implausible assumptions. This article extends nonparametric methods developed by Simar et al. and Hafner et al. to allow multiple inputs and outputs in an almost fully nonparametric framework while avoiding endogeneity problems. We discuss properties of the resulting estimators, and examine their finite-sample performance through Monte Carlo experiments. Practical implementation of the method is illustrated using data on U.S. commercial banks.

Asymptotic statistical inference on productivity and production efficiency, using nonparametric envelopment estimators, is now available thanks to the basic central limit theorems (CLTs) developed in Kneip, Simar, and Wilson (2015). They provide asymptotic distributions of averages of Data Envelopment Analysis (DEA) and Free Disposal Hull (FDH) estimators of production efficiency. As shown in their Monte-Carlo experiments, due to the curse of dimensionality, the accuracy of the normal approximation is disappoint- ing when the sample size is not large enough. Simar and Zelenyuk (2020) have suggested a simple way to improve the approximation by using a more appropriate estimator of the variances. In this paper we suggest a novel way to improve the approximation, by smoothing out the spurious values of efficiency estimates when they are in a neighborhood of 1. This results in sharpening the data for observations near the estimated efficient frontier. The method is very easy to implement and does not require more computations than the original method. We compare our approach using Monte-Carlo experiments, both with the basic method and with the improved method suggested in Simar & Zelenyuk (2020) and in both cases we observe significant improvements. We show also that the Simar & Zelenyuk (2020) idea of the variance correction can also be adapted to our sharpening method, bringing additional improvements. We illustrate the method with some real data sets.

International audience; The aim of this paper is to construct a robust nonparametric estimator for the production frontier. We study this problem under a regression model with one-sided errors where the regression function defines the achievable maximum output, for a given level of inputs-usage, and the regression error defines the inefficiency term. The main tool is a concept of partial regression boundary defined as a special probability-weighted moment. This concept motivates a robustified unconditional alternative to the pioneering class of nonparametric conditional expected maximum production functions. We prove that both the resulting benchmark partial frontier and its estimator share the desirable monotonicity of the true full frontier. We derive the asymptotic properties of the partial and full frontier estimators, and unravel their behavior from a robustness theory point of view. We provide numerical illustrations and Monte Carlo evidence that the presented concept of unconditional expected maximum production functions is more efficient and reliable in filtering out noise than the original conditional version. The methodology is very easy and fast to implement. Its usefulness is discussed through two concrete datasets from the sector of Delivery Services, where outliers are likely to affect the traditional conditional approach.

International audience; The main objective of this study is to assess the impact of gender on microfinance social efficiency. Our methodology is based on nonparametric techniques to estimate the gender effect. We use a conditional directional free disposal hull (FDH) approach as well as its robust version of order-; we study the effect of the heterogeneity factor on the difference of conditional and non conditional inefficiencies as well as on the inefficiency level using a local linear regression and we test the significance of its effect using a wild double bootstrap procedure. Using a cross-country sample of 680 microfinance institutes (MFIs) in 2011 from six main regions of the world, our findings suggest that gender diversity has globally a positive impact on the microfinance social efficiency. However, the nature of the effect depends on the considered heterogeneity factor and we find that the boardroom gender diversity effect is linear, whereas the effect of the percentage of women loan officers is non linear (U-shaped on the difference of inefficiencies and inverted U-shaped on the inefficiency levels). We assess the robustness of our findings on various subsamples (global or regional scale, and also depending on the considered profit oriented status). Our findings reinforce the importance of the role played by women in MFI social efficiency.

Despite the long and great history, developed institutions, and high level of physical and human capital, the Italian economy has been fairly stagnant during the last three decades. In this paper, we merge two streams of literature: nonparametric methods to estimate frontier efficiency of an economy, which allows us to develop a new measure of output gap, and nonparametric methods to estimate probability of an economic recession. To illustrate the new framework, we use quarterly data for Italy from 1995 to 2019 and find that our model, using either nonparametric or the linear probit model, is able to provide useful insights.

Although human capital has been recognized as playing important role in spurring productivity growth, its empirical effect remains ambiguous due to the possibility of latent heterogeneity. To reveal the impact of this important driver of economic growth, we propose an alternative empirical methodology, robust frontier in non parametric location-scale models for accommodating simultaneously the problem of model specification uncertainty, latent heterogeneity and cross-section dependence in modelling technical efficiency. We estimate a nonparametric frontier model to define the world technology frontier of 40 countries over the period 1970–2007. Conditional versions of the frontier enables us to investigate the effect of external factors on the production process. One of these factors will allow to measure the openness of the economy of the country and the second will be this latent factor of heterogeneity linked to the absorptive capability and hence to human capital of the country. We have to adjust the methodology to take into account the panel structure of our data set, i.e., handling the time dimension and the cross-sectional dependence affecting the process. Our findings prove that human capital plays an important role in accelerating the technological catch-up (increase in the efficiency) but not on the technological changes (shifts in the frontier). This result seems to confirm the theoretical hypothesis that countries benefit from new technology (technological catch-up) only when they have the ability to exploit it, hence only when they have high level of absorptive capability.

National audience; In stochastic frontier models, the regression function defines the production frontier and the regression errors are assumed to be composite. The actually observed outputs are assumed to be contaminated by a stochastic noise. The additive regression errors are composed from this noise term and the one-sided inefficiency term. The aim is to construct a robust nonparametric estimator for the production function. The main tool is a robust concept of partial, expected maximum production frontier, defined as a special probability-weighted moment. In contrast to the deterministic one-sided error model where robust partial frontier modeling is fruitful, the composite error problem requires a substantial different treatment based on deconvolution techniques. To ensure the identifiability of the model, it is sufficient to assume an independent Gaussian noise. In doing so, the frontier estimation necessitates the computation of a survival function estimator from an illposed equation. A Tikhonov regularized solution is constructed and nonparametric frontier estimation is performed. The asymptotic properties of the obtained survival function and frontier estimators are established. Practical guidelines to effect the necessary computations are described via a simulated example. The usefulness of the approach is discussed through two concrete data sets from the sector of Delivery Services.

© 2019, © 2019 American Statistical Association. Consider the model Y = X + ε with X = τ + Z, where τ is an unknown constant (the boundary of X), Z is a random variable defined on R + , ε is a symmetric error, and ε and Z are independent. Based on an iid sample of Y, we aim at identifying and estimating the boundary τ when the law of ε is unknown (apart from symmetry) and in particular its variance is unknown. We propose an estimation procedure based on a minimal distance approach and by making use of Laguerre polynomials. Asymptotic results as well as finite sample simulations are shown. The paper also proposes an extension to stochastic frontier analysis, where the model is conditional to observed variables. The model becomes Y = τ(w1,w2) + Z + ε, where Y is a cost, w 1 are the observed outputs and w 2 represents the observed values of other conditioning variables, so Z is the cost inefficiency. Some simulations illustrate again how the approach works in finite samples, and the proposed procedure is illustrated with data coming from post offices in France. ispartof: Journal Of The American Statistical Association vol:115 issue:529 pages:425-441 status: published

A rich literature on the analysis of efficiency in production has developed since pioneering work of Tjalling Koopmans and George Debreu in the 1950s. This literature includes work by researchers in economics, econometrics, management science, operations research, mathematical statistics and other fields. The focus in this survey is on nonparametric approaches for estimation and inference, with particular emphasis on inference since nothing can be learned from estimation without inference. The statistical problem amounts to estimating the support of a multivariate random variable, subject to some shape constraints, in multiple dimensions. New results that enable inference about mean efficiency as well as tests of convexity, returns to scale, differences in mean efficiency and the separability condition described by Simar and Wilson (Journal of Econometrics 136:31–64, 2007) are discussed. The well-known curse of dimensionality presents additional challenges, but recent work indicates that reducing dimensionality using eigensystem techniques may improve estimation accuracy. Remaining challenges and open issues are also discussed.

How to exit the economic recession and improve the economic condition in a globalized world is a central policy issue of importance. This paper examines whether countries that pursue outward orientation policies and that are increasingly economically integrated with the rest of the world have seen an increase in economic performance. Increasing globalization and interconnection among countries generates spatial and temporal dependence which will affect the production process of each country. We extend existing methodological tools – robust frontier in non parametric location-scale models - to estimate the world frontier of 44 countries over 1970–2007 and to obtain more reliable measure of productivity and efficiency to better investigate the driven forces behind the catching-up productivity process among countries. We explore the channels under which Foreign Direct Investment (FDI) and time affect the production process and its components: impact on the attainable production set (input-output space), and the impact on the distribution of efficiencies.