Consistent estimation of technical and allocative efficiencies for a semiparametric stochastic cost frontier with shadow input prices

Conventional parametric stochastic cost frontier models are likely to suffer from biased inferences due to misspecification and the ignorance of allocative efficiency (AE). To fill up the gap in the literature, this article proposes a semiparametric stochastic cost frontier with shadow input prices that combines a parametric portion with a nonparametric portion and that allows for the presence of both technical efficiency (TE) and AE. The introduction of AE and the nonparametric function into the cost function complicates substantially the estimation procedure. We develop a new estimation procedure that leads to consistent estimators and valid TE and AE measures, which are proved by conducting Monte Carlo simulations.     

Parametric approximations of nonparametric frontiers

The maximal achievable level of output for a given level of inputs defines the production frontier that can serve as benchmark to evaluate individual firm efficiencies. Nonparametric envelopment estimators (free disposal hull, data envelopment analysis) have been mostly used because they rely on very few assumptions, whereas parametric forms for the frontier allow for richer economic interpretation. Most of the parametric approaches rely on standard regression fitting the shape of the center of the cloud of points. In this paper, we investigated a new approach, which captures the shape of the cloud points near its boundary. It offers parametric approximations of nonparametric frontiers. We provide the statistical theory (asymptotic). Some simulated examples show the advantages of our method compared with the usual regression-type estimators.     

Statistical Inference in Nonparametric Frontier Models: The State of the Art

Efficiency scores of firms are measured by their distance to an estimated production frontier. The economic literature proposes several nonparametric frontier estimators based on the idea of enveloping the data (FDH and DEA-type estimators). Many have claimed that FDH and DEA techniques are non-statistical, as opposed to econometric approaches where particular parametric expressions are posited to model the frontier. We can now define a statistical model allowing determination of the statistical properties of the nonparametric estimators in the multi-output and multi-input case. New results provide the asymptotic sampling distribution of the FDH estimator in a multivariate setting and of the DEA estimator in the bivariate case. Sampling distributions may also be approximated by bootstrap distributions in very general situations. Consequently, statistical inference based on DEA/FDH-type estimators is now possible. These techniques allow correction for the bias of the efficiency estimators and estimation of confidence intervals for the efficiency measures. This paper summarizes the results which are now available, and provides a brief guide to the existing literature. Emphasizing the role of hypotheses and inference, we show how the results can be used or adapted for practical purposes.     

Robustness and inference in nonparametric partial frontier modeling

A major aim in recent nonparametric frontier modeling is to estimate a partial frontier well inside the sample of production units but near the optimal boundary. Two concepts of partial boundaries of the production set have been proposed: an expected maximum output frontier of order m=1,2,… and a conditional quantile-type frontier of order α∈]0,1]. In this paper, we answer the important question of how the two families are linked. For each m, we specify the order α for which both partial production frontiers can be compared. We show that even one perturbation in data is sufficient for breakdown of the nonparametric order-m frontiers, whereas the global robustness of the order-α frontiers attains a higher breakdown value. Nevertheless, once the α frontiers break down, they become less resistant to outliers than the order-m frontiers. Moreover, the m frontiers have the advantage to be statistically more efficient. Based on these findings, we suggest a methodology for identifying outlying data points. We establish some asymptotic results, contributing to important gaps in the literature. The theoretical findings are illustrated via simulations and real data.     

Stochastic non-smooth envelopment of data: semi-parametric frontier estimation subject to shape constraints

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 DEA-type nonparametric frontier, which satisfies monotonicity and concavity, with the SFA-style stochastic homoskedastic composite error term. To estimate this model, a new two-stage method is proposed, referred to as Stochastic Non-smooth 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 cross-sectional 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.     

Allowing for inefficiency in parametric estimation of production functions for urban transit firms

Efficient versus inefficient observations are first identified and evaluated numerically by the nonparametric free disposal hull (FDH) method. Next, parametric production frontiers are obtained by means of estimating translog production functions through OLS applied to the subset of efficient observations only. Technical progress is included at both stages. Monthly data from three urban transit firms in Belgium, to which this two-stage technique is applied, show widely varying degrees of efficiency over time and across firms, and much less technical progress than standard (i.e., non frontier) econometric estimates suggest.     

Stochastic FDH/DEA estimators for frontier analysis

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

Estimating global frontier shifts and global Malmquist indices

The Malmquist index is a measure of productivity changes, of which an important component is the frontier shift or technological change. Often technological change can be viewed as a global phenomenon, and therefore individual or local measures of technological changes are aggregated into an overall measure, traditionally using geometric means. In this paper we propose a way of calculating global Malmquist indices and global frontier shift indices which provides a better estimation of the true frontier shift and furthermore is easy to calculate. Using simulation studies we show how this method outperforms the traditional aggregation approach, especially for sparsely populated production possibility sets and for frontiers that also change shape over time. Furthermore, our global indices can be used for unbalanced panels without disregarding any information. Finally, we show how the global indices are meaningful for calculating differences between frontiers from different groups rather than different time periods as illustrated in a small case study of bank branches in different countries.      

Estimating production frontiers and efficiency when output is a discretely distributed economic bad

This article studies the estimation of production frontiers and efficiency scores when the commodity of interest is an economic bad with a discrete distribution. Existing parametric econometric techniques (stochastic frontier methods) assume that output is a continuous random variable but, if output is discretely distributed, then one faces a scenario of model misspecification. Therefore a new class of econometric models has been developed to overcome this problem. The Delaporte subclass of models is studied in detail, and tests of hypotheses are proposed to discriminate among parametric models. In particular, Pearson’s chi-squared test is adapted to construct a new kernel-based consistent Pearson test. A Monte Carlo experiment evaluates the merits of the new model and methods, and these are used to estimate the frontier and efficiency scores of the production of infant deaths in England. Extensions to the model are discussed.     

Nonparametric estimation and testing of fixed effects panel data models

In this paper we consider the problem of estimating nonparametric panel data models with fixed effects. We introduce an iterative nonparametric kernel estimator. We also extend the estimation method to the case of a semiparametric partially linear fixed effects model. To determine whether a parametric, semiparametric or nonparametric model is appropriate, we propose test statistics to test between the three alternatives in practice. We further propose a test statistic for testing the null hypothesis of random effects against fixed effects in a nonparametric panel data regression model. Simulations are used to examine the finite sample performance of the proposed estimators and the test statistics.     

Nonparametric likelihood ratio model selection tests between parametric likelihood and moment condition models

We propose a nonparametric likelihood ratio testing procedure for choosing between a parametric (likelihood) model and a moment condition model when both models could be misspecified. Our procedure is based on comparing the Kullback–Leibler Information Criterion (KLIC) between the parametric model and moment condition model. We construct the KLIC for the parametric model using the difference between the parametric log likelihood and a sieve nonparametric estimate of population entropy, and obtain the KLIC for the moment model using the empirical likelihood statistic. We also consider multiple (>2)$(>2)$ model comparison tests, when all the competing models could be misspecified, and some models are parametric while others are moment-based. We evaluate the performance of our tests in a Monte Carlo study, and apply the tests to an example from industrial organization.     

Adjusting MV-efficient portfolio frontier bias for skewed and non-mesokurtic returns

This study investigates the bias adjustment for mean–variance efficient portfolio frontiers due to population mean and variance estimation error in Taiwan stock market. Although Siegel and Woodgate (2007; Management Science, 53, 1005–1015) and Kan and Smith (2008; Management Science, 54, 1364–1380) suggested two portfolio frontiers that improved upon the out-of-sample performance of a traditional sample portfolio frontier. However, this study shows that, using the copula function and Gram-Charlier series, the two frontiers are theoretically biased toward the actual frontier unless returns behave normally, and the bias is related to the return skewness and kurtosis. Indeed, the two frontiers are empirically biased to the lower-left side of the actual ones, because the Taiwan stock returns are right-skewed and highly leptokurtic. Thus, this study thus proposes revised portfolio frontiers that are closer to the actual frontier than unrevised ones. This improvement may enhance the estimation accuracy of the capital market line, and hence this study can provide an effective investment reference.     

Productive performance of the French insurance industry

The purpose of this paper is to provide for both life and non-life insurance an assessment of the relative productive performance of French companies. We use parametric and nonparametric approaches to construct a frontier to be used as a yardstick of productive efficiency. Our data basis covers 84 life and 243 non-life companies for the period 1984–1989. The main findings show a high correlation between parametric and nonparametric results and a wide dispersion in the rates of inefficiency across companies. This dispersion can be reduced when controlling for variations in scale, ownership, distribution, reinsurance, and claims ratios.     

Kernel estimation for disequilibrium models for floorspace efficiency in retailing

Conclusions In this paper we have proposed new techniques for simplifying the estimation of disequilibrium models by avoiding constrained maximum likelihood methods (which cannot avoid numerous theoretical and practical difficulties mentioned above) including an unrealistic assumption of the independence of errors in demand and supply system of equations. In the proposed first stage, one estimates the relative magnitude of the residuals from the demand and supply equations nonparametrically, even though they suffer from omitted variables bias, because the coefficient of the omitted variable is known to be the same in both equations. The reason for using nonparametric methods is that they do not depend on parametric functional forms of biased (bent inward) demand and supply equations. The first stage compares the absolute values of residuals from conditional expectations in order to classify the data points as belonging to the demand or the supply curve. We estimate the economically meaningful scale elasticity and distribution parameters at the second stage from classified (separated) data.We extend nonparametric kernel estimation to the r = 4 case to improve the speed of convergence, as predicted by Singh's [1981] theory. In the first stage, r = 4 results give generally improved R² and ¦t¦ values in our study of the Dutch data—used by many authors concerned with the estimation of floorspace productivity. We find that one can obtain reasonable results by our approximate but simpler two stage methods. Detailed results are reported for four types of Dutch retail establishments. More research is needed to gain further experience and to extend the methodology to other disequilibrium models and other productivity estimation problems.This paper was processed by W. Eichhorn.     

Multiple Objective Approach as an Alternative to Radial Projection in DEA

Radial projection is a standard technique applied in data envelopment analysis (DEA) to calculate efficiency scores for input and/or output variables. In this paper, we have studied the appropriateness of radial projection for target setting. We have created a situation where the decision making units (DMUs) are free to choose their own target values on the efficient frontier and then compared the results to those of radial projection. In practice, target values are primarily used for future goal attainment; hence, not only preferences but also, and on the whole, change in time frame, affect the choice of target values. Based on that, we conducted an empirical experiment with an aim to study how the DMUs choose their most preferred target values on the efficient frontier. The subjects, who all were students of the Helsinki School of Economics, were given the freedom to explore their personalized efficient frontiers by using a multiple objective linear programming (MOLP) approach. To study various and relevant scenarios, the personalized efficient frontiers for all students were constructed in such a way that the current position of each student in relation to the frontier made him/her inefficient, efficient, or super-efficient. The results show that the use of radial projection for target setting is too restrictive.     

Estimation in a generalization of bivariate probit models with dummy endogenous regressors

The purpose of this paper is to provide guidelines for empirical researchers who use a class of bivariate threshold crossing models with dummy endogenous variables. A common practice employed by the researchers is the specification of the joint distribution of unobservables as a bivariate normal distribution, which results in a bivariate probit model. To address the problem of misspecification in this practice, we propose an easy‐to‐implement semiparametric estimation framework with parametric copula and nonparametric marginal distributions. We establish asymptotic theory, including root‐n normality, for the sieve maximum likelihood estimators that can be used to conduct inference on the individual structural parameters and the average treatment effect (ATE). In order to show the practical relevance of the proposed framework, we conduct a sensitivity analysis via extensive Monte Carlo simulation exercises. The results suggest that estimates of the parameters, especially the ATE, are sensitive to parametric specification, while semiparametric estimation exhibits robustness to underlying data‐generating processes. We then provide an empirical illustration where we estimate the effect of health insurance on doctor visits. In this paper, we also show that the absence of excluded instruments may result in identification failure, in contrast to what some practitioners believe.     

Count data stochastic frontier models, with an application to the patents–R&D relationship

This article introduces a new count data stochastic frontier model that researchers can use in order to study efficiency in production when the output variable is a count (so that its conditional distribution is discrete). We discuss parametric and nonparametric estimation of the model, and a Monte Carlo study is presented in order to evaluate the merits and applicability of the new model in small samples. Finally, we use the methods discussed in this article to estimate a production function for the number of patents awarded to a firm given expenditure on R&D.     

Semiparametric three‐step estimation methods for simultaneous equation systems

This paper proposes a new method for estimating a structural model of labour supply in which hours of work depend on (log) wages and the wage rate is considered endogenous. The main innovation with respect to other related estimation procedures is that a nonparametric additive structure in the hours of work equation is permitted. Though the focus of the paper is on this particular application, a three‐step methodology for estimating models in the presence of the above econometric problems is described. In the first step the reduced form parameters of the participation equation are estimated by a maximum likelihood procedure adapted for estimation of an additive nonparametric function. In the second step the structural parameters of the wage equation are estimated after obtaining the selection‐corrected conditional mean function. Finally, in the third step the structural parameters of the labour supply equation are estimated using local maximum likelihood estimation techniques. The paper concludes with an application to illustrate the feasibility, performance and possible gain of using this method. Copyright © 2005 John Wiley & Sons, Ltd.     

Economic Efficiency and Frontier Techniques

Abstract. Most of the literature related to the measurement of economic efficiency has based its analysis either on parametric or on non‐parametric frontier methods. The choice of estimation method has been an issue of debate, with some researchers preferring the parametric and others the non‐parametric approach. The aim of this paper is to provide a critical and detailed review of both core frontier methods. In our opinion, no approach is strictly preferable to any other. Moreover, a careful consideration of their main advantages and disadvantages, of the data set utilized, and of the intrinsic characteristics of the framework under analysis will help us in the correct implementation of these techniques. Recent developments in frontier techniques and economic efficiency measurement such as Bayesian techniques, bootstrapping, duality theory and the analysis of sampling asymptotic properties are also considered in this paper.     

应该用哪一个模型来描述中国货币市场利率的动态变化

本文论证了双曲模型是描述中国货币市场利率动态变化的最佳单因子利率模型。由极大似然估计可以得到单因子利率模型的边际密度函数。双曲模型的边际密度和非参数估计得到的边际密度函数拟合较好，其表现远远优于几个常见的利率模型（CIR、CKLS和AG模型）。与较一般的Ait-Sahalia模型相比差别很小，但参数形式得到简化，似然比检验也支持这一点。双曲模型在刻画利率的均值回复特征方面还克服了AG模型的不足。