Fixed and Random Effects in Stochastic Frontier Models

Received stochastic frontier analyses with panel data have relied on traditional fixed and random effects models. We propose extensions that circumvent two shortcomings of these approaches. The conventional panel data estimators assume that technical or cost inefficiency is time invariant. Second, the fixed and random effects estimators force any time invariant cross unit heterogeneity into the same term that is being used to capture the inefficiency. Inefficiency measures in these models may be picking up heterogeneity in addition to or even instead of inefficiency. A fixed effects model is extended to the stochastic frontier model using results that specifically employ the nonlinear specification. The random effects model is reformulated as a special case of the random parameters model. The techniques are illustrated in applications to the U.S. banking industry and a cross country comparison of the efficiency of health care delivery.JEL classification: C1, C4     

FIRM HETEROGENEITY,PERSISTENT AND TRANSIENT TECHNICAL INEFFICIENCY: A GENERALIZED TRUE RANDOM‐EFFECTS model

This paper considers a panel data stochastic frontier model that disentangles unobserved firm effects (firm heterogeneity) from persistent (time‐invariant/long‐term) and transient (time‐varying/short‐term) technical inefficiency. The model gives us a four‐way error component model, viz., persistent and time‐varying inefficiency, random firm effects and noise. We use Bayesian methods of inference to provide robust and efficient methods of estimating inefficiency components in this four‐way error component model. Monte Carlo results are provided to validate its performance. We also present results from an empirical application that uses a large panel of US commercial banks. Copyright © 2012 John Wiley & Sons, Ltd.     

Technical efficiency in competing panel data models: a study of Norwegian grain farming

Estimation of technical efficiency is widely used in empirical research using both cross-sectional and panel data. Although several stochastic frontier models for panel data are available, only a few of them are normally applied in empirical research. In this article we chose a broad selection of such models based on different assumptions and specifications of heterogeneity, heteroskedasticity and technical inefficiency. We applied these models to a single dataset from Norwegian grain farmers for the period 2004–2008. We also introduced a new model that disentangles firm effects from persistent (time-invariant) and residual (time-varying) technical inefficiency. We found that efficiency results are quite sensitive to how inefficiency is modeled and interpreted. Consequently, we recommend that future empirical research should pay more attention to modeling and interpreting inefficiency as well as to the assumptions underlying each model when using panel data.     

Adjustment and unobserved heterogeneity in dynamic stochastic frontier models

Stochastic frontier models with autocorrelated inefficiency have been proposed in the past as a way of addressing the issue of temporal variation in firm-level efficiency scores. They are justified using an underlying model of dynamic firm behavior. In this paper we argue that these models could have radically different implications for the expected long-run efficiency scores in the presence of unobserved heterogeneity. The possibility of accounting for unobserved heterogeneity is explored. Random- and correlated random-effects dynamic stochastic frontier models are proposed and applied to a panel of US electric utilities.     

Inference in dynamic stochastic frontier models

An important issue in models of technical efficiency measurement concerns the temporal behaviour of inefficiency. Consideration of dynamic models is necessary but inference in such models is complicated. In this paper we propose a stochastic frontier model that allows for technical inefficiency effects and dynamic technical inefficiency, and use Bayesian inference procedures organized around data augmentation techniques to provide inferences. Also provided are firm‐specific efficiency measures. The new methods are applied to a panel of large US commercial banks over the period 1989–2000. Copyright © 2006 John Wiley & Sons, Ltd.     

Bayesian estimation of inefficiency heterogeneity in stochastic frontier models

Estimation of the one sided error component in stochastic frontier models may erroneously attribute firm characteristics to inefficiency if heterogeneity is unaccounted for. However, unobserved inefficiency heterogeneity has been little explored. In this work, we propose to capture it through a random parameter which may affect the location, scale, or both parameters of a truncated normal inefficiency distribution using a Bayesian approach. Our findings using two real data sets, suggest that the inclusion of a random parameter in the inefficiency distribution is able to capture latent heterogeneity and can be used to validate the suitability of observed covariates to distinguish heterogeneity from inefficiency. Relevant effects are also found on separating and shrinking individual posterior efficiency distributions when heterogeneity affects the location and scale parameters of the one-sided error distribution, and consequently affecting the estimated mean efficiency scores and rankings. In particular, including heterogeneity simultaneously in both parameters of the inefficiency distribution in models that satisfy the scaling property leads to a decrease in the uncertainty around the mean scores and less overlapping of the posterior efficiency distributions, which provides both more reliable efficiency scores and rankings.     

Persistent and transient cost efficiency—an application to the Swiss hydropower sector

Electricity prices on the European market have decreased significantly over the past few years, resulting in a deterioration of Swiss hydropower firms’ competitiveness and profitability. One option to improve the sector’s competitiveness is to increase cost efficiency. The goal of this study is to quantify the level of persistent and transient cost efficiency of individual firms by applying the generalized true random effects (GTRE) model introduced by Colombi et al. (Journal of Productivity Analysis 42(2): 123–136, 2014) and Filippini and Greene (Journal of Productivity Analysis 45(2): 187–196, 2016). Applying this newly developed GTRE model to a total cost function, the level of cost efficiency of 65 Swiss hydropower firms is analyzed for the period between 2000 and 2013. A true random effects specification is estimated as a benchmark for the transient level of cost efficiency. The results show the presence of both transient as well as persistent cost inefficiencies. The GTREM predicts the aggregate level of cost inefficiency to amount to 21.8% (8.0% transient, 13.8% persistent) on average between 2000 and 2013. These two components differ in interpretation and implication. From an individual firm’s perspective, the two types of cost inefficiencies might require a firm’s management to respond with different improvement strategies. The existing level of persistent inefficiency could prevent the hydropower firms from adjusting their production processes to new market environments. From a regulatory point of view, the results of this study could be used in the scope and determination of the amount of financial support given to struggling firms.     

Functional Forms of Stochastic Frontier Production Functions and Models for Technical Inefficiency Effects: A Comparative Study for Wheat Farmers in Pakistan

This paper considers translog and Cobb-Douglas stochastic frontiers in which the technical inefficiency effects are defined by three different models. The models involved are the time-varying inefficiency model, proposed by Battese and Coelli (1992), the inefficiency effects model for panel data, proposed by Battese and Coelli (1995), and the non-neutral frontier model, proposed by Huang and Liu (1994). Technical change is also accounted for in the frontier models. Predicted technical efficiencies of the wheat farmers and estimates of the elasticities of wheat production with respect to the different inputs and the returns-to-scale parameter are compared under the different model specifications.     

Cost efficiency of Slovenian water distribution utilities: an application of stochastic frontier methods

This study estimates cost inefficiency and economies of scale of Slovenian water distribution utilities over the 1997–2003 period by employing several different stochastic frontier methods. The results indicate that significant cost inefficiencies are present in the utilities. An introduction of incentive-based price regulation scheme might help resolve this problem. However, the inefficiency scores obtained from different cost frontier models are not found to be robust. The levels of inefficiency estimates as well as the rankings depend on the econometric specification of the model. The established lack of robustness can be at least partly explained by different ability of the models to separate unobserved heterogeneity from inefficiency. Newly proposed true fixed effects model (Greene, J Econom 126:269–303, 2005; J Prod Anal 23(1):7–32, 2005) appears to perform better than the conventional panel data models with respect to distinguishing between unobserved heterogeneity and inefficiency. On the other hand, different models produce fairly robust results with respect to estimates of economies of output density, customer density and economies of scale. The optimal size of a company is found to closely corresponds to the sample median. Economies of scale are found in small-sized utilities, while large companies exhibit diseconomies of scale.

Estimating fixed-effect panel stochastic frontier models by model transformation

Traditional panel stochastic frontier models do not distinguish between unobserved individual heterogeneity and inefficiency. They thus force all time-invariant individual heterogeneity into the estimated inefficiency. Greene (2005) proposes a true fixed-effect stochastic frontier model which, in theory, may be biased by the incidental parameters problem. The problem usually cannot be dealt with by model transformations owing to the nonlinearity of the stochastic frontier model. In this paper, we propose a class of panel stochastic frontier models which create an exception. We show that first-difference and within-transformation can be analytically performed on this model to remove the fixed individual effects, and thus the estimator is immune to the incidental parameters problem. Consistency of the estimator is obtained by either N→∞$N\to \infty$ or T→∞$T\to \infty$, which is an attractive property for empirical researchers.     

Firm efficiency: Domestic owners,coalitions, and FDI

In this paper we analyze the evolution of firm efficiency in the Czech Republic. Using a large panel of more than 190,000 Czech firm/years we study whether firms fully utilize their resources, how firm efficiency evolves over time, and how firm efficiency is determined by ownership structure. We employ a panel version of a stochastic production frontier model for the period 1996–2007 with time-varying efficiency. We differentiate among various degrees of ownership concentration and domestic or foreign origin. In a two-stage set-up we first estimate the degree of firm inefficiency and then the effect of ownership structure on the distance from the efficiency frontier. Our results support the hypothesis that concentrated ownership is positively related to efficiency. FDI has beneficial effects at the microeconomic level. However, we show that a simple majority is not necessarily the best structure to improve efficiency. We further analyze the effects of ownership coalitions and shed light on many other subtleties of how ownership and the specific industry affect firm efficiency.     

Public utility planning and cost efficiency in a decentralized regulation context: the case of the Italian integrated water service

The reformed Italian water and sewerage industry has several distinctive features. It is based on a decentralized structure where local authorities are entitled to define detailed long-term budget plans that claim to provide efficiency incentives to operating suppliers. Unlike previous studies, this paper analyzes the cost efficiency embedded in these budget plans to evaluate the actual capability of local regulators to adequately orientate firm performance. Several panel data cost frontier models were estimated that incorporate diverse specifications for inefficiency and unobserved heterogeneity terms. The results indicate that the decentralized planning mechanism applied in Italian water and sewerage industry regulation failed in fulfilling the declared goal and further highlights that the time-invariant terms are the prevailing source of cost differences, which may conceal a structural component attributable to persistent inefficiency.     

Alternative technical efficiency measures: Skew,bias and scale

In the fixed‐effects stochastic frontier model an efficiency measure relative to the best firm in the sample is universally employed. This paper considers a new measure relative to the worst firm in the sample. We find that estimates of this measure have smaller bias than those of the traditional measure when the sample consists of many firms near the efficient frontier. Moreover, a two‐sided measure relative to both the best and the worst firms is proposed. Simulations suggest that the new measures may be preferred depending on the skewness of the inefficiency distribution and the scale of efficiency differences. Copyright © 2010 John Wiley & Sons, Ltd.     

Estimation and testing of stochastic frontier models using variational Bayes

We show how a wide range of stochastic frontier models can be estimated relatively easily using variational Bayes. We derive approximate posterior distributions and point estimates for parameters and inefficiency effects for (a) time invariant models with several alternative inefficiency distributions, (b) models with time varying effects, (c) models incorporating environmental effects, and (d) models with more flexible forms for the regression function and error terms. Despite the abundance of stochastic frontier models, there have been few attempts to test the various models against each other, probably due to the difficulty of performing such tests. One advantage of the variational Bayes approximation is that it facilitates the computation of marginal likelihoods that can be used to compare models. We apply this idea to test stochastic frontier models with different inefficiency distributions. Estimation and testing is illustrated using three examples.     

Closed-skew normality in stochastic frontiers with individual effects and long/short-run efficiency

This paper considers the estimation of Kumbhakar et al. (J Prod Anal. doi:10.1007/s11123-012-0303-1, 2012) (KLH) four random components stochastic frontier (SF) model using MLE techniques. We derive the log-likelihood function of the model using results from the closed-skew normal distribution. Our Monte Carlo analysis shows that MLE is more efficient and less biased than the multi-step KLH estimator. Moreover, we obtain closed-form expressions for the posterior expected values of the random effects, used to estimate short-run and long-run (in)efficiency as well as random-firm effects. The model is general enough to nest most of the currently used panel SF models; hence, its appropriateness can be tested. This is exemplified by analyzing empirical results from three different applications.     

Estimation and efficiency measurement in stochastic production frontiers with ordinal outcomes

We consider Bayesian estimation of a stochastic production frontier with ordered categorical output, where the inefficiency error is assumed to follow an exponential distribution, and where output, conditional on the inefficiency error, is modelled as an ordered probit model. Gibbs sampling algorithms are provided for estimation with both cross-sectional and panel data, with panel data being our main focus. A Monte Carlo study and a comparison of results from an example where data are used in both continuous and categorical form supports the usefulness of the approach. New efficiency measures are suggested to overcome a lack-of-invariance problem suffered by traditional efficiency measures. Potential applications include health and happiness production, university research output, financial credit ratings, and agricultural output recorded in broad bands. In our application to individual health production we use data from an Australian panel survey to compute posterior densities for marginal effects, outcome probabilities, and a number of within-sample and out-of-sample efficiency measures.     

Estimation of Technical Inefficiencies with Heterogeneous Technologies

This paper considers the measurement of firm's specific (in)efficiency while allows for the possible heterogeneous technologies adopted by different firms. A flexible stochastic frontier model with random coefficients is proposed to distinguish technical inefficiency from technological differences across firms. Posterior inference of the model is made possible via the simulation-based approach, namely, Markov chain Monte Carlo method. The model is applied to a real data set which has also been considered in Christensen and Greene (1976), Greene (1990), Tsionas (2002), among others. Empirical results show that the regression coefficients can vary across firms, indicating the adoption of heterogeneous technologies by different firms. More importantly, we find that, without considering this possible heterogeneity, the inefficiency of firms can be over-estimated.     

Consistent inference in fixed-effects stochastic frontier models

The classical stochastic frontier panel data models provide no mechanism to disentangle individual time invariant unobserved heterogeneity from inefficiency. Greene (2005a, b) proposed the so-called “true” fixed-effects specification that distinguishes these two latent components. However, due to the incidental parameters problem, his maximum likelihood estimator may lead to biased variance estimates. We propose two alternative estimators that achieve consistency for $n\to \infty$ with fixed $T$. Furthermore, we extend the Chen et al. (2014) results providing a feasible estimator when the inefficiency is heteroskedastic and follows a first-order autoregressive process. We investigate the behavior of the proposed estimators through Monte Carlo simulations showing good finite sample properties, especially in small samples. An application to hospitals’ technical efficiency illustrates the usefulness of the new approach.     

Panel estimates of a two-tiered earnings frontier

This paper uses panel data to estimate a two-tiered instead of a one-tiered frontier model. The innovation is to develop a two-step maximum likelihood procedure yielding consistent estimates of inefficiency, while at the same time accounting for heterogeneity. The model is applied by estimating a ‘two-tiered’ earnings function to obtain indices of worker and firm incomplete labour market wage information using panel data from the Panel Study of Income Dynamics (1969–84). The estimation preserves the traditional quadratic age-earnings profile, but measures the extent to which employers often pay more than necessary to hire a worker (incomplete employer information), while at the same time, employees often accept wages less than they could otherwise command (incomplete employee information). The results indicate that employees acquire less information than employers. © 1996 John Wiley & Sons, Ltd.     

Estimation of a dynamic stochastic frontier model using likelihood-based approaches

This paper considers a panel stochastic production frontier model that allows the dynamic adjustment of technical inefficiency. In particular, we assume that inefficiency follows an AR(1) process. That is, the current year's inefficiency for a firm depends on its past inefficiency plus a transient inefficiency incurred in the current year. Interfirm variations in the transient inefficiency are explained by some firm-specific covariates. We consider four likelihood-based approaches to estimate the model: the full maximum likelihood, pairwise composite likelihood, marginal composite likelihood, and quasi-maximum likelihood approaches. Moreover, we provide Monte Carlo simulation results to examine and compare the finite-sample performances of the four above-mentioned likelihood-based estimators of the parameters. Finally, we provide an empirical application of a panel of 73 Finnish electricity distribution companies observed during 2008–2014 to illustrate the working of our proposed models.