On maximum likelihood estimation of stochastic frontier production models

The iterative algorithm suggested by Greene (1982) for the estimation of stochastic frontier production models does not necessarily solve the likelihood equations. Corrected iterative algorithms which generalize Fair's method (1977) and solve the likelihood equations are derived. These algorithms are compared with the Newton method in an empirical case. The Newton method is more time saving than these algorithms.     

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.     

Estimation of a non-neutral stochastic frontier production function

This article proposed a hybrid of a stochastic frontier regression. The proposed model and estimation differ from the conventional model of Aigner, Lovell, and Schmidt. The model combines a stochastic frontier regression and a truncated regression to estimate the production frontier with non-neutral shifting of the average production function. The truncated regression identifies the sources of efficiency. The article presents empirical evidence of non-neutral effects of the firm's characteristics—the age of the firms, the export ratio, and the R&D expenditure—on the frontier production function and production efficiency in the Taiwan's electronics industry.We would like to express appreciation to George E. Battese, the associate editor, and anonymous referees for various comments and suggestions. The research was partially supported by the University Research Council of Vanderbilt University, and the Sun Yat-Sen Institute for Social Sciences and Philosophy of Academia Sinica, Taiwan. Residual errors are ours alone.     

Maximum likelihood estimation of stochastic frontier models by the Fourier transform

The paper is concerned with several kinds of stochastic frontier models whose likelihood function is not available in closed form. First, with output-oriented stochastic frontier models whose one-sided errors have a distribution other than the standard ones (exponential or half-normal). The gamma and beta distributions are leading examples. Second, with input-oriented stochastic frontier models which are common in theoretical discussions but not in econometric applications. Third, with two-tiered stochastic frontier models when the one-sided error components follow gamma distributions. Fourth, with latent class models with gamma distributed one-sided error terms. Fifth, with models whose two-sided error component is distributed as stable Paretian and the one-sided error is gamma. The principal aim is to propose approximations to the density of the composed error based on the inversion of the characteristic function (which turns out to be manageable) using the Fourier transform. Procedures that are based on the asymptotic normal form of the log-likelihood function and have arbitrary degrees of asymptotic efficiency are also proposed, implemented and evaluated in connection with output-oriented stochastic frontiers. The new methods are illustrated using data for US commercial banks, electric utilities, and a sample from the National Youth Longitudinal Survey.     

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.     

The stochastic frontier production function and average efficiency: An empirical analysis

This paper provides an empirical estimation of a stochastic frontier Cobb-Douglas production function using micro data from a cross-section of Brazilian manufacturing firms. Following a procedure developed by Aigner, Lovell and Schmidt incorporating both stochastic and efficiency disturbance terms in the estimating model, maximum likelihood techniques are used for the estimation of the stochastic frontier. A measure of mean technical efficiency is also developed and employed with the Brazilian data. Unlike the previous empirical exercises carried out with aggregated data, the efficiency disturbance with the Brazilian micro data estimates is not swamped by the stochastic disturbance.     

Modeling dependence in two-tier stochastic frontier models

The two-tier stochastic frontier model has seen widespread application across a range of social science domains. It is particularly useful in examining bilateral exchanges where unobserved side-specific information exists on both sides of the transaction. These buyer and seller specific informational aspects offer opportunities to extract surplus from the other side of the market, in combination also with uneven relative bargaining power. Currently, this model is hindered by the fact that identification and estimation relies on the potentially restrictive assumption that these factors are statistically independent. We present three different models for empirical application that allow for varying degrees of dependence across these latent informational/bargaining factors.

     

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.     

Goodness of fit tests in stochastic frontier models

In this paper we discuss goodness of fit tests for the distribution of technical inefficiency in stochastic frontier models. If we maintain the hypothesis that the assumed normal distribution for statistical noise is correct, the assumed distribution for technical inefficiency is testable. We show that a goodness of fit test can be based on the distribution of estimated technical efficiency, or equivalently on the distribution of the composed error term. We consider both the Pearson χ ² test and the Kolmogorov–Smirnov test. We provide simulation results to show the extent to which the tests are reliable in finite samples.     

Money and production: A stochastic frontier approach

Considerable controversy surrounds the role of money in the production of goods and services. Previous empirical research has appeared to find that the real money stock affects aggregate output, holding other, more conventional inputs constant. However, the theoretical literature offers no convincing explanation for this empirical finding. One interpretation is that real money balances reduce the extent to which labor and capital are diverted into exchange-related activities instead of being used in production defined in a more narrow sense. To investigate this hypothesis, we estimate a production function augmented with real money balances as an input, using time-series data for the aggregate U.S. economy. A stochastic production frontier is then estimated without real money balances. We use these estimates to establish the presence of technical inefficiency. Finally, we show that the extent of technical inefficiency is negatively correlated with the real money stock. Our results provide a reconciliation between the empirical literature, which finds that real money balances affect output in a production function framework, and the theoretical literature, which suggests that real money balances enhance the technical efficiency of the economy.     

Maximum likelihood estimation of seemingly unrelated stochastic frontier regressions

This paper explores the changes in value added (VA) of a sample of schools for cohorts of students finishing secondary education between 2005 and 2008. VA estimates are based on distance measures obtained from DEA models. These measures are computed for each pupil in each school, and evaluate the distance between the school frontier in a given year and a pooled frontier comprising all schools analysed. The school VA is then computed by aggregating the VA scores for the cohort of pupils attending that school in a given year. The ratio between VA estimates for two consecutive cohorts, that attended the school in different years, is taken as the index of VA change. However, the evolution of school performance over time should consider not only the movements of the school frontier, but should also take into account other effects, such as the proximity of the students to the best-practices, represented by the school frontier, observed over time. For that purpose we developed an enhanced Malmquist index to evaluate the evolution of school performance over time. One of the components of the Malmquist index proposed measures VA change, and the other measures the ability of all school students to move closer to their own school best practices over time. The approach developed is applied to a sample of Portuguese secondary schools.     

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.     

Consistent estimation of the fixed effects stochastic frontier model

In this paper we consider a fixed-effects stochastic frontier model. That is, we have panel data, fixed individual (firm) effects, and the usual stochastic frontier analysis (SFA) composed error.     

Estimates of technical inefficiency in stochastic frontier models with panel data: generalized panel jackknife estimation

Estimates of technical inefficiency based on fixed effects estimation of the stochastic frontier model with panel data are biased upward. Previous work has attempted to correct this bias using the bootstrap, but in simulations the bootstrap corrects only part of the bias. The usual panel jackknife is based on the assumption that the bias is of order T ⁻¹ and is similar to the bootstrap. We show that when there is a tie or a near tie for the best firm, the bias is of order T ^−1/2, not T ⁻¹, and this calls for a different form of the jackknife. The generalized panel jackknife is quite successful in removing the bias. However, the resulting estimates have a large variance.     

Nonparametric least squares methods for stochastic frontier models

When analyzing productivity and efficiency of firms, stochastic frontier models are very attractive because they allow, as in typical regression models, to introduce some noise in the Data Generating Process . Most of the approaches so far have been using very restrictive fully parametric specified models, both for the frontier function and for the components of the stochastic terms. Recently, local MLE approaches were introduced to relax these parametric hypotheses. In this work we show that most of the benefits of the local MLE approach can be obtained with less assumptions and involving much easier, faster and numerically more robust computations, by using nonparametric least-squares methods. Our approach can also be viewed as a semi-parametric generalization of the so-called “modified OLS” that was introduced in the parametric setup. If the final evaluation of individual efficiencies requires, as in the local MLE approach, the local specification of the distributions of noise and inefficiencies, it is shown that a lot can be learned on the production process without such specifications. Even elasticities of the mean inefficiency can be analyzed with unspecified noise distribution and a general class of local one-parameter scale family for inefficiencies. This allows to discuss the variation in inefficiency levels with respect to explanatory variables with minimal assumptions on the Data Generating Process.     

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.     

Simple tests of alternative specifications in stochastic frontier models

Previous work on stochastic production frontiers has generated a family of models, of varying degrees of complexity. Since this family is nested (in the sense that the more general models contain the less general), we can test the restrictions that distinguish the model. In this paper we provide tests of these restrictions, based on the results of estimating the simpler (restricted) models. Some of our tests are LM tests. However, in other cases the LM test fails, so we provide alternative simple tests.     

Estimation of stochastic frontier models based on multimodel inference

In most empirical studies, once the best model has been selected according to a certain criterion, subsequent analysis is conducted conditionally on the chosen model. In other words, the uncertainty of model selection is ignored once the best model has been chosen. However, the true data-generating process is in general unknown and may not be consistent with the chosen model. In the analysis of productivity and technical efficiencies in the stochastic frontier settings, if the estimated parameters or the predicted efficiencies differ across competing models, then it is risky to base the prediction on the selected model. Buckland et al. (Biometrics 53:603?C618, 1997) have shown that if model selection uncertainty is ignored, the precision of the estimate is likely to be overestimated, the estimated confidence intervals of the parameters are often below the nominal level, and consequently, the prediction may be less accurate than expected. In this paper, we suggest using the model-averaged estimator based on the multimodel inference to estimate stochastic frontier models. The potential advantages of the proposed approach are twofold: incorporating the model selection uncertainty into statistical inference; reducing the model selection bias and variance of the frontier and technical efficiency estimators. The approach is demonstrated empirically via the estimation of an Indian farm data set.