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Stochastic Frontier Analysis: Foundations and Advances I

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Handbook of Production Economics

Abstract

This chapter (as well as Chap. 11) reviews some of the most important developments in the econometric estimation of productivity and efficiency surrounding the stochastic frontier model.

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Notes

  1. 1.

    Despite the variety of definitions, intuitively, production efficiency can be understood as a relative measure of productivity. In other words, production efficiency is a productivity measure that is being normalized (e.g., to be between 0 and 1 to reflect percentages) relative to some benchmark, such as the corresponding frontier outcome, optimal with respect to some criteria: e.g., maximal output given certain level of input and technology in the case of technical efficiency or minimal cost given certain level of output and technology in the case of cost efficiency.

  2. 2.

    Battese and Corra [16] and Meeusen and van den Broeck [73], while appearing in the same year, are applications of the methods.

  3. 3.

    Our discussion in both chapters will focus on a production frontier, as it is the most popular object of study, while the framework for dual characterizations (e.g., cost, revenue, profit) or other frontiers is similar and follows with only minor changes in notation.

  4. 4.

    See chapter “Nonparametric Estimation of the SFM” for a discussion on relaxing parametric restrictions on the production frontier in the SFM.

  5. 5.

    Prior to the development of the SFM, approaches which intended to model inefficiency typically ignored vi leading to estimators of the SFM with less desirable statistical properties: see the work of [1, 3, 29, 84, 87, 99].

  6. 6.

    See section “Handling Endogeneity in the SFM” for a discussion on estimation of the SFM when some inputs are allowed to be endogenous.

  7. 7.

    See section “Modeling Determinants of Inefficiency” for models handling determinants of inefficiency.

  8. 8.

    ALS also briefly discussed the exponential distribution, but its use and development is mainly attributed to MvB.

  9. 9.

    The pdf of a skew normal random variable x is f(x) = 2ϕ(x) Φ(αx). The distribution is right skewed if α > 0 and is left skewed if α < 0. We can also place the normal, truncated-normal pair of distributional assumptions in this class. The pdf of x with location ξ, scale ω, and skew parameter α is \(f(x)=\frac {2}{\omega }\phi \left (\frac {x-\xi }{\omega }\right )\Phi \left (\alpha \left (\frac {x-\xi }{\omega }\right )\right )\). See [12, 76] for more details.

  10. 10.

    This in no way suggests that inference cannot be undertaken when the DEA estimator is deployed; rather, the DEA estimator has an asymptotic distribution which is much more complicated that the MLE for the SFM, and so direct asymptotic inference is not available; bootstrapping techniques are required for many of the most popular DEA estimators [93, 94].

  11. 11.

    See [79] for a more detailed analysis of the SFM with u distributed exponentially.

  12. 12.

    Prior to [68] all of the previously proposed distributions always produced a composed error density that was theoretically negatively skewed. Note that if u is distributed uniformly over the interval [0, b], inefficiency is equally likely to be either 0 or b.

  13. 13.

    Note that the likelihood function for the normal-half-normal pair is dependent upon the cdf of the normal distribution, Φ(⋅) which contains an integral, but this can be quickly and easily evaluated across all modern software platforms.

  14. 14.

    See also [34, pp. 31–32]. Richmond [84] also proposed adjusting the intercept from OLS estimation, however, his model differs from that of [77] by assuming the presence of inefficiency (which follows a gamma distribution) but no noise.

  15. 15.

    An alternative approach would be to estimate a weighted average efficiency of an industry, as described theoretically in chapter 25.

  16. 16.

    There exists some confusion over the terminology COLS as it relates to another method, modified OLS (MOLS). Beginning with [111] and discussed in [31] and [34, pp. 32–34], MOLS shifts the estimated OLS production function until all of the observations lie on or below the “frontier.” At issue is the appropriate name of these two techniques. Greene [38] called the bounding approach COLS, crediting [70, p. 21] with the initial nomenclature, and referred to MOLS as the method in which one bias corrects the intercept based on a specific set of distributional assumptions. Further, [59, pp. 70–71] also adopted this terminology. However, given that [77, p. 69] explicitly used the terminology COLS, in our review we will adopt COLS to imply bias correction of the OLS intercept and MOLS as a procedure that shifts up (or down) the intercept to bound all of the data. The truth is both COLS and MOLS are the same in the sense that the OLS intercept is augmented, it is just in how each method corrects, or modifies, the intercept that is important. While we are departing from the more mainstream use of COLS and MOLS currently deployed, given the original use of COLS, coupled with myriad papers written by Peter Schmidt and coauthors that we discuss here, we will use the COLS acronym to imply a bias corrected intercept.

  17. 17.

    The current literature is fairly rich on various examples of empirical values of SFA for the estimation and use of efficiency estimates in different fields of research. For example, in the context of electricity providers, see [42, 54, 62]; for banking efficiency, see [23] and references cited therein; for the analysis of the efficiency of national healthcare systems, see [33] and a review by [45]; for analyzing efficiency in agriculture, see [14, 15, 21, 69], to mention just a few.

  18. 18.

    Jondrow et al. [50] also suggested an alternative estimator based on the conditional mode.

  19. 19.

    In principle, these individual efficiency scores can then be used for estimating weighted average efficiencies of an industry or a group within it, as described theoretically in chapter 25, which seems novel for SFA context.

  20. 20.

    One could test if other moments of the distribution were 0 as well, but most of the SFMs parameterize the distribution of u with σu and so this seems the most natural.

  21. 21.

    The JLMS efficiency estimator is known as a shrinkage estimator; on average, it understates the efficiency level of a firm with small ui while it overstates efficiency for a firm with large ui.

  22. 22.

    See also [66] for a different test based on the Pearson distributional assumption for u.

  23. 23.

    In a limited Monte Carlo analysis, [86] compared rank correlations of stochastic frontier estimates assuming that inefficiency was either half-normal (which was the true distribution) or exponential (a misspecified distribution) and found very little evidence that misspecification impacted the rank correlations in any meaningful fashion; [46] conducted a similar set of experiments and found essentially the same results.

  24. 24.

    Note that the estimator of the skewness coefficient is distributed asymptotically standard normal, so it is feasible to have either negative or positive skewness in any finite sample.

  25. 25.

    Typically the standard errors can be obtained either through use of the outer product of gradients (OPG) or direct estimation of the Hessian matrix of the log-likelihood function. Given the nascency of these methods, it has yet to be determined which of these two methods is more reliable in practice, though in other settings both tend to work well. One caveat for promoting the use of the OPG is that since this only requires calculation of the first derivatives, it can be more stable (and more likely to be invertible) than calculation of the Hessian. Also note that in finite samples, the different estimators of covariance of MLE estimator can give different numerical estimates, even suggesting different implications on the inference (reject or do not reject the null hypothesis). So, for small samples, it is often advised to check all feasible estimates whenever there is suspicion of ambiguity in the conclusions (e.g., when a hypothesis is rejected only at say around the 10% of significance level).

  26. 26.

    It is possible to treat a subset of x as endogenous; i.e., x = (x1, x2), where x1 is endogenous and x2 is exogenous.

  27. 27.

    Reifschneider and Stevenson [83] used the term “inefficiency explanatory variables,” while others call them “environmental variables,” but it is now common to refer to these variables as “determinants of inefficiency.” A variety of approaches have been proposed to model the determinants of inefficiency with the first pertaining to panel data models [14, 55] (see chapter 11).

  28. 28.

    Caudill and Ford [24], Huang and Liu [49], Battese and Coelli [15], Caudill et al. [25], Hadri [39], and Wang [105] present alternative specifications as well.

  29. 29.

    It is also possible to model \(\sigma _v^2\) as a function of variables, but this poses fewer problems, and we omit the details here. See [79] and [91] for more discussion.

  30. 30.

    Actually, given the reparameterization of the log-likelihood function, the specification for σu implies a particular specification for both λ and σ.

  31. 31.

    See also [49] and [15] for early approaches following this strategy.

  32. 32.

    Other early approaches that followed this route include [25] and [39].

  33. 33.

    Note here that we are making the implicit assumption that z is different from x. The nonlinearity of the scaling function does allow z and x to overlap however.

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Authors and Affiliations

  1. Department of Economics, State University of New York at Binghamton, Binghamton, NY, USA

    Subal C. Kumbhakar

  2. Department of Economics, University of Miami, Miami, FL, USA

    Christopher F. Parmeter

  3. School of Economics and Centre for Efficiency and Productivity Analysis, University of Queensland, Brisbane, QLD, Australia

    Valentin Zelenyuk

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  1. Subal C. Kumbhakar
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Editors and Affiliations

  1. Department of Economics, University of Connecticut, Storrs, CT, USA

    Subhash C. Ray

  2. Department of Agricultural and Resource Economics, University of Maryland, College Park, MD, USA

    Robert Chambers

  3. Department of Economics, Binghamton University, Binghamton, NY, USA

    Subal Kumbhakar

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Kumbhakar, S.C., Parmeter, C.F., Zelenyuk, V. (2021). Stochastic Frontier Analysis: Foundations and Advances I. In: Ray, S.C., Chambers, R., Kumbhakar, S. (eds) Handbook of Production Economics. Springer, Singapore. https://doi.org/10.1007/978-981-10-3450-3_9-2

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  • DOI: https://doi.org/10.1007/978-981-10-3450-3_9-2

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-3450-3

  • Online ISBN: 978-981-10-3450-3

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Chapter history

  1. Latest

    Stochastic Frontier Analysis: Foundations and Advances I
    Published:
    19 November 2020

    DOI: https://doi.org/10.1007/978-981-10-3450-3_9-2

  2. Original

    Stochastic Frontier Analysis: Foundations and Advances I
    Published:
    28 July 2020

    DOI: https://doi.org/10.1007/978-981-10-3450-3_9-1

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