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Stochastic Frontier Analysis: Foundations and Advances II

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Handbook of Production Economics

Abstract

This chapter continues to review some of the most important developments in the econometric estimation of productivity and efficiency surrounding the stochastic frontier model. As in the previous chapter, we continue to place an emphasis on highlighting recent research and providing broad coverage, while details are left for further reading in the rich (although not exhaustive) references at the end of this chapter.

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Notes

  1. 1.

    If firms maximize profit, and inefficiency is known to the firm, then this assumption is unlikely to be true as firms may adjust their inputs to account for inefficiency (e.g., see [75]).

  2. 2.

    A sufficient statistic contains all the information needed to compute any estimate of the parameter.

  3. 3.

    Colombi et al. [24] showed that the likelihood function has a closed-form expression. Chen et al. [20] considered a special case of [24] and derived a closed-form expression.

  4. 4.

    This is the model proposed by [56,57,58], among others.

  5. 5.

    Which is the standard practice in any two- or multi-step procedure.

  6. 6.

    The skew normal distribution is a more general distribution than the normal distribution, allowing for asymmetry [5].

  7. 7.

    They used a local constant (Nadaraya-Watson) regression, although other consistent nonparametric estimators can be used there too.

  8. 8.

    In their work, the normal-half-normal assumption was used, but other assumptions as discussed above can be used there too. Note, however, that for some alternative distributional assumptions on u, for example, exponential or truncated-normal, a concentrated version of the log-likelihood function may not exist, causing identification problems.

  9. 9.

    See [86] for a more comprehensive review of this topic.

  10. 10.

    One could also use a quadratic approximation, but note that even in this local-linear case, there are already 3 + 3q parameters to estimate (i.e., optimize over) at each point of interest x: these are the three functional estimates, \(\ddot {m}_{0}\), \(\ddot {\sigma }_{0}^{2}\) and \(\ddot {\lambda }_{0}\) and the 3q derivative estimates of the functions, \(\ddot {m}_{1}\), \(\ddot {\sigma }_{1}^{2}\) and \(\ddot {\lambda }_{1}\).

  11. 11.

    For more discussions on the pros and cons, as well as references on this approach in general, see [42].

  12. 12.

    As with our earlier discussion, SVKZ referred to this approach as nonparametric MOLS, but cite [79], who used the term COLS and so we refer to it as COLS here.

  13. 13.

    Wang [101] documents non-monotonic efficiency effects in a panel of Philippine rice farmers based on the age of the farmer.

  14. 14.

    Hall and Simar [41] discussed nonparametric identification of the mean of inefficiency subject to the variance of the noise distribution diminishing as n →∞. Horrace and Parmeter [43] showed how to nonparametrically identify the full distribution of inefficiency if one assumes that v is distributed normal.

  15. 15.

    The approach of SVKZ allows for both x and z to influence both the frontier and inefficiency, and as such the separability assumption is not required. Yet, one may say that there is also a kind of “separability” structure involved implicitly: (x, z) is assumed to influence the frontier via the first moment, while for the inefficiency term, u, the same (x, z) is modeled through the skedastic function defining the second moment. Besides helping with statistical identification, such structure can be viewed as quite natural to the context of measurement. Indeed, one often thinks of the frontier as the level, and so using the (conditional) first moment, measuring the (conditional) average level of outputs, would be very natural. Meanwhile, the inefficiency is often understood as the deviation from the frontier, so it would be a more natural way to model it with the second moment. In addition, one could also think of the inefficiency as a reflection of the uncertainty and related “risk” to produce less than the potential and beyond the usual (and symmetric) noise, and it is very common to model risk through the second moment.

  16. 16.

    The frontier package accesses the Frontier V4.1 Fortran codes originally developed by Tim Coelli, which is also freely available (at http://www.uq.edu.au/economics/cepa/frontier.php), although fairly outdated by now (see also https://cran.r-project.org/web/packages/frontier/frontier.pdf).

  17. 17.

    One can install these commands via net install sfcross, all from (http://www.econometrics.it/stata) net install sfpanel, all from (http://www.econometrics.it/stata), see also https://sites.google.com/site/productivityefficiency/home1 and for details refer to Chapter 17 of [93]

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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. Inland Norway University of Applied Sciences, Lillehammer, Norway

    Subal C. Kumbhakar

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

    Christopher F. Parmeter

  4. School of Economics and Centre for Efficiency and Productivity Analysis (CEPA), The University of Queensland, Brisbane, QLD, Australia

    Valentin Zelenyuk

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  1. Subal C. Kumbhakar
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Correspondence to 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 G. Chambers

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

    Subal C. Kumbhakar

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Kumbhakar, S.C., Parmeter, C.F., Zelenyuk, V. (2022). Stochastic Frontier Analysis: Foundations and Advances II. In: Ray, S.C., Chambers, R.G., Kumbhakar, S.C. (eds) Handbook of Production Economics. Springer, Singapore. https://doi.org/10.1007/978-981-10-3455-8_11

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