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Expected efficiency ranks from parametric stochastic frontier models

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Abstract

In the stochastic frontier model, we extend the multivariate probability statements of Horrace (J Econom, 126:335–354, 2005) to calculate the conditional probability that a firm is any particular efficiency rank in the sample. From this, we construct the conditional expected efficiency rank for each firm. Compared to the traditional ranked efficiency point estimates, firm-level conditional expected ranks are more informative about the degree of uncertainty of the ranking. The conditional expected ranks may be useful for empiricists. A Monte Carlo study and an empirical example are provided.

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Notes

  1. A fixed effect model is also considered in Schmidt and Sickles (1984).

  2. However, our results, the results of Flores-Lagunes et al. (2007) and the results of Horrace (2005) are easily generalizable to any parametric stochastic frontier model that produces a conditional inefficiency distribution of known form (based on assumptions on the error components).

  3. See, for example, Sect. 5.2.2 of Reingold et al. (1977) for an efficient algorithm that serves to find \(N_i^{l-}(r)\) and \(N_i^{l+}(r)\) for \(l=1,\ldots ,{}_{n-1}C_{r-1}\).

  4. One could supplement the conditional means with the conditional prediction intervals of Horrace and Schmidt (1996), to judge how much the marginal distributions overlap. The degree of overlap may correspond to the extent to which the conditional probabilities and expected ranks are close to their unconditional counterparts, but this might be highly subjective and (perhaps) lead to an inaccurate assessment of the nature of efficiency in the population.

  5. Feng and Horrace are only concerned with detecting the best firm. We want to detect the rank of all firms.

  6. The nomenclature “mostly stars and dogs” is due to Almanidis et al. (2014).

  7. The skew of a truncated normal is necessarily positive. We use the “standardized” definition of skew where the 3rd central moment is divided by the third power of the standard deviation.

  8. Again, the probabilities in (2) could be easily simulated for large \(n\), but for the purposes of illustration, small \(n\) is sufficient.

  9. These results are consistent with the Feng and Horrace (2012).

  10. We thank an anonymous referee for these insights into the information contained in the quantiles, which may imply that a resampling method, such as bootstrap, may provide better estimates of expected rank.

  11. The results of the estimation are not reproduced here to focus attention on the different characterization of efficiency ranks and the importance of the proposed conditional expected rank statistic.

  12. The expected ranks are calculated using the conditional efficiency rank probabilities for only these five vessels in each tier. In particular we did not calculate the efficiency rank probabilities for all vessels, and calculate expected rank only for these five, based on the probabilities from all vessels.

  13. When conditional expected ranks are based on simulated probabilities, the simulation sample size is always 5,000.

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Acknowledgments

The authors are grateful to Peng Liu for excellent research assistance and to several anonymous referees for their suggestions.

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Authors and Affiliations

  1. Department of Economics, Syracuse University, Syracuse, NY , 13244, USA

    William C. Horrace & Ian A. Wright

  2. H. John Heinz III College, School of Public Policy and Management, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA , 15213, USA

    Seth Richards-Shubik

  3. NBER, Cambridge, MA , USA

    Seth Richards-Shubik

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  1. William C. Horrace
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  2. Seth Richards-Shubik
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  3. Ian A. Wright
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Correspondence to Seth Richards-Shubik.

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Horrace, W.C., Richards-Shubik, S. & Wright, I.A. Expected efficiency ranks from parametric stochastic frontier models. Empir Econ 48, 829–848 (2015). https://doi.org/10.1007/s00181-014-0808-8

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