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Stochastic non-smooth envelopment of data for multi-dimensional output

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Abstract

The proposed method of Stochastic Non-smooth Envelopment of Data (StoNED) for measuring efficiency has to date mainly found application in the analysis of production systems which have exactly one output. Therefore, the objective of this paper is to examine the applicability of StoNED when a ray production function models a production technology with multi-dimensional input and output. In addition to a general analysis of properties required by a ray production function for StoNED to be applicable, we conduct a Monte Carlo simulation in order to evaluate the quality of the frontier and efficiencies estimated by StoNED. The results are compared with those derived via Stochastic Frontier Analysis (SFA) and Data Envelopment Analysis (DEA). We show that StoNED provides competitive estimates in regard to other methods and especially in regard to the real functional form and efficiency.

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Notes

  1. A detailed explanation of the basic properties of directional distance functions can be found in Kuosmanen et al. (2015b).

  2. Until now, there has been little scientific investigation of the differences that exist between modeling multiple outputs with a directional distant function or a ray production function. For instance, Henningsen et al. (2015) present a Monte Carlo simulation that investigates the qualitative differences between modeling multi-dimensional output with an output distance function and with a ray production function when measuring the efficiency by SFA. In the scenarios considered by their analysis, neither distance function nor ray production function could be shown to be superior to the other.

  3. But other assumptions are necessary. For example, discontinuity at zero for the ineffciency distribution is needed and the variance of the noise term has to converge to zero if the sample size goes to infinity.

  4. The distributional assumptions could be modified, for example, by implying an exponential distribution for the inefficiency term (see Kumbhakar and Lovell 2003).

  5. We used GAMS for the implementation of the simulation design. The code is available upon request from us.

  6. The three technologies used here are consistent with the convexity and free disposability axioms. To check this, we first performed a pre-simulation to generate data without inefficiency and noise. These data were used to show that StoNED and DEA BCC estimate both the frontier and efficiencies exactly and thus that each technology satisfies the required properties.

  7. Since the stochastic methods StoNED and SFA minimize the sum of squared errors, these methods should perform better on the MSE criteria than on the MAD criteria. If we regard the following analyses concerning the MSE criteria the measures are indeed better than the MAD measures. However, the following general statements are also valid when regarding MSE measures. Therefore, we abstain from presenting the MSE results in detail.

  8. When additionally taking the MD into account, the deviations are almost zero. This shows that under- and overestimations of the true frontier mutually balance each other out. StoNED and SFA CD tend to slightly underestimate the frontier in scenarios without noise, and slightly overestimate them in scenarios with noise. Also the MSE values are small, which suggests that the stochastic methods tend to generate many small deviations but no large ones.

  9. Additionally, the MSE shows that DEA tend to generate high deviations of single DMUs (see Table 11), which is plausible considering the deterministic nature of DEA.

  10. In 43 of 60 scenarios SFA TL is ranked on the first place.

  11. The MAD results of the efficiency estimation are shown in Tables 12 and 13 in the appendix.

  12. However, this is only valid if the assumptions required for pseudolikelihood estimations are fulfilled. The method of moments is likely more robust to violations of these assumptions.

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Acknowledgements

The authors would like to express their gratitude to the editor and two anonymous reviewers for their constructive comments which contributed to the improvement of the paper. Additionally, the authors would like to thank Sebastian Gutgesell for his work on a former version of this paper.

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Authors and Affiliations

  1. School of Business and Economics, Chair of Business Theory, RWTH Aachen University, Templergraben 64, 52056, Aachen, Germany

    Julia Schaefer

  2. Duale Hochschule Gera-Eisenach, Am Wartenberg 2, 99817, Eisenach, Germany

    Marcel Clermont

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  1. Julia Schaefer
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Correspondence to Marcel Clermont.

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Appendix

Appendix

Tables 9–14

Table 9 Overview of the assessment criteria for all settings
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Table 10 Overview of the assessment criteria for all settings without noise
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Table 11 Overview of the assessment criteria for all settings with noise
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Table 12 Variation of sample size in regard to mean absolute deviation and the efficiency
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Table 13 Variation of noise-to-signal-ratio in regard to mean absolute deviation and the efficiency
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Table 14 Variation of functional form in regard to mean absolute deviation and the efficiency
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Schaefer, J., Clermont, M. Stochastic non-smooth envelopment of data for multi-dimensional output. J Prod Anal 50, 139–154 (2018). https://doi.org/10.1007/s11123-018-0539-5

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