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Categorical and continuous non-discretionary variables in data envelopment analysis: a comparison of two single-stage models

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Abstract

Non-discretionary or environmental variables are regarded as important in the evaluation of efficiency in Data Envelopment Analysis (DEA), but there is no consensus on the correct treatment of these variables. This paper compares the performance of the standard BCC model as a base case with two single-stage models: the Banker and Morey (1986a) model, which incorporates continuous environmental variables and the Banker and Morey (1986b) model, which incorporates categorical environmental variables. Simulation analyses are conducted using a shifted Cobb-Douglas function, with one output, one non-discretionary input, and two discretionary inputs. The production function is constructed to separate environmental impact from managerial inefficiency, while providing measures of both for comparative purposes. Tests are performed to evaluate the accuracy of each model. The distribution of the inputs, the sample size and the number of categories for the categorical model are varied in the simulations to determine their impact on the performance of each model. The results show that the Banker and Morey models should be used in preference to the standard BCC model when the environmental impact is moderate to high. Both the continuous and categorical models perform equally well but the latter may be better suited to some applications with larger sample sizes. Even when the environmental impact is slight, the use of a simple two-way split of the sample data can produce significantly better results under the Categorical model in comparison to the BCC model.

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Notes

  1. While nondiscretionary inputs occur most of the time, there are situations where nondiscretionary outputs can arise. These are not addressed in this paper. We use the terms nondiscretionary and environmental interchangeably in this paper.

  2. Førsund (2002) describes an approach where the categorical variables are not ordered hierarchically, i.e., they are not ordinal. In our study, any variable described as ‘categorical’ is ordinal.

  3. In the Continuous Model the measure of efficiency θ is confined to the constraints relating to the discretionary inputs. \( \sum\nolimits_{n = 1}^{N} {\lambda_{n} X_{pm} \le \theta X_{po} } ,\quad p = 1, \ldots ,P \).

  4. Again with reference to the Continuous Model, there is no efficiency θ variable in the constraints for the non-discretionary factors. \( \sum\nolimits_{n = 1}^{N} {\lambda_{n} Z_{rn} \le Z_{rn} } ,\quad r = 1, \ldots ,R \).

  5. It is not obvious how they support this statement in their paper, although in the introduction they refer to a search using www.isknowledge.com to identify the number of citations to the two Banker and Morey papers (1986a, b). Our own search for applications of the categorical model using the Scopus Database identified 11 empirical applications of the Categorical Model compared to 39 applications of the Continuous Model.

  6. ‘True’ efficiency as determined by an artificial data set when the production function is known.

  7. Continuous variables can be given ordinal characteristics through using weight restrictions on factors to force such behaviour (Golany 1988).

  8. It is important to note that these measures respond differently to changes in sample size and the number of inputs. Measures (iii) and (iv) can cope with an increase in the number of inputs so long as sample size also increases. Measure (i) can cope with large increases in dimensionality with only relatively small increases in sample size. The reverse is true for measure (ii) (Pedraja-Chaparro et al. 1999).

  9. Perelman and Santín (2009) hold that Cobb-Douglas technology does not allow for the evaluation of performance with respect to scale.

  10. As an example of this type of effect in real world situations, Harrison (2011) notes that high schools operating in unfavourable socio-economic environments have lower academic outputs.

  11. We are assuming that the nondiscretionary variable always contributes to output but does so in varying degrees e.g., rainfall, soil types or hours of sunshine in a farming context. In other contexts output may range from being affected detrimentally to beneficially which could be modelled by using a mean of one and some appropriate standard deviation.

  12. For example, Banker et al. (1989) recommend as a rule of thumb the number of DMUs should be equal to at least three times the sum of the inputs and outputs.

  13. We also examined the performance of the BCC model relative to ‘observed efficiency’, which combines environmental impact and managerial efficiency. The BCC model performed better in relation to this measure suggesting that this model captures the combined effect. Future research will examine how these combined effects could be separately identified using multi-stage models.

  14. The z test of differences in proportions could not be used where the proportion exceeded unity. This is denoted by a superscript “+”.

References

  • Asava-vallobh N, Gronberg TJ, Jansen DW (2008) An alternative DEA methodology for non-controllable (environmental) inputs. Working paper: Texas A&M University

  • Banker RD (1992) Selection of efficiency evaluation models. Contemp Acc Res 9(1):343–355

    Article  Google Scholar 

  • Banker RD, Chang H (2006) The super-efficiency procedure for outlier identification, not for ranking efficient units. Eur J Oper Res 175:1311–1320

    Article  Google Scholar 

  • Banker RD, Morey RC (1986a) Efficiency analysis for exogenously fixed inputs and outputs. Oper Res 34(4):513–521

    Article  Google Scholar 

  • Banker RD, Morey RC (1986b) The use of categorical variables in data envelopment analysis. Manag Sci 32(12):1613–1627

    Article  Google Scholar 

  • Banker RD, Charnes A, Cooper WW (1984) Models for the estimation of technical and scale inefficiencies in data envelopment analysis. Manag Sci 30(9):1078–1092

    Article  Google Scholar 

  • Banker RD, Charnes A, Cooper WW, Swarts J, Thomas DA (1989) An introduction to data envelopment analysis with some of its models and their uses. Res Gov Nonprofit Acc 5:125–163

    Google Scholar 

  • Barr R (2004) DEA software tools and technology. A state-of-the-art survey. In: Cooper WW, Seiford L, Zhu J (eds) Handbook on data envelopment analysis. Kluwer, Boston

    Google Scholar 

  • Basso A, Funari S (2003) Measuring the performance of ethical mutual funds: a DEA approach. J Oper Res Soc 54:521–531

    Article  Google Scholar 

  • Battese GE, Coelli TJ (1995) A model of technical inefficiency effects in a stochastic frontier production function for panel data. Empir Econ 20:325–332

    Article  Google Scholar 

  • Bhattacharya A, Lovell CAK, Sahay P (1997) The impact of liberalisation on the productive efficiency of Indian commercial banks. Eur J Oper Res 98(2):332–347

    Article  Google Scholar 

  • Charnes A, Cooper WW (1985) Preface to topics in data envelopment analysis. Ann Oper Res 2:59–94

    Article  Google Scholar 

  • Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision making units. Eur J Oper Res 2:429–444

    Article  Google Scholar 

  • Christensen LR, Jorgenson DW, Lau LJ (1971) Conjugate duality and the transcendental logarithmic production function. Econometrica 39(2):255–256

    Google Scholar 

  • Cook WD, Kress M, Seiford LM (1996) Data envelopment analysis in the presence of both quantitative and qualitative factors. J Oper Res Soc 47(7):945–953

    Google Scholar 

  • Cordero JM, Pedraja F, Santín D (2009) Alternative approaches to include exogenous variables in DEA measures: a comparison using Monte Carlo. Comput Oper Res 36:2699–2706

    Article  Google Scholar 

  • Daraio C, Simar L (2005) Introducing environmental variables in nonparametric frontier models: a probabilistic approach. J Prod Anal 24:93–121

    Article  Google Scholar 

  • Diewert E (1993) Data envelopment analysis: a practical alternative? In: Working paper presented at Swan Consultants (Canberra) Pty Ltd conference

  • Førsund FR (2002) Categorical variables in DEA. Int J Bus Econ 1(1):33–43

    Google Scholar 

  • Fried HO, Lovell CAK (1996) Searching the Zeds. In: Working paper presented at II Georgia productivity workshop

  • Fried HO, Schmidt SS, Yaisawarng S (1999) Incorporating the operating environment into a nonparametric measure of technical efficiency. J Prod Anal 12:249–267

    Article  Google Scholar 

  • Fried H, Lovell CAK, Schmidt S, Yaisawarng S (2002) Accounting for environmental effects and statistical noise in data envelopment analysis. J Prod Anal 17(1/2):157–174

    Article  Google Scholar 

  • Golany B (1988) A note on including ordinal relations among multipliers in data envelopment analysis. Manag Sci 34(8):1029–1033

    Article  Google Scholar 

  • Haas DA, Murphy FH (2003) Compensating for non-homogeneity in decision-making units in data envelopment analysis. Eur J Oper Res 144:530–544

    Article  Google Scholar 

  • Harrison JA (2011) Results or processes? A triangulation of alternative approaches to school performance measurement. Working paper: The University of Auckland

  • Kumbhakar SC, Lovell CAK (2000) Stochastic frontier analysis. Cambridge University Press, Cambridge

    Google Scholar 

  • Löber G, Staat M (2010) Integrating categorical variables in data envelopment analysis models: a simple solution technique. Eur J Oper Res 202:810–818

    Article  Google Scholar 

  • McCarty TA, Yaisawarng S (1993) Technical efficiency in New Jersey school districts. In: Fried H, Lovell CAK, Schmidt SS (eds) The measurement of productive efficiency: techniques and applications. Oxford University Press, Oxford, pp 271–287

    Google Scholar 

  • Mehrez A, Brown JR, Khouja M (1992) Aggregate efficiency measures and Simpson’s paradox. Contemp Acc Res 9(1):329–342

    Article  Google Scholar 

  • Muñiz M (2002) Separating managerial inefficiency and external conditions in data envelopment analysis. Eur J Oper Res 143:625–643

    Article  Google Scholar 

  • Muñiz M, Paradi J, Ruggiero J, Yang Z (2006) Evaluating alternative data DEA models used to control for non-discretionary inputs. Comput Oper Res 33(5):1173–1183

    Article  Google Scholar 

  • Orme C, Smith P (1996) The potential for endogeneity bias in data envelopment analysis. J Oper Res Soc 47(1):73–83

    Google Scholar 

  • Pastor JT (1994) How to discount environmental effects in DEA: an application to bank branches. Working Paper: Instituto Valenciano de Investigaciones Economicas

  • Pedraja-Chaparro F, Salinas-Jiménez J, Smith P (1999) On the quality of data envelopment analysis. J Oper Res Soc 50:636–644

    Google Scholar 

  • Perelman S, Santín D (2009) How to generate regularly behaved production data? Monte Carlo experimentation on DEA scale efficiency measurement. Eur J Oper Res 199:303–310

    Article  Google Scholar 

  • Ray SC (1988) Data envelopment analysis, nondiscretionary inputs and efficiency: an alternative interpretation. Soc-Econ Plan Sci 22(4):167–176

    Article  Google Scholar 

  • Ray S (1991) Resource use efficiency in public schools. A study of Connecticut data. Manag Sci 37:1620–1628

    Article  Google Scholar 

  • Rouse P, Putterill M, Ryan D (1997) Towards a general managerial framework for performance measurement: a comprehensive highway maintenance application’. J Prod Anal 8(2):127–149

    Article  Google Scholar 

  • Ruggiero J (1996) On the measurement of technical efficiency in the public sector. Eur J Oper Res 90(3):553–566

    Article  Google Scholar 

  • Ruggiero J (1998) Non-discretionary inputs in data envelopment analysis. Eur J Oper Res 111:461–469

    Article  Google Scholar 

  • Ruggiero J (2004) Performance evaluation when non-discretionary factors correlate with technical efficiency. Eur J Oper Res 159:250–257

    Article  Google Scholar 

  • Staat M (2001) The effect of sample size on the mean efficiency in DEA: comment. J Prod Anal 15:129–137

    Article  Google Scholar 

  • Steiger JH (1980) Test for comparing elements of a correlation matrix. Psychol Bull 87:245–251

    Article  Google Scholar 

  • Syrjänen MJ (2004) Non-discretionary and discretionary factors and scale in data envelopment analysis. Eur J Oper Res 158:20–30

    Article  Google Scholar 

  • Timmer CP (1971) Using a probabilistic frontier production function to measure technical efficiency. J Pol Econ 79(4):776–794

    Article  Google Scholar 

  • Yang Z, Paradi JC (2006) Cross firm bank branch benchmarking using “handicapped” data envelopment analysis to adjust for corporate strategic effects. In: Proceedings of the 39th Hawaii international conference on system sciences, pp 1–10

  • Yu C (1998) The effects of exogenous variables in efficiency measurement: a Monte Carlo study. Eur J Oper Res 105:569–580

    Article  Google Scholar 

  • Zhang Y, Bartels R (1998) The effect of sample size on the mean efficiency in DEA with an application to electricity distribution in Australia, Sweden and New Zealand. J Prod Anal 9:187–204

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Shawna Grosskopf and other participants at the Efficiency and Productivity Workshop, 20th Meeting of the New Zealand Econometric Study Group, 2010, Auckland New Zealand, for their helpful comments. We thank Robin Sickles for suggesting at this conference that we take a closer look at the relationship between environmental variables and inputs/outputs. We thank Matt Regan, Department of Statistics, The University of Auckland for his assistance. We also thank two anonymous reviewers for their helpful comments and insights. Jamie Armstrong wishes to acknowledge the financial assistance of The University of Auckland TREPA Scholarship.

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  1. Department of Accounting and Finance, The University of Auckland Business School, Private Bag 92019, Auckland, New Zealand

    Julie Harrison, Paul Rouse & Jamie Armstrong

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  1. Julie Harrison
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Correspondence to Julie Harrison.

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Harrison, J., Rouse, P. & Armstrong, J. Categorical and continuous non-discretionary variables in data envelopment analysis: a comparison of two single-stage models. J Prod Anal 37, 261–276 (2012). https://doi.org/10.1007/s11123-011-0239-x

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