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Stochastic vs. deterministic frontier distance output function: Evidence from Brazilian higher education institutions

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Using data from the Brazilian Higher Education Census and other public institutions, this study aims to obtain and compare efficiency scores from stochastic frontier analysis (SFA) and data envelopment analysis (DEA) models for 56 Brazilian federal universities for the period of 2010 to 2016. The output distance function includes financial and human resources as inputs, and teaching, research, patents and third mission activities as outputs. The research is innovative considering: (i) the estimation of SFA for Brazilian universities as whole institutions, (ii) its comparison with DEA; and (iii) the inclusion of patents and third mission variables. The findings suggest there is inefficiency in Brazilian higher education production, with a very small increase through time and with some influence from universities and environmental characteristics. Thus, consolidated traditional institutions with university hospitals tend to be more efficient than the younger ones. The values and the rank of the efficiencies are sensitive to the model/method employed, presenting highly significant although modest correlations. In general, the inclusion of third mission activities improves the efficiencies for both approaches, mainly for DEA. Hence, as advised in other international comparative analyses, caution is required when deriving management and policy recommendations from the analytical results.

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  1. Until 2016 there were 7 other universities, 4 completely new ones, and 3 others created by disaggregation. The new ones were: UFFS (in 2009), UNILA (in 2010), UNILAB (in 2010) and UFESBA (in 2013). The disaggregated ones were UFCA (in 2013, originally from UFC), UFOB (in 2013, originally from UFBA) and UNIFESSPA (in 2013, originally from UFPA).

  2. According to the recommendation of TCU (2000), the problem is how to disentangle the share of expenses incurred in the university hospitals that corresponds to teaching and research activities, in addition to the regular services provided by these hospitals. Amaral (1998 apud TCU 2000) considers that 35% of the expenses of the university hospitals can be appropriated in the cost of teaching. According to Silva et al. (2004), this proportion of 35% seems to be based on Jones and Korn (1997).

  3. Measured in R$ of year 2000, deflated by the GDP implicit index.

  4. It is important to clarify (or remember) that the Brazilian regions are very heterogeneous regarding their natural, social and economic characteristics.

  5. Decision Making Unit (DMU) in this context is a synonymous to HEI, or University.

  6. It could be considered the standard primal problem in contemporary DEA literature using VRS model and output orientation (Forsund 2018, p. 4; Thanassoulis et al. 2011, p. 1297)

  7. These procedures were developed focusing in solving specifically DEA limitations regarding outlier DMU(s).

  8. The Cobb-Douglas functional form was tested versus the complete Translog functional form (the former is nested in the latter). According to the LR test, the Translog model has a better fit.

  9. According to Coelli et al. (2005) and O’Donnell (2014), cited by Johnes (2013), endogeneity could exist, caused by correlation of the explanatory variable and the error term (eit = vi + ui). However, Coelli and Perelman (2000, apud Johnes 2013, p. 5) argue that this “bias is not a problem in an output distance function which [as here] uses a translog functional form”.

  10. PROFES presented better results than PROFEQ according to AIC and BIC criteria.

  11. Some of the recent federal universities present different years of creation and federalization, that is, they were created originally as other type of institution and worked for a time until the federal government ‘federalize’ them, that is, they became managed by the federal government. Then, both ‘year of creation’ and ‘year of federalization’ were tested in the models and only the latter presents statistical significance in the models.

  12. The sensitivity of the results was checked by using the other outputs as numeraire; results then showed the insensitivity of the change, as expected, according to Coelli and Perelman (2000) and Johnes (2013).

  13. To estimate ûit of uit, the largely used strategy is to look at the conditional distribution of uit given eit and use the conditional expectation EV(uit | eit) as an estimator of uit. The details of this procedure, following the seminal work of Jondrow et al. (1982, p. 238) and Battese and Coelli (1988, p. 392), are described and commented with details in Bogetoft and Otto (2011, pp. 217–219).

  14. Battese and Coelli (1995, p. 327) argue this assumption is a simplifying, but restrictive, condition but Gómez and Pérez (2017) found that the consideration of independent error terms results in overestimated cost efficiencies in a general magnitude lower than 5%. Because of this lower value and the novelty of this work to the Brazilian case, in the present research we chose to consider the ‘classical’ assumption of independent and identically distributed error terms to all models estimated.

  15. Johnes and Schwarzenberger (2011, p. 498) pertinently observe that in empirical applications, when sigma and gamma terms are statistically different from zero, it suggests an appropriate approach in relation to efficiency distribution.

  16. The log-likelihood function of this model is presented in the appendix of Battese and Coelli (1992).

  17. The log-likelihood function of this model is presented in the appendix of Battese and Coelli (1993).

  18. In an attempt to choose the final specification of the BC92 models, first we estimated eight models considering different outputs and only CCCHU as input. Then, the same eight models were estimated considering only PROFEQ and FUNCEQSHU as inputs and, also, the same eight models were estimated considering the three inputs. Then the results were compared by using the LR test and all models with the three inputs presented the best fit. Only then, we compared these eight models with three inputs and different outputs among themselves, and we selected the two models with the best fit.

  19. Regarding the estimation method and the software used, all DEA procedures were done using the R (2017) and the package ‘Benchmarking’ developed by Bogetoft and Otto (2018). The package ‘FEAR’ (Wilson 2008) was used to apply the Wilson (1993)’s procedures to identify potential HEI outliers to DEA analysis. The SFA estimations were done by maximum likelihood estimation using the package ‘frontier’ developed by Coelli and Henningsen (2017). It is an R version of the classical FRONTIER 4.1 software developed by Tim Coelli and presented in Coelli (1996).

  20. Only if considering the model with THIRDM.


  • Abbott M, Doucouliagos C (2009) Competition and efficiency: Overseas students and technical efficiency in Australian and New Zealand universities. Edu Econ 17:31–57

    Article  Google Scholar 

  • Agasisti T, Barra C, Zotti R (2016) Evaluating the efficiency of Italian public universities (2008-2011) in presence of (unobserved) heterogeneity. Socio-Econ Planning Sci 55:47–58

    Article  Google Scholar 

  • Aigner DJ, Lovell CAK, Schmidt P (1977) Formulation and estimation of stochastic frontier production models. J Econom 6:21–37

    Article  Google Scholar 

  • Aleskerov FT, Belousova VY, Petrushchenko VV (2017) Models of data envelopment analysis and stochastic frontier analysis in the efficiency assessment of universities. Automation Remote Control 78:902–923

    Article  Google Scholar 

  • Andrews DF, Pregibon D (1978) Finding the outliers that matter. J Royal Statistical Soc 40:85–93

    Google Scholar 

  • Battese GE, Coelli TJ (1988) Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data. J Econom 38:387–399

    Article  Google Scholar 

  • Battese GE, Coelli TJ (1992) Frontier production functions, technical efficiency and panel data: with application to paddy farmers in India. J Prod Anal 3:153–169

    Article  Google Scholar 

  • Battese GE, Coelli TJ (1993) A stochastic frontier production function incorporating a model for technical inefficiency effects. Department of Econometrics. University of New England. Working papers in Econometrics and Applied Estatistics, n. 69.

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

    Article  Google Scholar 

  • Blanchard O (2004) The economic future of Europe, National Bureau of Economic Research, NBER Working Paper, n. 10310.

  • Bogetoft P, Otto L (2011) Benchmarking with DEA, SFA, and R. Springer, New York, NY

    Book  Google Scholar 

  • Bogetoft P, Otto L (2018) Benchmarking with DEA and SFA, R package version 0.27.

  • CAPES (2018) GEOCAPES – Sistema de Informaçãoes Georreferenciadas.

  • Castano M, Cabanda E (2007) Performance evaluation of the efficiency of Philippine Private Higher Educational Institutions: application of frontier approaches. Int Transactions Operational Res 14:431–444

    Article  Google Scholar 

  • Chapple W, Lockett A, Siegel D, Wright M (2005) Assessing the relative performance of U.K. university technology transfer offices: parametric and non-parametric evidence. Res Policy 34:369–384

    Article  Google Scholar 

  • Coelli T (1996) A Guide to FRONTIER Version 4.1: A Computer Program for Stochastic Frontier Production and Cost Function Estimation, Working Paper. Centre for Efficiency and Productivity Analysis (CEPA), Department of Econometrics, University of New England, Armidale, Australia, p n. 7/96

    Google Scholar 

  • Coelli T, Henningsen A (2017) frontier: Stochastic Frontier Analysis. R package version 1.1-2

  • Coelli T, Perelman S (2000) Technical efficiency of European railways: a distance function approach. Appl Econ 32:1967–1976

    Article  Google Scholar 

  • Coelli T, Rao DSP, O’Donnel CJ, Battese GE (2005) An introduction to efficiency and productivity analysis, 2nd edn. Springer, New York, NY

    Google Scholar 

  • Farrel M (1957) The measurement of productive efficiency. J Royal Statistical Soc 120:253–281

    Article  Google Scholar 

  • Forsund FR (2018) Economic interpretation of DEA. Socio-Econ Planning Sci 61:9–15

    Article  Google Scholar 

  • Forsund FR, Lovell CA, Schmidt PA (1980) A survey of frontier production functions and of their relationship to efficiency measurement. J Econom 13:5–25

    Article  Google Scholar 

  • Gómez-Déniz E, Pérez-Rodriguez JV (2017) Stochastic frontier models with dependent errors based on normal and exponential margins. Revista de Métodos Quantitativos para la Economía y la Empresa 23:3–23

    Google Scholar 

  • Gralka S (2018) Stochastic Frontier Analysis in higher education: a systematic review. Technische Universitat Dresden. Center of Public and International Economics – CEPIE. Working Paper, n. 5.

  • Henningsen A (2018) Introduction to econometric production analysis with R. Collection of Lecture Notes. 2nd Draft Version. Department of Food and Resource Economics, University of Copenhagen.

  • IBGE–Instituto Brasileiro de Geografia e Estatística (2010) Censo Demográfico 2010. Rio de Janeiro.

  • INEP–Instituto Nacional de Pesquisas Educacionais Anísio Teixeira (2017) Indicadores Financeiros Educacionais.

  • INEP–Instituto Nacional de Pesquisas Educacionais Anísio Teixeira (2018). Censo da Educação Superior. Microdados.

  • INPI-Instituto Nacional de Propriedade Industrial (2018). Estatística. Indicadores de Propriedade Industrial. Dowload das tabelas completas.

  • Jones RF, Korn D (1997) On the cost of educating a medical student. Acad Med 72:200–210

    Article  Google Scholar 

  • Johnes J (2004) Efficiency measurement. In: Johnes G, Johnes J eds International Handbook on the Economics of Education. Edward Elgar Publishing, Cheltenham-UK, p 613–742

    Google Scholar 

  • Johnes J (2006) Data envelopment analysis and its application to the measurement of efficiency in higher education. Econ Educ Rev 25:273–288

    Article  Google Scholar 

  • Johnes J (2013) Efficiency and mergers in English higher education 1996/97 to 2008/09: parametric and non-parametric estimation of the multi-input multi-output distance function. Department of Economics, Discussion Papers, Lancaster University, The Manchester School, v. 84, n. 4, 465-487.

  • Johnes J, Johnes G (2013) Efficiency in the higher education sector: a technical exploration. BIS Research Paper, n. 113.

  • Johnes G, Schwarzenberger A (2011) Differences in cost structure and the evaluation of efficiency: the case of German universities. Edu Econ 19:487–499

    Article  Google Scholar 

  • Jondrow J, Lovel CAK, Materov IS, Schmidt P (1982) On the estimation of technical inefficiency in the stochastic frontier production function model. J Econom 19:233–238

    Article  Google Scholar 

  • Katharakisa G, Katostaras T (2013) SFA vs. DEA for measuring helthcare efficiency: a systematic review. Int J Statistics Medical Res 2:152–166

    Google Scholar 

  • Kempkes G, Pohl C (2010) The efficiency of German universities – some evidence from nonparametric and parametric methods. Appl Econ 42:2063–2079

    Article  Google Scholar 

  • Letti AG, Bittencourt MVL, Vila LE (2018a) The relative efficiency of Brazilian Public Universities (2010-2016): an analysis through time and space. 49th European Regional Science Association (ERSA) Congress, Cork, Ireland

    Google Scholar 

  • Letti AG, Bittencourt MVL, Vila LE (2018b) Searching for the lost efficiency: A review about Brazilian university performance evaluation. 18th Colóquio Internacional de Gestão Universitária (CIGU), Loja, Equador

    Google Scholar 

  • Lindsay AW (1982) Institutional performance in higher education: The efficiency dimension. Rev Educational Res 52:175–199

    Article  Google Scholar 

  • Mcmillan M, Chan W (2006) University efficiency: a comparison and consolidation of results from stochastic and non-stochastic methods. Edu Econ 14:1–30

    Article  Google Scholar 

  • Miranda R, Gramani MC, Andrade E (2012) Technical efficiency of business administration courses: A simultaneous analysis using DEA and SFA. Int Transactions Operational Res 19:847–862

    Article  Google Scholar 

  • O’Donnel CJ (2014) Econometric estimation of distance functions and associated measures of productivity and efficiency change. J Prod Anal 41:187–200

    Article  Google Scholar 

  • OECD (2015) Education at a Glance. OECD Indicators. OECD Publishing,

    Google Scholar 

  • R (2017) R Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.

  • Salerno C (2003) What we know about the efficiency of higher education institutions: The best evidence. Zoetermeer, Ministry of Education, Culture and Science, Netherlands, p 2003

    Google Scholar 

  • Sampaio CEM (2017) Diretoria de Estatísticas Educacionais.

  • SESu/MEC (2018) Orientações para o cálculo dos indicadores de gestão. Decisão TCU n. 408/2002-Plenário. Versão revisada em março/2004.

  • Shepard RW (1970) Theory of cost and production functions. Princeton University Press, New Jersey

    Google Scholar 

  • Siegel DS, Wright M, Chapple W, Locket A (2008) Assessing the relative performance of university technology transfer in the US and UK: a stochastic distance function approach. Econ Innov New Techn 18:717–729

    Article  Google Scholar 

  • Silva CAT, Costa PDS, Morgan BF (2004) Aplicabilidade das informações de custo em hospitais universitários: o caso do hospital universitário de Brasília. XI Congresso Brasileiro de Custos, Porto Seguro, BA, Brasil

    Google Scholar 

  • Stevens (2004) Accounting for Background Variables in Stochastic Frontier Analysis, National Institute of Economic and Social Research (NIESR), Discussion Papers 239

  • Stevens PA (2005) A stochastic frontier analysis of English and Welsh universities. Edu Econ 13:355–374

    Article  Google Scholar 

  • Tabak BM, Cajueiro DO, Dias MVB (2014a) The efficiency of Chinese local banks: A comparison of DEA and SFA. Banco Central do Brasil, Working Papers Series, n. 346, 1–15.

  • Tabak BM, Cajueiro DO, Dias MVB (2014b) The adequacy of deterministic and parametric frontiers to analyze the efficiency of Indian commercial banks. Banco Central do Brasil, Working Papers Series, n. 350, 1–19.

  • Thanassoulis E, Kortelainen M, Johnes G, Johnes J (2011) Cost and efficiency of higher education institutions in England: a DEA analysis. J Operational Res Soc 62:1282–1297

    Article  Google Scholar 

  • TCU–Tribunal de Contas da União (2000) Decisão n. 358/2000 – Plenário. Relatório de Acompanhamento (RACOM). Relatório de Auditoria, Brasília,

  • TCU–Tribunal de Contas da União (2002) Decisão n. 408/2002 – Plenário. Relatório de Acompanhamento (RACOM). Relatório consolidado de Auditoria Operacional. Brasília.

  • TCU–Tribunal de Contas da União (2010) Portaria TCU n.° 277, de 7 de dezembro de 2010. PARTE C, ITEM 7, DO ANEXO II.

  • TCU–Tribunal de Contas da União (2018a) Contas e relatórios de gestão. Consultar relatórios de gestão a partir de 2014.

  • TCU–Tribunal de Contas da União (2018b). Contas e relatórios de gestão. Consultar relatórios de gestão de 2008 a 2013.

  • Thursby JG, Kemp S (2002) Growth and productive efficiency of university intellectual property licensing. Res Policy 31:109–124

    Article  Google Scholar 

  • Vila LE (2000) The non-monetary benefits of education. Eur J Edu 35:21–32

    Article  Google Scholar 

  • Wang HJ, Ho CW (2010) Estimating fixed-effect panel stochastic frontier models by model transformation. J Econom 157:286–296

    Article  Google Scholar 

  • Wilson PW (1993) Detecting outliers in deterministic nonparametric frontier models with multiple outputs. J Business Econ Statistics 11:319–323

    Google Scholar 

  • Wilson PW (2010) Detecting outliers in deterministic nonparametric frontier models with multiple outputs: correction. Unpublished working paper, Department of Economics, Clemson University. 2010.

  • Wilson PW (2008) FEAR: A software package for frontier efficiency analysis with R. Socio-Economic Planning. Sciences 42:247–254

    Google Scholar 

  • Worthington A (2001) An empirical survey of frontier efficiency measurement techniques in education. Edu Econ 9:245–265

    Article  Google Scholar 

  • Zoghbi A, Rocha F, Mattos E (2013) Education production efficiency: evidence from Brazilian universities. Econ Modelling 31:94–103

    Article  Google Scholar 

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This work received a grant by CAPES, Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brazil, from Sep 2017 to Aug 2018; and it was also supported by CNPq, Conselho Nacional de Desenvolvimento Científico e Tecnológico (grant 312369/2018-2).

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Letti, A.G., Bittencourt, M.V.L. & Vila, L.E. Stochastic vs. deterministic frontier distance output function: Evidence from Brazilian higher education institutions. J Prod Anal 58, 55–74 (2022).

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