Inferência estatística dos estimadores de eficiência obtidos com a técnica fronteira não paramétrica de DEA. Uma metodologia de Bootstrap

Rui Cunha Marques ^†

Duarte Silva ^‡

^† Centro de Sistemas Urbanos e Regionais

Instituto Superior Técnico

Universidade Técnica de Lisboa

rcmar@civil.ist.utl.pt

^‡ Rave

dnsilva@rave.pt

 

 

 Statistical inference of efficiency estimators obtained with the DEA nonparametric frontier technique. A Bootstrap methodology

Abstract

The efficiency measurement of decision making units is a key-task in contemporary societies. Despite the existence of several methodologies with this aim none of them is clearly superior. One of the most used is the non-parametric technique of data envelopment analysis (DEA). The DEA method has many advantages, therefore, being very popular and widely adopted. However, it doesn’t allow for the results statistical inference, which can constrain its use. This document provides some insights for that debate, discussing and applying the DEA technique to 70 water services in Portugal. Later, a re-sampling (bootstrap) methodology is applied to the results attained by DEA, enabling its statistical inference. The quality of the values obtained by the bootstrap depends much on the average efficiency level and on its dispersion. That is, if the average inefficiency gap and its dispersion are considerable, the results will be far from the ones wished. Nevertheless, under the opposite circumstances, the results are interesting and satisfactory, surpassing the traditional drawbacks pointed at DEA. 

Keywords: Efficiency, DEA, Statistical Inference, Bootstrap, Water Services

 

 

Resumo

A medição de eficiência de unidades organizatórias ou unidades de decisão constitui uma tarefa-chave nas sociedades contemporâneas. Para a sua prossecução existem várias metodologias disponíveis embora nenhuma seja claramente superior. Um dos métodos mais conhecidos, e aqui analisado, consiste na técnica não paramétrica data envelopment analysis (DEA). Esta técnica possui diversas vantagens, sendo, portanto muito popular e largamente adoptada. Todavia, a DEA não permite a inferência estatística dos resultados obtidos, o que condiciona o seu uso empírico. Este documento fornece alguns contributos para esse debate, discutindo e aplicando a técnica DEA a 70 serviços de água em Portugal. Posteriormente, é empregue uma metodologia de bootstrap (reamostragem) aos estimadores de eficiência obtidos com a DEA, permitindo a sua inferência estatística. A qualidade dos resultados alcançados com o bootstrap depende muito do nível de eficiência médio e da sua dispersão. Isto é, quando a ineficiência média é considerável, tal como a sua dispersão, os resultados estão longe do desejável. Porém, quando o valor da ineficiência é reduzido e a sua amplitude é baixa os resultados são satisfatórios e bem interessantes, ultrapassando alguns dos problemas tradicionais apontados à DEA.

 

Texto completo apenas disponível em PDF.

Full text only in PDF.

 

Referências

Banker, R. (1993) Maximum Likelihood, Consistency and Data Envelopment Analysis: a Statistical Foundation, Management Science, Vol 39, No 10, pp. 1265-1273.        [ Links ]

Banker, R. (1996) Hypothesis Tests using Data Envelopment Analysis, Journal of Productivity Analysis, Vol 7, No 2/3, pp. 139-159.

Banker, R., Charnes, A. e Cooper, W. (1984) Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis, Management Science, Vol 30, No 9, pp. 1078-1092.

Bosworth, D., Stoneman, P. e Thanassoulis, E. (1996) The Measurement of Comparative Total Efficiency in the Sewerage and Water industry: an Exploratory Study, Office of Water Services, UK.

Burns, P., Huggins, M., Riechmann, C. e Weyman-Jones, T. (2000) Choice of Model and Availability of Data for the Efficiency Analysis of Dutch Network and Supply Businesses in the Electricity Sector, Frontier Economics, Dienst uitvoering en toezicht Elektriciteitswet, Holanda.

Charnes, A., Cooper, W. e Rhodes, E. (1978) Measuring the Efficiency of Decision Making Units, European Journal of Operational Research, Vol 2, No 6, pp. 429-444.

Charnes, A., Cooper, W., Lewin, A. e Seiford, L. (1994) Data Envelopment Analysis. Theory, methodology and applications, Kluwer Academic Publishers, USA.

Coelli, T., Prasada, R. e Battese, G. (1998) An Introduction to Efficiency and Productivity Analysis, Kluwer Academic Publishers, USA.

Cooper, W., Seiford, L. e Tone, K. (2000) Data Envelopment Analysis. A Comprehensive Text with Models, Applications, References and DEA-Solver Software, Kluwer Academic Publishers, USA.

Efron, B. (1979) Bootstrap Methods: another Look at the Jackknife, Annals of Statistics, Vol 7, No 1, pp. 1-26.

Efron, B. (1982) The Jacknife, the Bootstrap and other Resampling Plans, Monograph No. 38, Society for Industrial and Applied Mathematics, USA.

Efron, B. e Tibshirani, J. (1993) An Introduction to the Bootstrap, Chapman and Hall, UK.

Färe, R. e Primont, D. (1987) Efficiency Measures for Multiplant Firms with Limited Data, In Measurement in Economics: Theory and Applications in Economic Indices, Ed. por W. Eichhorn e E. Diewert, Physica-Verlag, Alemanha, pp. 177-186.

Farrell, M. (1957) The Measurement of Productive Efficiency, Journal of the Royal Statistical Society, Vol 120A, No 3, pp. 353-290.

Ferrier, G. e Hirschberg, J. (1997) Bootstrap Confidence Intervals for Linear Programming Efficiency Scores: with an Illustration using Italian Banking Data, Journal of Productivity Analysis, Vol 8, No 1, pp. 19-33.

Grosskopf, S. (1986) The Role of the Reference Technology in Measuring Productive Efficiency, Economic Journal, Vol 96, No 382, pp. 499-513.

Grosskopf, S. (1996) Statistical Inference and Nonparametric Efficiency: a Selective Survey, Journal of Productivity Analysis, Vol 7, No 2/3, pp. 161-176.

Hall, P. (1986) On the Number of Bootstrap Simulations Required to Construct a Confidence Interval, Annals of Statistics, Vol 14, No 4, pp. 1453-1462.

Hall, P. (1988) Theoretical Comparison of Bootstrap Confidence Intervals, Annals of Statistics, Vol 16, No 3, pp. 927-953.

Hall, P., Hardle, W. e Simar, L. (1995) Iterated Bootstrap with Applications to Frontier Models, Journal of Productivity Analysis, Vol 6, No 1, pp. 63-76.

Kittelsen, S. (1998) Stepwise DEA: Choosing Variables for Measuring Technical Efficiency in Norwegian Electricity Distribution, Memorandum, No. 6, Oslo University.

Kittelsen, S. (1999) Monte Carlo Simulations of DEA Efficiency Scores and Hypothesis Tests, Memorandum, No. 9, Oslo University.

Löthgren, M. (1998) How to Bootstrap DEA Estimators: a Monte Carlo Comparison, Working Paper, No. 223, Stockholm University.

Marques, R. (2005a) A Regulação dos Serviços de Água e de Águas Residuais. Aplicação de Técnicas Não Paramétricas de Benchmarking, Dissertação de Doutoramento, Instituto Superior Técnico, Universidade Técnica de Lisboa.

Marques, R. (2005b) A Regulação dos Serviços Públicos, Edições Sílabo, Lisboa.

Marques, R. e Monteiro, A. (2004) Benchmarking the Economic Performance of Portuguese Water and Sewerage Services, In Data Envelopment Analysis and Performance Measurement, Ed. por Ali Emrouznejad e Victor Podinovski, Warwick Print, UK, pp. 65-72.

Pedraja-Chaparro, F., Salinas-Jiménez, J. e Smith, P. (1999) On the Quality of the Data Envelopment Analysis Model, Journal of the Operations Research Society, Vol 50, No 6, pp. 636-644.

Sarafidis, V. (2002) An Assessment of Comparative Efficiency Measurement Techniques, Europe Economics, Office of Water Services, UK.

Silverman, B. (1986) Density Estimation of Statistics and Data Analysis, Chapman and Hall, UK.

Silverman, B. e Young, G. (1987) The Bootstrap: to Smooth or not to Smooth?, Biometrika, Vol 74, No 3, pp. 469-479.

Simar, L. (1992) Estimating Efficiencies from Frontier Models with Panel Data: a Comparison of Parametric, Non-parametric and Semi-parametric Methods with Bootstrapping, Journal of Productivity Analysis, Vol 3, No 1/2, pp. 171-191.

Simar, L. e Wilson, P. (1998) Sensitivity Analysis of Efficiency Scores: How to Bootstrap in Nonparametric Models, Management Science, Vol 44, No 1, pp. 46-61.

Simar, L. e Wilson, P. (2000a) Statistical Inference in Non-parametric Frontier Models: the State of the Art, Journal of Productivity Analysis, Vol 13, No 1, pp. 49-78.

Simar, L. e Wilson, P. (2000b) A General Methodology for Bootstrapping in Nonparametric Frontier Models, Journal of Applied Statistics, Vol 27, No 6, pp. 779-802.

Simar, L. e Wilson, P. (2002) Non-parametric Tests of Returns to Scale, European Journal of Operational Research, Vol 139, No 1, pp. 115-132.

Tavares, G. (2002) A bibliography of data envelopment analysis (1978-2001), Rutcor Research Report, Rutgers University.