A Survey of Stochastic Frontier Models and Likely Future Developments
JEL Classification: C10, C 20, D24
Abstract
This paper summarizes the literature on stochastic frontier production function models. It covers the definition of technical efficiency, the basic cross-sectional stochastic frontier model, and the stochastic frontier model with panel data and time-invariant as well as time-varying technical inefficiency. It also discusses models in which technical inefficiency depends on explanatory variables. Finally, it discusses the problem of inference on the inefficiencies and makes some predictions about likely future developments in the field.
Keywords:
Stochastic frontiers, Production functions, Technical efficiency, Panel data, Efficiency measurementAcknowledgments
The second author acknowledges that this research is financially supported by the 2008 Sogang University Research Fund (10045). Paper presented at the 16th Seoul Journal of Economics International Symposium held at Seoul National University, Seoul, 27 November 2008.
References
- Ahn, S. C., Lee, Y. H., and Schmidt, P. “GMM Estimation of Linear Panel Data Models with Time-Varying Individual Effects.” Journal of Econometrics 101 (No. 2 2001): 219-55. [https://doi.org/10.1016/S0304-4076(00)00083-X]
- Ahn, S. C., Lee, Y. H., and Schmidt, P. “Stochastic Frontier Models with Multiple Time-Varying Individual Effects.” Journal of Productivity Analysis 27 (No. 1 2007): 1-12. [https://doi.org/10.1007/s11123-006-0020-8]
- Aigner, D. J., and Chu, S. F. “On Estimating the Industry Production Function.” American Economic Review 58 (1968): 226-39.
- Aigner, D. J., Lovell, C. A. K., and Schmidt, P. “Formulation and Estimation of Stochastic Frontier Production Function Models.” Journal of Econometrics 6 (No. 1 1977): 21-37. [https://doi.org/10.1016/0304-4076(77)90052-5]
- Álvarez, A., Amsler, C., Orea, L., and Schmidt, P. “Interpreting and Testing the Scaling Property in Models Where Inefficiency Depends on Firm Characteristics.” Journal of Productivity Analysis 25 (No. 3 2006): 201-12. [https://doi.org/10.1007/s11123-006-7639-3]
- Bai, J. “Inferential Theory for Factor Models of Large Dimensions.” Econometrica 71 (No. 1 2003): 135-71. [https://doi.org/10.1111/1468-0262.00392]
- Bai, J., and Ng, S. “Determining the Number of Factors in Approximate Factor Models.” Econometrica 70 (No. 1 2002): 191-221. [https://doi.org/10.1111/1468-0262.00273]
- Battese, G. E., and Coelli, T. J. “Prediction of Firm-Level Technical Efficiency with a Generalized Frontier Production Function and Panel Data.” Journal of Econometrics 38 (No. 3 1988): 387-99. [https://doi.org/10.1016/0304-4076(88)90053-X]
- Battese, G. E., and Coelli, T. J. “Frontier Production Functions, Technical Efficiency and Panel Data with Application to Paddy Farmers in India.” Journal of Productivity Analysis 3 (No. 2 1992): 153-69. [https://doi.org/10.1007/BF00158774]
- Battese, G. E., and Coelli, T. J. “A Model for Technical Inefficiency Effects in a Stochastic Frontier Production Function for Panel Data.” Empirical Economics 20 (No. 2 1995): 325-32. [https://doi.org/10.1007/BF01205442]
- Caudill, S. B., and Ford, J. M. “Biases in Frontier Estimation Due to Heteroskedasticity.” Economics Letters 41 (No. 1 1993): 17-20. [https://doi.org/10.1016/0165-1765(93)90104-K]
- Caudill, S. B., Ford, J. M., and Gropper, D. M. “Frontier Estimation and Firm-Specific Inefficiency Measures in the Presence of Heteroskedasticity.” Journal of Business and Economic Statistics 13 (No. 1 1995): 105-11. [https://doi.org/10.2307/1392525]
- Cornwell, C., Schmidt, P., and Sickles, R. “Production Frontiers with Cross-Sectional and Time-Series Variation in Efficiency Levels.” Journal of Econometrics 46 (Nos. 1-2 1990): 185-200. [https://doi.org/10.1016/0304-4076(90)90054-W]
- Cuesta, R. A. “A Production Model with Firm-Specific Temporal Variation in Technical Inefficiency: with Application to Spanish Dairy Farms.” Journal of Productivity Analysis 13 (No. 2 2000): 139-49. [https://doi.org/10.1023/A:1017297831646]
- Edwards, D. G., and Hsu, J. C. “Multiple Comparisons with the Best Treatment.” Journal of the American Statistical Association 78 (1983): 965-71. [https://doi.org/10.1080/01621459.1983.10477047]
- Farrell, M. J. “The Measurement of Productive Efficiency.” Journal of the Royal Statistal Society, Series A. 120 (No. 3 1957): 253-90. [https://doi.org/10.2307/2343100]
- Greene, W. H. “Maximum Likelihood Estimation of Econometric Frontier Functions.” Journal of Econometrics 13 (No. 1 1980): 27-56. [https://doi.org/10.1016/0304-4076(80)90041-X]
- Greene, W. H. “A Gamma-Distributed Stochastic Frontier Model.” Journal of Econometrics 46 (Nos. 1-2 1990): 141-63. [https://doi.org/10.1016/0304-4076(90)90052-U]
- Hall, P., Härdle, W., and Simar, L. “Iterated Bootstrap with Applications to Frontier Models.” Journal of Productivity Analysis 6 (No. 1 1995): 63-76. [https://doi.org/10.1007/BF01073495]
- Han, C., Orea, L., and Schmidt, P. “Estimation of a Panel Data Model with Parametric Temporal Variation in Individual Effects.” Journal of Econometrics 126 (No. 2 2005): 241-67. [https://doi.org/10.1016/j.jeconom.2004.05.002]
- Hochberg, Y., and Tamhane, A. C. Multiple Comparison Procedures. New York: John Wiley and Sons, 1987. [https://doi.org/10.1002/9780470316672]
- Horrace, W. C., and Schmidt, P. “Confidence Statements for Efficiency Estimates from Stochastic Frontier Models.” Journal of Productivity Analysis 7 (No. 3 1996): 257-82. [https://doi.org/10.1007/BF00157044]
- Horrace, W. C., and Schmidt, P. “Multiple Comparisons with the Best with Economic Applications.” Journal of Applied Econometrics 15 (No. 1 2000): 1-26. [https://doi.org/10.1002/(SICI)1099-1255(200001/02)15:1<1::AID-JAE551>3.0.CO;2-Y]
- Hsu, J. “Simultaneous Confidence Intervals for All Distances from the Best.” Annals of Statistics 9 (1981): 1026-34. [https://doi.org/10.1214/aos/1176345582]
- Hsu, J. “Constrained Simultaneous Confidence Intervals for Multiple Comparisons with the Best.” Annals of Statistics 12 (1984): 1145-50. [https://doi.org/10.1214/aos/1176346732]
- Hsu, J. Multiple Comparisons: Theory and Methods. London: Chapman and Hall, 1996. [https://doi.org/10.1201/b15074]
- Huang, C. J., and Liu, J. T. “Estimation of a Non-Neutral Stochastic Frontier Production Function.” Journal of Productivity Analysis 5 (No. 2 1994): 171-80. [https://doi.org/10.1007/BF01073853]
- Jondrow, J., Lovell, C. A. K., Materov, I. S., and Schmidt, P. “On the Estimation of Technical Efficiency in the Stochastic Frontier Production Model.” Journal of Econometrics 19 (Nos. 2-3 1982): 233-38. [https://doi.org/10.1016/0304-4076(82)90004-5]
- Kim, M., Kim, Y., and Schmidt, P. “On the Accuracy of Bootstrap Confidence Intervals for Efficiency Levels in Stochastic Frontier Models with Panel Data.” Journal of Productivity Analysis 28 (No. 3 2007): 165-81. [https://doi.org/10.1007/s11123-007-0058-2]
- Kim, Y., and Schmidt, P. “A Review and Empirical Comparison of Bayesian and Classical Approaches to Inference on Efficiency Levels in Stochastic Frontier Models with Panel Data.” Journal of Productivity Analysis 14 (No. 2 2000): 91-118.
- Koop, G., Osiewalski, J., and Steele, M. F. “Bayesian Efficiency Analysis through Individual Effects: Hospital Cost Frontiers.” Journal of Econometrics 76 (Nos. 1-2 1997): 77-105. [https://doi.org/10.1016/0304-4076(95)01783-6]
- Koop, G., Steele, M. F., and Osiewalski, J. “Posterior Analysis of Stochastic Frontier Models Using Gibbs Sampling.” Computational Statistics 10 (1995): 353-73.
- Kumbhakar, S. C. “Production Frontiers, Panel Data, and Time-Varying Technical Inefficiency.” Journal of Econometrics 46 (Nos. 1-2 1990): 201-11. [https://doi.org/10.1016/0304-4076(90)90055-X]
- Kumbhakar, S. C., Ghosh, S., and McGuckin, J. T. “A Generalized Production Approach for Estimating Determinants of Inefficiency in U.S. Dairy Farms.” Journal of Business and Economic Statistics 9 (No. 3 1991): 279-86. [https://doi.org/10.1080/07350015.1991.10509853]
- Kumbhakar, S. C., and Lovell, C. A. K. Stochastic Frontier Analysis. Cambridge: Cambridge University Press, 2000. [https://doi.org/10.1017/CBO9781139174411]
- Kuosmanen, T. Stochastic Nonparametric Envelopment of Data: Combining Virtues of SFA and DEA in a Unified Framework. Discussion Paper, MTT Agrifood Research Finland, 2006. [https://doi.org/10.2139/ssrn.905758]
- Kuosmanen, T. “Representation Theorem for Convex Nonparametric Least Squares.” The Econometrics Journal 11 (No. 2 2008): 308-25. [https://doi.org/10.1111/j.1368-423X.2008.00239.x]
- Lee, Y. H. “A Stochastic Production Frontier Model with Group-Specific Temporal Variation in Technical Efficiency.” European Journal of Operational Research 174 (No. 3 2006): 1616-30. [https://doi.org/10.1016/j.ejor.2005.03.044]
- Lee, Y. H. “The Group-Specific Stochastic Frontier Models with Parametric Specifications.” Forthcoming in European Journal of Operational Research, 2009.
- Lee, Y. H., and Schmidt, P. “A Production Frontier Model with Flexible Temporal Variation in Technical Inefficiency.” In H. Fried, C. A. K. Lovell, and S. Schmidt (eds.), The Measurement of Productive Efficiency: Techniques and Applications. Oxford: Oxford University Press, 1993.
- Meeusen, W., and van den Broeck, J. “Efficiency Estimation from Cobb-Douglas Production Functions with Composed Error.” International Economic Review 18 (No. 2 1977): 435-44. [https://doi.org/10.2307/2525757]
- Park, B. U. and Simar, L. “Efficient Semiparametric Estimation in a Stochastic Frontier Model.” Journal of American Statistics Association 89 (1994): 929-36. [https://doi.org/10.1080/01621459.1994.10476826]
- Pitt, M., and Lee, L. F. “The Measurement and Sources of Technical Inefficiency in the Indonesian Weaving Industry.” Journal of Development Economics 9 (No. 1 1981): 43-64. [https://doi.org/10.1016/0304-3878(81)90004-3]
- Reifschneider, D., and Stevenson, R. “Systematic Departures from the Frontier: A Framework for the Analysis of Firm Inefficiency.” International Economic Review 32 (No. 3 1991): 715-23. [https://doi.org/10.2307/2527115]
- Schmidt, P., and Sickles, R. C. “Production Frontiers and Panel Data.” Journal of Business and Economic Statistics 2 (No. 4 1984): 367-74. [https://doi.org/10.1080/07350015.1984.10509410]
- Simar, L. “Estimating Efficiencies from Frontier Models with Panel Data: a Comparison of Parametric, Non-Parametric and Semi-Parametric Methods with Bootstrapping.” Journal of Productivity Analysis 3 (No. 2 1992): 171-203. [https://doi.org/10.1007/BF00158775]
- Simar, L., and Wilson, P. “Sensitivity of Efficiency Scores: How to Bootstrap in Nonparametric Frontier Models.” Management Science 44 (No. 1 1998): 49-61. [https://doi.org/10.1287/mnsc.44.1.49]
- Stevenson, R. E. “Likelihood Functions for Generalized Stochastic Frontier Estimation.” Journal of Econometrics 13 (No. 1 1980): 57-66. [https://doi.org/10.1016/0304-4076(80)90042-1]
- Tripathi, G. “Local Semiparametric Efficiency Bounds under Shape Restrictions.” Econometric Theory 16 (No. 5 2000): 729-39. [https://doi.org/10.1017/S0266466600165053]
- Tripathi, G. and Kim, Woocheol. “Nonparametric Estimation of Homogeneous Functions.” Econometric Theory 19 (No. 4 2003): 640-63. [https://doi.org/10.1017/S026646660319408X]
- Wang, H. J. “Heteroscedasticity and Non-Monotonic Efficiency Effects in a Stochastic Frontier Model.” Journal of Productivity Analysis 18 (No. 3 2002): 241-53.
- Wang, H. J. and Schmidt, P. “One-Step and Two-Step Estimation of the Effects of Exogenous Variables on Technical Efficiency Levels.” Journal of Productivity Analysis 18 (No. 2 2002): 129-44.
- Wang, W. S., Amsler, C., and Schmidt, P. Goodness of Fit Tests in Stochastic Frontier Models. Unpublished Manuscript, Michigan State University, 2008.
- Wang, W. S., and Schmidt, P. “On the Distribution of Estimated Technical Efficiency in Stochastic Frontier Models.” Journal of Econometrics 148 (No. 1 2009): 36-45. [https://doi.org/10.1016/j.jeconom.2008.08.025]