Abstract
This chapter (as well as Chap. 11) reviews some of the most important developments in the econometric estimation of productivity and efficiency surrounding the stochastic frontier model.
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Notes
- 1.
Despite the variety of definitions, intuitively, production efficiency can be understood as a relative measure of productivity. In other words, production efficiency is a productivity measure that is being normalized (e.g., to be between 0 and 1 to reflect percentages) relative to some benchmark, such as the corresponding frontier outcome, optimal with respect to some criteria: e.g., maximal output given certain level of input and technology in the case of technical efficiency or minimal cost given certain level of output and technology in the case of cost efficiency.
- 2.
- 3.
Our discussion in both chapters will focus on a production frontier, as it is the most popular object of study, while the framework for dual characterizations (e.g., cost, revenue, profit) or other frontiers is similar and follows with only minor changes in notation.
- 4.
See chapter “Nonparametric Estimation of the SFM” for a discussion on relaxing parametric restrictions on the production frontier in the SFM.
- 5.
- 6.
See section “Handling Endogeneity in the SFM” for a discussion on estimation of the SFM when some inputs are allowed to be endogenous.
- 7.
See section “Modeling Determinants of Inefficiency” for models handling determinants of inefficiency.
- 8.
ALS also briefly discussed the exponential distribution, but its use and development is mainly attributed to MvB.
- 9.
The pdf of a skew normal random variable x is f(x) = 2ϕ(x) Φ(αx). The distribution is right skewed if α > 0 and is left skewed if α < 0. We can also place the normal, truncated-normal pair of distributional assumptions in this class. The pdf of x with location ξ, scale ω, and skew parameter α is \(f(x)=\frac {2}{\omega }\phi \left (\frac {x-\xi }{\omega }\right )\Phi \left (\alpha \left (\frac {x-\xi }{\omega }\right )\right )\). See [76, 12] for more details.
- 10.
This in no way suggests that inference cannot be undertaken when the DEA estimator is deployed; rather, the DEA estimator has an asymptotic distribution which is much more complicated that the MLE for the SFM, and so direct asymptotic inference is not available; bootstrapping techniques are required for many of the most popular DEA estimators [93, 94].
- 11.
See [79] for a more detailed analysis of the SFM with u distributed exponentially.
- 12.
Prior to [68] all of the previously proposed distributions always produced a composed error density that was theoretically negatively skewed. Note that if u is distributed uniformly over the interval [0, b], inefficiency is equally likely to be either 0 or b.
- 13.
Note that the likelihood function for the normal-half-normal pair is dependent upon the cdf of the normal distribution, Φ(⋅) which contains an integral, but this can be quickly and easily evaluated across all modern software platforms.
- 14.
- 15.
An alternative approach would be to estimate a weighted average efficiency of an industry, as described theoretically in chapter 25.
- 16.
There exists some confusion over the terminology COLS as it relates to another method, modified OLS (MOLS). Beginning with [111] and discussed in [31] and [34, pp. 32–34], MOLS shifts the estimated OLS production function until all of the observations lie on or below the “frontier.” At issue is the appropriate name of these two techniques. Greene [38] called the bounding approach COLS, crediting [70, p. 21] with the initial nomenclature, and referred to MOLS as the method in which one bias corrects the intercept based on a specific set of distributional assumptions. Further, [59, pp. 70–71] also adopted this terminology. However, given that [77, p. 69] explicitly used the terminology COLS, in our review we will adopt COLS to imply bias correction of the OLS intercept and MOLS as a procedure that shifts up (or down) the intercept to bound all of the data. The truth is both COLS and MOLS are the same in the sense that the OLS intercept is augmented, it is just in how each method corrects, or modifies, the intercept that is important. While we are departing from the more mainstream use of COLS and MOLS currently deployed, given the original use of COLS, coupled with myriad papers written by Peter Schmidt and coauthors that we discuss here, we will use the COLS acronym to imply a bias corrected intercept.
- 17.
The current literature is fairly rich on various examples of empirical values of SFA for the estimation and use of efficiency estimates in different fields of research. For example, in the context of electricity providers, see [54, 42, 62]; for banking efficiency, see [23] and references cited therein; for the analysis of the efficiency of national healthcare systems, see [33] and a review by [45]; for analyzing efficiency in agriculture, see [21, 14, 15, 69], to mention just a few.
- 18.
Jondrow et al. [50] also suggested an alternative estimator based on the conditional mode.
- 19.
In principle, these individual efficiency scores can then be used for estimating weighted average efficiencies of an industry or a group within it, as described theoretically in chapter 25, which seems novel for SFA context.
- 20.
One could test if other moments of the distribution were 0 as well, but most of the SFMs parameterize the distribution of u with σu and so this seems the most natural.
- 21.
The JLMS efficiency estimator is known as a shrinkage estimator; on average, it understates the efficiency level of a firm with small ui while it overstates efficiency for a firm with large ui.
- 22.
See also [66] for a different test based on the Pearson distributional assumption for u.
- 23.
In a limited Monte Carlo analysis, [86] compared rank correlations of stochastic frontier estimates assuming that inefficiency was either half-normal (which was the true distribution) or exponential (a misspecified distribution) and found very little evidence that misspecification impacted the rank correlations in any meaningful fashion; [46] conducted a similar set of experiments and found essentially the same results.
- 24.
Note that the estimator of the skewness coefficient is distributed asymptotically standard normal, so it is feasible to have either negative or positive skewness in any finite sample.
- 25.
Typically the standard errors can be obtained either through use of the outer product of gradients (OPG) or direct estimation of the Hessian matrix of the log-likelihood function. Given the nascency of these methods, it has yet to be determined which of these two methods is more reliable in practice, though in other settings both tend to work well. One caveat for promoting the use of the OPG is that since this only requires calculation of the first derivatives, it can be more stable (and more likely to be invertible) than calculation of the Hessian. Also note that in finite samples, the different estimators of covariance of MLE estimator can give different numerical estimates, even suggesting different implications on the inference (reject or do not reject the null hypothesis). So, for small samples, it is often advised to check all feasible estimates whenever there is suspicion of ambiguity in the conclusions (e.g., when a hypothesis is rejected only at say around the 10% of significance level).
- 26.
It is possible to treat a subset of x as endogenous; i.e., x = (x1, x2), where x1 is endogenous and x2 is exogenous.
- 27.
Reifschneider and Stevenson [83] used the term “inefficiency explanatory variables,” while others call them “environmental variables,” but it is now common to refer to these variables as “determinants of inefficiency.” A variety of approaches have been proposed to model the determinants of inefficiency with the first pertaining to panel data models [55, 14] (see chapter 11).
- 28.
- 29.
- 30.
Actually, given the reparameterization of the log-likelihood function, the specification for σu implies a particular specification for both λ and σ.
- 31.
- 32.
- 33.
Note here that we are making the implicit assumption that z is different from x. The nonlinearity of the scaling function does allow z and x to overlap however.
References
Afriat SN (1972) Efficiency estimation of production functions. Int Econ Rev 13(3):568–598
Ahmad IA, Li Q (1997) Testing symmetry of an unknown density function by kernel method. J Nonparametric Stat 7:279–293
Aigner D, Chu S (1968) On estimating the industry production function. Am Econ Rev 58:826–839
Aigner DJ, Lovell CAK, Schmidt P (1977) Formulation and estimation of stochastic frontier production functions. J Econ 6(1):21–37
Ali M, Flinn JC (1989) Profit efficiency among Basmati rice producers in Pakistan Punjab. Am J Agric Econ 71(2):303–310
Almanidis P, Qian J, Sickles RC (2014) Stochastic frontier models with bounded inefficiency. In: Sickles RC, Horrace WC (eds) Festschrift in honor of Peter Schmidt econometric methods and applications. Springer, New York, pp 47–82
Almanidis P, Sickles RC (2011) The skewness issue in stochastic frontier models: fact or fiction? In: van Keilegom I, Wilson PW (eds) Exploring research frontiers in contemporary statistics and econometrics. Springer, Berlin
Alvarez A, Amsler C, Orea L, Schmidt P (2006) Interpreting and testing the scaling property in models where inefficiency depends on firm characteristics. J Prod Anal 25(2):201–212
Amemiya T (1974) The nonlinear two-stage least-squares estimator. J Econ 2:105–111
Amsler C, Prokhorov A, Schmidt P (2016) Endogeneity in stochastic frontier models. J Econ 190:280–288
Amsler C, Prokhorov A, Schmidt P (2017) Endogeneity environmental variables in stochastic frontier models. J Econ 199:131–140
Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12(2):171–178
Battese GE, Coelli TJ (1988) Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data. J Econ 38:387–399
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
Battese GE, Coelli TJ (1995) A model for technical inefficiency effects in a stochastic frontier production function for panel data. Empir Econ 20(1):325–332
Battese GE, Corra GS (1977) Estimation of a production frontier model: with application to the pastoral zone off Eastern Australia. Aust J Agric Econ 21(3):169–179
Benabou R, Tirole J (2016) Mindful economics: the production, consumption, and value of beliefs. J Econ Perspect 30(3):141–164
Bera AK, Sharma SC (1999) Estimating production uncertainty in stochastic frontier production function models. J Prod Anal 12(2):187–210
Bloom N, Lemos R, Sadun R, Scur D, Van Reenen J (2016) International data on measuring management practices. Am Econ Rev 106(5):152–156
Bonanno G, De Giovanni D, Domma F (2017) The ‘wrong skewness’ problem: a re-specification of stochastic frontiers. J Prod Anal 47(1):49–64
Bravo-Ureta BE, Rieger L (1991) Dairy farm efficiency measurement using stochastic frontiers and neoclassical duality. Am J Agric Econ 73(2):421–428
Carree MA (2002) Technological inefficiency and the skewness of the error component in stochastic frontier analysis. Econ Lett 77(1):101–107
Case B, Ferrari A, Zhao T (2013) Regulatory reform and productivity change in Indian banking. Rev Econ Stat 95(3):1066–1077
Caudill SB, Ford JM (1993) Biases in frontier estimation due to heteroskedasticity. Econ Lett 41(1):17–20
Caudill SB, Ford JM, Gropper DM (1995) Frontier estimation and firm-specific inefficiency measures in the presence of heteroskedasticity. J Bus Econ Stat 13(1):105–111
Chamberlain G (1987) Asymptotic efficiency in estimation with conditional moment restrictions. J Econ 34(2):305–334
Chen Y-Y, Schmidt P, Wang H-J (2014) Consistent estimation of the fixed effects stochastic frontier model. J Econ 181(2):65–76
Coelli TJ (1995) Estimators and hypothesis tests for a stochastic frontier function: a Monte Carlo analysis. J Prod Anal 6(4):247–268
Dugger R (1974) An application of bounded nonparametric estimating functions to the analysis of bank cost and production functions, Ph.D. thesis, University of North Carolina, Chapel Hill
Feng Q, Horrace WC, Wu GL (2015) Wrong skewness and finite sample correction in parametric stochastic frontier models. Center for Policy Research – The Maxwell School, working paper N. 154
Gabrielsen A (1975) On estimating efficient production functions. Working Paper No. A-85, Chr. Michelsen Institute, Department of Humanities and Social Sciences, Bergen
Gagnepain P, Ivaldi M (2002) Stochastic frontiers and asymmetric information models. J Prod Anal 18(2):145–159
Greene W (2004) Distinguishing between heterogeneity and inefficiency: stochastic frontier analysis of the World Health Organization’s panel data on national health care systems. Health Econ 13(9):959–980
Greene WH (1980a) Maximum likelihood estimation of econometric frontier functions. J Econ 13(1):27–56
Greene WH (1980b) On the estimation of a flexible frontier production model. J Econ 13(1):101–115
Greene WH (1990) A gamma-distributed stochastic frontier model. J Econ 46(1–2):141–164
Greene WH (2003) Simulated likelihood estimation of the normal-gamma stochastic frontier function. J Prod Anal 19(2):179–190
Greene WH (2008) The econometric approach to efficiency analysis. In: Knox Lovell CA, Fried HO, Schmidt SS (eds) The measurement of productive efficiency and productivity change, chapter 2. Oxford University Press, Oxford, UK
Hadri K (1999) Estimation of a doubly heteroscedastic stochastic frontier cost function. J Bus Econ Stat 17(4):359–363
Hafner C, Manner H, Simar L (2016) The “wrong skewness” problem in stochastic frontier model: a new approach. Econometric Reviews. forthcoming
Hansen C, McDonald JB, Newey WK (2010) Instrumental variables estimation with flexible distributions. J Bus Econ Stat 28:13–25
Hattori T (2002) Relative performance of U.S. and Japanese electricity distribution: an application of stochastic frontier analysis. J Prod Anal 18(3):269–284
Henderson DJ, Parmeter CF (2015) A consistent bootstrap procedure for nonparametric symmetry tests. Econ Lett 131:78–82
Hjalmarsson L, Kumbhakar SC, Heshmati A (1996) DEA, DFA, and SFA: a comparison. J Prod Anal 7(2):303–327
Hollingsworth B (2008) The measurement of efficiency and productivity of health care delivery. Health Econ 17(10):1107–1128
Horrace WC, Parmeter CF (2014) A Laplace stochastic frontier model. University of Miami Working Paper
Horrace WC, Schmidt P (1996) Confidence statements for efficiency estimates from stochastic frontier models. J Prod Anal 7:257–282
Horrace WC, Wright IA (2016) Stationary points for parametric stochastic frontier models. Center for Policy Research – The Maxwell School, working paper N. 196
Huang CJ, Liu J-T (1994) Estimation of a non-neutral stochastic frontier production function. J Prod Anal 5(1):171–180
Jondrow J, Lovell CAK, Materov IS, Schmidt P (1982) On the estimation of technical efficiency in the stochastic frontier production function model. J Econ 19(2/3):233–238
Kalirajan KP (1990) On measuring economic efficiency. J Appl Econ 5(1):75–85
Karakaplan MU, Kutlu L (2013) Handling endogeneity in stochastic frontier analysis. Unpublished manuscript
Kim M, Schmidt P (2008) Valid test of whether technical inefficiency depends on firm characteristics. J Econ 144(2):409–427
Knittel CR (2002) Alternative regulatory methods and firm efficiency: stochastic frontier evidence form the U.S. electricity industry. Rev Econ Stat 84(3):530–540
Kumbhakar SC (1987) The specification of technical and allocative inefficiency in stochastic production and profit frontiers. J Econ 34(1):335–348
Kumbhakar SC (2011) Estimation of production technology when the objective is to maximize return to the outlay. Eur J Oper Res 208:170–176
Kumbhakar SC (2013) Specification and estimation of multiple output technologies: a primal approach. Eur J Oper Res 231:465–473
Kumbhakar SC, Ghosh S, McGuckin JT (1991) A generalized production frontier approach for estimating determinants of inefficiency in US diary farms. J Bus Econ Stat 9(1):279–286
Kumbhakar SC, Lovell CAK (2000) Stochastic frontier analysis. Cambridge University Press, Cambridge
Kumbhakar SC, Wang H-J (2006) Estimation of technical and allocative inefficiency: a primal system approach. J Econ 134(3):419–440
Kumbhakar SC, Wang H-J, Horncastle A (2015) A practitioners guide to stochastic frontier analysis using stata. Cambridge University Press, Cambridge, UK
Kuosmanen T (2012) Stochastic semi-nonparametric frontier estimation of electricity distribution networks: application of the StoNED method in the Finnish regulatory model. Energy Econ 34:2189–2199
Kuosmanen T, Fosgerau M (2009) Neoclassical versus frontier production models? Testing for the skewness of regression residuals. Scand J Econ 111(2):351–367
Kutlu L (2010) Battese-Coelli estimator with endogenous regressors. Econ Lett 109:79–81
Latruffe L, Bravo-Ureta BE, Carpentier A, Desjeux Y, Moreira VH (2017) Subsidies and technical efficiency in agriculture: evidence from European dairy farms. Am J Agric Econ 99:783–799
Lee L (1983) A test for distributional assumptions for the stochastic frontier function. J Econ 22(2):245–267
Lee L-F, Tyler WG (1978) The stochastic frontier production function and average efficiency: an empirical analysis. J Econ 7:385–389
Li Q (1996) Estimating a stochastic production frontier when the adjusted error is symmetric. Econ Lett 52(3):221–228
Lien G, Kumbhakar SC, Hardaker JB (2017) Accounting for risk in productivity analysis: an application to Norwegian dairy farming. J Prod Anal 47(3):247–257
Lovell CAK (1993) Production frontiers and productive efficiency. In: Knox Lovell CA, Fried HO, Schmidt SS (eds) The measurement of productive efficiency, chapter 1. Oxford University Press, Oxford, UK
McFadden D (1989) A method of simulated moments for estimation of discrete response models without numerical integration. Econometrica 57(5):995–1026
Meeusen W, van den Broeck J (1977a) Efficiency estimation from Cobb-Douglas production functions with composed error. Int Econ Rev 18(2):435–444
Meeusen W, van den Broeck J (1977b) Technical efficiency and dimension of the firm: some results on the use of frontier production functions. Empir Econ 2(2):109–122
Mutter RL, Greene WH, Spector W, Rosko MD, Mukamel DB (2013) Investigating the impact of endogeneity on inefficiency estimates in the application of stochastic frontier analysis to nursing homes. J Prod Anal 39(1):101–110
Nguyen NB (2010) Estimation of technical efficiency in stochastic frontier analysis. Ph.D. thesis, Bowling Green State University
O’Hagan A, Leonard T (1976) Bayes estimation subject to uncertainty about parameter constraints. Biometrika 63(1):201–203
Olson JA, Schmidt P, Waldman DA (1980) A Monte Carlo study of estimators of stochastic frontier production functions. J Econ 13:67–82
Ondrich J, Ruggiero J (2001) Efficiency measurement in the stochastic frontier model. Eur J Oper Res 129(3):434–442
Parmeter CF, Kumbhakar SC (2014) Efficiency analysis: a primer on recent advances. Found Trends Econ 7(3–4):191–385
Parmeter CF, Zelenyuk V (2016) A bridge too far? the state of the art in combining the virtues of stochastic frontier analysis and data envelopment analysis. University of Miami Working Paper 2016-10
Paul S, Shankar S (2017) An alternative specification for technical efficiency effects in a stochastic frontier production function. Crawford School Working Paper 1703
Pitt MM, Lee L-F (1981) The measurement and sources of technical inefficiency in the Indonesian weaving industry. J Dev Econ 9(1):43–64
Reifschneider D, Stevenson R (1991) Systematic departures from the frontier: a framework for the analysis of firm inefficiency. Int Econ Rev 32(1):715–723
Richmond J (1974) Estimating the efficiency of production. Int Econ Rev 15(2):515–521
Ritter C, Simar L (1997) Pitfalls of normal-gamma stochastic frontier models. J Prod Anal 8(2):167–182
Ruggiero J (1999) Efficiency estimation and error decomposition in the stochastic frontier model: a Monte Carlo analysis. Eur J Oper Res 115(6):555–563
Schmidt P (1976) On the statistical estimation of parametric frontier production functions. Rev Econ Stat 58(2):238–239
Schmidt P (2011) One-step and two-step estimation in SFA models. J Prod Anal 36(2):201–203
Silvapulle M, Sen P (2005) Constrained statistical inference. Wiley, Hoboken
Simar L, Lovell CAK, van den Eeckaut P (1994) Stochastic frontiers incorporating exogenous influences on efficiency. Discussion Papers No. 9403, Institut de Statistique, Universite de Louvain
Simar L, Van Keilegom I, Zelenyuk V (2017) Nonparametric least squares methods for stochastic frontier models. J Prod Anal 47(3):189–204
Simar L, Wilson PW (2010) Inferences from cross-sectional, stochastic frontier models. Econ Rev 29(1):62–98
Simar L, Wilson PW (2013) Estimation and inference in nonparametric frontier models: recent developments and perspectives. Found Trends Econ 5(2):183–337
Simar L, Wilson PW (2015) Statistical approaches for nonparametric frontier models: a guided tour. Int Stat Rev 83(1):77–110
Solow R (1957) Technical change and the aggregate production function. Rev Econ Stat 39(3):312–320
Stevenson R (1980) Likelihood functions for generalized stochastic frontier estimation. J Econ 13(1):58–66
Stiglitz JE, Greenwald BC (1986) Externalities in economies with imperfect information and incomplete markets. Q J Econ 101(2):229–264
Taube R (1988) Möglichkeiten der effizienzmess ung von öffentlichen verwaltungen. Duncker & Humbolt GmbH, Berlin
Timmer CP (1971) Using a probabilistic frontier production function to measure technical efficiency. J Polit Econ 79(4):776–794
Tran KC, Tsionas EG (2013) GMM estimation of stochastic frontier models with endogenous regressors. Econ Lett 118:233–236
Tsionas EG (2007) Efficiency measurement with the Weibull stochastic frontier. Oxf Bull Econ Stat 69(5):693–706
Tsionas EG (2012) Maximum likelihood estimation of stochastic frontier models by the Fourier transform. J Econ 170(2):234–248
Uekusa M, Torii A (1985) Stochastic production functions: an application to Japanese manufacturing industry (in Japanese). Keizaigaku Ronsyu (Journal of Economics) 51(1):2–23
Waldman DM (1982) A stationary point for the stochastic frontier likelihood. J Econ 18(1):275–279
Wang H-J (2002) Heteroscedasticity and non-monotonic efficiency effects of a stochastic frontier model. J Prod Anal 18(2):241–253
Wang H-J, Schmidt P (2002) One-step and two-step estimation of the effects of exogenous variables on technical efficiency levels. J Prod Anal 18:129–144
Wang WS, Amsler C, Schmidt P (2011) Goodness of fit tests in stochastic frontier models. J Prod Anal 35(1):95–118
Wang WS, Schmidt P (2009) On the distribution of estimated technical efficiency in stochastic frontier models. J Econ 148(1):36–45
Wheat P, Greene B, Smith A (2014) Understanding prediction intervals for firm specific inefficiency scores from parametric stochastic frontier models. J Prod Anal 42:55–65
White H (1980) A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48:817–838
Winsten CB (1957) Discussion on Mr. Farrell’s paper. J R Stat Soc Ser A Gen 120(3):282–284
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Kumbhakar, S.C., Parmeter, C.F., Zelenyuk, V. (2020). Stochastic Frontier Analysis: Foundations and Advances I. In: Ray, S., Chambers, R., Kumbhakar, S. (eds) Handbook of Production Economics. Springer, Singapore. https://doi.org/10.1007/978-981-10-3450-3_9-1
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