Abstract
This chapter is a survey of developments in stochastic frontier modelling. The literature on stochastic frontiers has grown substantially in the 42 years since the seminal work by Aigner et al. (J. Econom 6 (1): 21–37, 1977). There exist many surveys of this literature that cover a broad range of contribution pathways in the field. In this chapter, we present a review of key developments in distributional specifications in stochastic frontier models, with a particular emphasis on innovations that address practical issues identified by practitioners.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In the SF literature, ‘truncated normal’ refers specifically to the left truncation at zero of a normal distribution with mean \(\mu\) and variance \(\sigma_{u}^{2}\).
- 2.
- 3.
Such ‘tolerance’ does not necessarily reflect the technical competence or experience of regulators per se. It could reflect the perceived limitations on the robustness of the analysis (e.g. data quality), which necessitates a risk averse efficiency finding from a regulatory review.
- 4.
If we view the normal-half normal model as a skew-normal regression model in which we expect (but do not restrict) the skewness parameter \(\sigma_{u} /\sigma_{v}\) to be positive, then we view the test for the presence of inefficiency as a one-tailed test of the H0 that \(\sigma_{u} \le 0\), or equivalently that \(\sigma_{u} /\sigma_{v} = 0\), rather than as a test involving a boundary issue. Comparing the case of one inequality constraint in Gouriéroux et al. (1982) to Case 5 in Self and Liang (1987), we see the same result.
- 5.
- 6.
N being the total number of observations, so that N = IT in the case of a balanced panel.
- 7.
Clearly, this model is far from parsimonious, since g(t) includes 2I parameters. In fact, the authors apply a simpler model, \(g\left( t \right) = \exp \left[ {\lambda_{1i} \left( {t - \lambda_{2} } \right)} \right]\) after failing to reject \(H_{0} : \lambda_{2i} = \lambda_{2}\).
- 8.
The authors instead estimate a system of T equations via the seemingly unrelated regressions (SUR) model proposed by Zellner (1962). However, this approach offers no way of predicting observation-specific efficiencies.
- 9.
However, in the context of a log-linear model, the estimate of the intercept will be biased in either case.
- 10.
Despite this, Tsionas (2002) does interpret the models as incorporating technological heterogeneity.
- 11.
- 12.
The univariate skew normal distribution is a special case of the closed skew-normal distribution. To see that \(\varepsilon_{it}\) is the sum of two closed skew-normal random variables, therefore, consider that \(v_{it}^{*} = v_{it} + w_{it}\). and \(a_{i}^{*} = a_{i} + w_{i}\) both follow skew-normal distributions. For details on the closed skew-normal distribution, see González-Farías et al. (2004).
- 13.
Note that the authors in fact proposed a deterministic frontier model in which \(E\left( {u_{i} |z_{i} } \right) = \exp \left( {z_{i} \delta } \right),\) but if we interpret the random error as vi rather than a component of ui, we have an SF model with a deterministic ui.
- 14.
- 15.
Note the two similar but subtly different parameterisations, \(\sigma_{ui} = \exp \left( {z_{i} \gamma } \right)\) and \(\sigma_{ui}^{2} = \exp \left( {z_{i} \gamma } \right)\).
- 16.
Note the similarity of the issues here to those around ‘confidence intervals’ and prediction intervals for \(E\left( {u_{i} |\varepsilon_{i} } \right)\), discussed by Wheat et al. (2014).
- 17.
However, the authors’ discussion overstates the simplicity of marginal effects in this case, since it focuses on \(\partial \ln \hat{u}_{i} /\partial z_{li}\), which is \(\eta_{l}\) regardless of the distribution of \(u_{i}^{*}\) (or indeed the choice of predictor). However, \(\partial \hat{u}_{i} /\partial z_{li}\) is more complex, and as previously noted, the translation into efficiency space via \(\partial \exp \left( { - \hat{u}_{i} } \right)/\partial z_{li}\) adds additional complexity.
- 18.
For an explanation of shrinkage in the context of the predictor \(E\left( {u_{i} |\varepsilon_{i} } \right)\), see Wang and Schmidt (2009).
- 19.
Holding \(\beta\) constant.
- 20.
In keeping with previous terminology, ‘truncated’ (without further qualification) refers specifically to the left truncation at zero of a distribution with mean \(\mu\), and ‘half’ refers to the special case where \(\mu = 0\). Note that truncating the Laplace distribution thus yields the exponential distribution whenever \(\mu \le 0\) due to the memorylessness property of the exponential distribution.
- 21.
Or more specifically, the scale contaminated normal distribution.
- 22.
- 23.
Again, as an exception to this, dependency between error components may be introduced via ‘environmental’ variables influencing the parameters of their distributions as discussed in Sect. 5.
- 24.
Schmidt and Lovell (1980) fold, rather than truncate.
- 25.
Kumbhakar et al. (2009), using panel data, also include a lagged regime membership (i.e. technology choice) dummy in their selection equation.
- 26.
The authors actually use ui to denote the noise term and vi and wi for the one-sided errors. In the interest of consistency and to avoid confusion, we use vi to refer to the noise term and ui and wi for the one-sided errors.
References
Afriat, S.N. 1972. Efficiency estimation of production functions. International Economic Review 13 (3): 568–598.
Ahn, S.C., Y.H. Lee, and P. Schmidt. 2007. Stochastic frontier models with multiple time-varying individual effects. Journal of Productivity Analysis 27 (1): 1–12.
Ahn, S.C., Y.H. Lee, and P. Schmidt. 2013. Panel data models with multiple time-varying individual effects. Journal of Econometrics 174 (1): 1–14.
Aigner, D.J., and S.F. Chu. 1968. On estimating the industry production function. The American Economic Review 58 (4): 826–839.
Aigner, D.J., T. Amemiya, and D.J. Poirier. 1976. On the estimation of production frontiers: Maximum likelihood estimation of the parameters of a discontinuous density function. International Economic Review 17 (2): 377–396.
Aigner, D., C.A.K. Lovell, and P. Schmidt. 1977. Formulation and estimation of stochastic frontier production function models. Journal of Econometrics 6 (1): 21–37.
Almanidis, P., and R.C. Sickles. 2012. The skewness issue in stochastic frontiers models: Fact or fiction? In Exploring research frontiers in contemporary statistics and econometrics: A festschrift for Léopold Simar, ed. I. Van Keilegom and W.P. Wilson, 201–227. Heidelberg: Physica-Verlag HD.
Almanidis, P., J. Qian, and R.C. Sickles. 2014. Stochastic frontier models with bounded inefficiency. In Festschrift in Honor of Peter Schmidt: Econometric Methods and Applications, ed. R.C. Sickles and W.C. Horrace, 47–81. New York, NY: Springer.
Alvarez, A., C. Amsler, L. Orea, and P. Schmidt. 2006. Interpreting and testing the scaling property in models where inefficiency depends on firm characteristics. Journal of Productivity Analysis 25 (3): 201–212.
Amemiya, T., and T.E. MaCurdy. 1986. Instrumental-variable estimation of an error-components model. Econometrica 54 (4): 869–880.
Amsler, C., A. Prokhorov, and P. Schmidt. 2014. Using copulas to model time dependence in stochastic frontier models. Econometric Reviews 33 (5–6): 497–522.
Amsler, C., P. Schmidt, and W.-J. Tsay. 2015. A post-truncation parameterization of truncated normal technical inefficiency. Journal of Productivity Analysis 44 (2): 209–220.
Andrews, D.W.K. 1993a. An introduction to econometric applications of empirical process theory for dependent random variables. Econometric Reviews 12 (2): 183–216.
Andrews, D.W.K. 1993b. Tests for parameter instability and structural change with unknown change point. Econometrica 61 (4): 821–856.
Azzalini, A. 1985. A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 12 (2): 171–178.
Azzalini, A., and A. Capitanio. 2003. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 65 (2): 367–389.
Badunenko, O., and S.C. Kumbhakar. 2016. When, where and how to estimate persistent and transient efficiency in stochastic frontier panel data models. European Journal of Operational Research 255 (1): 272–287.
Bandyopadhyay, D., and A. Das. 2006. On measures of technical inefficiency and production uncertainty in stochastic frontier production model with correlated error components. Journal of Productivity Analysis 26 (2): 165–180.
Barrow, D.F., and A.C. Cohen. 1954. On some functions involving Mill’s ratio. The Annals of Mathematical Statistics 25 (2): 405–408.
Battese, G.E., and T.J. Coelli. 1988. Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data. Journal of Econometrics 38 (3): 387–399.
Battese, G.E., and T.J. Coelli. 1992. Frontier production functions, technical efficiency and panel data: With application to paddy farmers in India. Journal of Productivity Analysis 3 (1): 153–169.
Battese, G.E., and T.J. Coelli. 1995. A model for technical inefficiency effects in a stochastic frontier production function for panel data. Empirical Economics 20 (2): 325–332.
Battese, G.E., and G.S. Corra. 1977. Estimation of a production frontier model: With application to the pastoral zone of Eastern Australia. Australian Journal of Agricultural Economics 21 (3): 169–179.
Battese, G.E., A.N. Rambaldi, and G.H. Wan. 1997. A stochastic frontier production function with flexible risk properties. Journal of Productivity Analysis 8 (3): 269–280.
Baumol, W.J. 1967. Business behavior, value and growth, revised ed. New York: Macmillan.
Beckers, D.E., and C.J. Hammond. 1987. A tractable likelihood function for the normal-gamma stochastic frontier model. Economics Letters 24 (1): 33–38.
Belotti, F., and G. Ilardi. 2018. Consistent inference in fixed-effects stochastic frontier models. Journal of Econometrics 202 (2): 161–177.
Bera, A.K., and S.C. Sharma. 1999. Estimating production uncertainty in stochastic frontier production function models. Journal of Productivity Analysis 12 (3): 187–210.
Blanco, G. 2017. Who benefits from job placement services? A two-sided analysis. Journal of Productivity Analysis 47 (1): 33–47.
Bonanno, G., D. De Giovanni, and F. Domma. 2017. The ‘wrong skewness’ problem: A re-specification of stochastic frontiers. Journal of Productivity Analysis 47 (1): 49–64.
Bracewell, R.N. 1978. The Fourier transform and its applications, 2nd ed. New York: McGraw-Hill.
Bradford, W.D., A.N. Kleit, M.A. Krousel-Wood, and R.N. Re. 2001. Stochastic frontier estimation of cost models within the hospital. The Review of Economics and Statistics 83 (2): 302–309.
Brorsen, B.W., and T. Kim. 2013. Data aggregation in stochastic frontier models: the closed skew normal distribution. Journal of Productivity Analysis 39 (1): 27–34.
Carree, M.A. 2002. Technological inefficiency and the skewness of the error component in stochastic frontier analysis. Economics Letters 77 (1): 101–107.
Caudill, S.B., and J.M. Ford. 1993. Biases in frontier estimation due to heteroscedasticity. Economics Letters 41 (1): 17–20.
Caudill, S.B., J.M. Ford, and D.M. Gropper. 1995. Frontier estimation and firm-specific inefficiency measures in the presence of heteroscedasticity. Journal of Business & Economic Statistics 13 (1): 105–111.
Chamberlain, G. 1984. Panel data. In Handbook of econometrics, ed. Z. Griliches and M.D. Intriligator, 1247–1318. Amsterdam: Elsevier.
Chen, Y.-Y., P. Schmidt, and H.-J. Wang. 2014. Consistent estimation of the fixed effects stochastic frontier model. Journal of Econometrics 181 (2): 65–76.
Coelli, T. 1995. Estimators and hypothesis tests for a stochastic frontier function: A Monte Carlo analysis. Journal of Productivity Analysis 6 (3): 247–268.
Coelli, T.J., D.S.P. Rao, and G.E. Battese. 2005. An introduction to efficiency and productivity analysis, 2nd ed. New York: Springer.
Colombi, R., S.C. Kumbhakar, G. Martini, and G. Vittadini. 2014. Closed-skew normality in stochastic frontiers with individual effects and long/short-run efficiency. Journal of Productivity Analysis 42 (2): 123–136.
Cornwell, C., P. Schmidt, and R.C. Sickles. 1990. Production frontiers with cross-sectional and time-series variation in efficiency levels. Journal of Econometrics 46 (1): 185–200.
Cuesta, R.A. 2000. A production model with firm-specific temporal variation in technical inefficiency: With application to Spanish dairy farms. Journal of Productivity Analysis 13 (2): 139–158.
Debreu, G. 1951. The coefficient of resource utilization. Econometrica 19 (3): 273–292.
Deprins, D., and L. Simar. 1989a. Estimating technical inefficiencies with correction for environmental conditions. Annals of Public and Cooperative Economics 60 (1): 81–102.
Deprins, D., and L. Simar. 1989b. Estimation de frontières déterministes avec facteurs exogènes d’inefficacitè. Annales d’Économie et de Statistique 14: 117–150.
El Mehdi, R., and C.M. Hafner. 2014. Inference in stochastic frontier analysis with dependent error terms. Mathematics and Computers in Simulation 102 (Suppl. C): 104–116.
Fan, J. 1991. On the optimal rates of convergence for nonparametric deconvolution problems. The Annals of Statistics 19 (3): 1257–1272.
Fan, J. 1992. Deconvolution with supersmooth distributions. The Canadian Journal of Statistics/La Revue Canadienne de Statistique 20 (2): 155–169.
Farrell, M.J. 1957. The measurement of productive efficiency. Journal of the Royal Statistical Society. Series A (General) 120 (3): 253–290.
Filippini, M., and W. Greene. 2016. Persistent and transient productive inefficiency: a maximum simulated likelihood approach. Journal of Productivity Analysis 45 (2): 187–196.
Gómez-Déniz, E., and J.V. Pérez-Rodríguez. 2015. Closed-form solution for a bivariate distribution in stochastic frontier models with dependent errors. Journal of Productivity Analysis 43 (2): 215–223.
González-Farı́as, G., J.A. Domı́nguez-Molina, and A.K. Gupta. 2004a. Additive properties of skew normal random vectors. Journal of Statistical Planning and Inference 126 (2): 521–534.
González-Farı́as, G., J.A. Domı́nguez-Molina, and A.K. Gupta. 2004b. The closed skew-normal distribution. In Skew-elliptical distributions and their applications, ed. M.G. Genton, 25–42. Boca Raton: Chapman and Hall/CRC.
Gouriéroux, C., A. Holly, and A. Monfort. 1982. Likelihood ratio test, Wald test, and Kuhn-Tucker test in linear models with inequality constraints on the regression parameters. Econometrica 50 (1): 63–80.
Greene, W.H. 1980. Maximum likelihood estimation of econometric frontier functions. Journal of Econometrics 13 (1): 27–56.
Greene, W.H. 1990. A gamma-distributed stochastic frontier model. Journal of Econometrics 46 (1): 141–163.
Greene, W.H. 2003. Simulated likelihood estimation of the normal-gamma stochastic frontier function. Journal of Productivity Analysis 19 (2): 179–190.
Greene, W.H. 2004. Distinguishing between heterogeneity and inefficiency: stochastic frontier analysis of the World Health Organization’s panel data on national health care systems. Health Economics 13 (10): 959–980.
Greene, W.H. 2005a. Fixed and random effects in stochastic frontier models. Journal of Productivity Analysis 23 (1): 7–32.
Greene, W.H. 2005b. Reconsidering heterogeneity in panel data estimators of the stochastic frontier model. Journal of Econometrics 126 (2): 269–303.
Greene, W.H. 2008. The econometric approach to efficiency analysis. In The measurement of productive efficiency and productivity growth, 2nd ed., ed. H.O. Fried, C.A.K. Lovell, and S.S. Schmidt, 92–159. Oxford: Oxford University Press.
Greene, W.H. 2010. A stochastic frontier model with correction for sample selection. Journal of Productivity Analysis 34 (1): 15–24.
Greene, W.H. 2016. LIMDEP Version 11.0 econometric modeling guide. Econometric Software.
Greene, W.H., and S. Misra. 2003. Simulated maximum likelihood estimation of general stochastic frontier regressions. Working Paper, William Simon School of Business, University of Rochester.
Griffin, J.E., and M.F.J. Steel. 2007. Bayesian stochastic frontier analysis using WinBUGS. Journal of Productivity Analysis 27 (3): 163–176.
Griffin, J.E., and M.F.J. Steel. 2008. Flexible mixture modelling of stochastic frontiers. Journal of Productivity Analysis 29 (1): 33–50.
Griffiths, W.E., and G. Hajargasht. 2016. Some models for stochastic frontiers with endogeneity. Journal of Econometrics. 190 (2): 341–348.
Groot, W., and H. Oosterbeek. 1994. Stochastic reservation and offer wages. Labour Economics 1 (3): 383–390.
Grushka, E. 1972. Characterization of exponentially modified Gaussian peaks in chromatography. Analytical Chemistry 44 (11): 1733–1738.
Gupta, A.K., and N. Nguyen. 2010. Stochastic frontier analysis with fat-tailed error models applied to WHO health data. International Journal of Innovative Management, Information & Production 1 (1): 43–48.
Hadri, K. 1999. Estimation of a doubly heteroscedastic stochastic frontier cost function. Journal of Business & Economic Statistics 17 (3): 359–363.
Hajargasht, G. 2014. The folded normal stochastic frontier model. Working Paper.
Hajargasht, G. 2015. Stochastic frontiers with a Rayleigh distribution. Journal of Productivity Analysis 44 (2): 199–208.
Hansen, B.E. 1996. Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64 (2): 413–430.
Hausman, J.A. 1978. Specification tests in econometrics. Econometrica 46 (6): 1251–1271.
Hausman, J.A., and W.E. Taylor. 1981. A generalized specification test. Economics Letters 8 (3): 239–245.
Heckman, J.J. 1976. The common structure of statistical models of truncation, sample selection and limited dependent variables and a simple estimator for such models. Annals of Economic and Social Measurement 5 (4): 17.
Heckman, J.J. 1978. Dummy endogenous variables in a simultaneous equation system. Econometrica 46 (4): 931–959.
Heckman, J.J. 1979. Sample selection bias as a specification error. Econometrica 47 (1): 153–161.
Heshmati, A., and S.C. Kumbhakar. 1994. Farm heterogeneity and technical efficiency: Some results from Swedish dairy farms. Journal of Productivity Analysis 5 (1): 45–61.
Hicks, J.R. 1935. Annual survey of economic theory: The theory of monopoly. Econometrica 3 (1): 1–20.
Hochberg, Y., and A.C. Tamhane. 1987. Multiple comparison procedures, 1st ed. New York: Wiley.
Horrace, W.C. 2005. Some results on the multivariate truncated normal distribution. Journal of Multivariate Analysis 94 (1): 209–221.
Horrace, W.C. 2015. Moments of the truncated normal distribution. Journal of Productivity Analysis 43 (2): 133–138.
Horrace, W.C., and C.F. Parmeter. 2011. Semiparametric deconvolution with unknown error variance. Journal of Productivity Analysis 35 (2): 129–141.
Horrace, W.C., and C.F. Parmeter. 2018. A Laplace stochastic frontier model. Econometric Reviews 37 (3): 260–280.
Horrace, W.C., and P. Schmidt. 2000. Multiple comparisons with the best, with economic applications. Journal of Applied Econometrics 15 (1): 1–26.
Hsu, J.C. 1981. Simultaneous confidence intervals for all distances from the “best”. The Annals of Statistics 9 (5): 1026–1034.
Hsu, J.C. 1984. Constrained simultaneous confidence intervals for multiple comparisons with the best. The Annals of Statistics 12 (3): 1136–1144.
Huang, C.J., and J.-T. Liu. 1994. Estimation of a non-neutral stochastic frontier production function. Journal of Productivity Analysis 5 (2): 171–180.
Johnston, J. 1960. Statistical cost analysis, 1st ed. New York: McGraw-Hill.
Jondrow, J., C.A. Knox Lovell, I.S. Materov, and P. Schmidt. 1982. On the estimation of technical inefficiency in the stochastic frontier production function model. Journal of Econometrics 19 (2): 233–238.
Jovanovic, B., and Y. Nyarko. 1996. Learning by doing and the choice of technology. Econometrica 64 (6): 1299–1310.
Jung, H. 2017. Adaptive LASSO for stochastic frontier models with many efficient firms. Working Paper, Maxwell School of Citizenship and Public Affairs, Syracuse University.
Just, R.E., and R.D. Pope. 1978. Stochastic specification of production functions and economic implications. Journal of Econometrics 7 (1): 67–86.
Kneip, A., R.C. Sickles, and W. Song. 2012. A new panel data treatment for heterogeneity in time trends. Econometric Theory 28 (3): 590–628.
Kodde, D.A., and F.C. Palm. 1986. Wald criteria for jointly testing equality and inequality restrictions. Econometrica 54 (5): 1243–1248.
Koopmans, T.C. 1951. Efficient allocation of resources. Econometrica 19 (4): 455–465.
Kumbhakar, S.C. 1990. Production frontiers, panel data, and time-varying technical inefficiency. Journal of Econometrics 46 (1): 201–211.
Kumbhakar, S.C. 1991. Estimation of technical inefficiency in panel data models with firm- and time-specific effects. Economics Letters 36 (1): 43–48.
Kumbhakar, S.C. 1993. Production risk, technical efficiency, and panel data. Economics Letters 41 (1): 11–16.
Kumbhakar, S.C., and A. Heshmati. 1995. Efficiency measurement in Swedish dairy farms: An application of rotating panel data, 1976–88. American Journal of Agricultural Economics 77 (3): 660–674.
Kumbhakar, S.C., and L. Hjalmarsson. 1995. Labour-use efficiency in Swedish social insurance offices. Journal of Applied Econometrics 10 (1): 33–47.
Kumbhakar, S.C., and C.A.K Lovell. 2000. Stochastic frontier analysis, 1st ed. Cambridge: Cambridge University Press.
Kumbhakar, S.C., and C.F. Parmeter. 2010. Estimation of hedonic price functions with incomplete information. Empirical Economics 39 (1): 1–25.
Kumbhakar, S.C., and K. Sun. 2013. Derivation of marginal effects of determinants of technical inefficiency. Economics Letters 120 (2): 249–253.
Kumbhakar, S.C., S. Ghosh, and J.T. McGuckin. 1991. A generalized production frontier approach for estimating determinants of inefficiency in U.S. dairy farms. Journal of Business & Economic Statistics 9 (3): 279–286.
Kumbhakar, S.C., E.G. Tsionas, and T. Sipiläinen. 2009. Joint estimation of technology choice and technical efficiency: an application to organic and conventional dairy farming. Journal of Productivity Analysis 31 (3): 151–161.
Kumbhakar, S.C., C.F. Parmeter, and E.G. Tsionas. 2013. A zero inefficiency stochastic frontier model. Journal of Econometrics 172 (1): 66–76.
Kumbhakar, S.C., G. Lien, and J.B. Hardaker. 2014. Technical efficiency in competing panel data models: A study of Norwegian grain farming. Journal of Productivity Analysis 41 (2): 321–337.
Lai, H.-P. 2015. Maximum likelihood estimation of the stochastic frontier model with endogenous switching or sample selection. Journal of Productivity Analysis 43 (1): 105–117.
Lai, H.-P., and C.J. Huang. 2013. Maximum likelihood estimation of seemingly unrelated stochastic frontier regressions. Journal of Productivity Analysis 40 (1): 1–14.
Lai, H.-P., S.W. Polachek, and H.-J. Wang. 2009. Estimation of a stochastic frontier model with a sample selection problem. Working Paper, Department of Economics, National Chung Cheng University.
Lee, L.-F. 1983. A test for distributional assumptions for the stochastic frontier functions. Journal of Econometrics 22 (3): 245–267.
Lee, L.-F. 1993. Asymptotic distribution of the maximum likelihood estimator for a stochastic frontier function model with a singular information matrix. Econometric Theory 9 (3): 413–430.
Lee, Y.H. 1996. Tail truncated stochastic frontier models. Journal of Economic Theory and Econometrics 2: 137–152.
Lee, L.-F., and A. Chesher. 1986. Specification testing when score test statistics are identically zero. Journal of Econometrics 31 (2): 121–149.
Lee, S., and Y.H. Lee. 2014. Stochastic frontier models with threshold efficiency. Journal of Productivity Analysis 42 (1): 45–54.
Lee, Y.H., and P. Schmidt. 1993. A production frontier model with flexible temporal variation in technical efficiency. In The measurement of productive efficiency: Techniques and applications, ed. H.O. Fried, S.S. Schmidt, and C.A.K. Lovell, 237–255. Oxford: Oxford University Press.
Lee, L.-F., and W.G. Tyler. 1978. The stochastic frontier production function and average efficiency. Journal of Econometrics 7 (3): 385–389.
Leibenstein, H. 1966. Allocative efficiency vs. “X-efficiency”. The American Economic Review 56(3), 392–415.
Leibenstein, H. 1975. Aspects of the X-efficiency theory of the firm. The Bell Journal of Economics 6 (2): 580–606.
Leibenstein, H. 1978. X-inefficiency Xists: Reply to an Xorcist. The American Economic Review 68 (1): 203–211.
Li, Q. 1996. Estimating a stochastic production frontier when the adjusted error is symmetric. Economics Letters 52 (3): 221–228.
Lukacs, E. and R.G. Laha. 1964. Applications of characteristic functions. London: Charles Griffin and Company.
Maddala, G.S. 1983. Limited-dependent and qualitative variables in econometrics. Cambridge: Cambridge University Press.
Marris, R.L. 1964. The economic theory of managerial capitalism, 1st ed. London: Macmillan.
Meeusen, W., and J. van Den Broeck. 1977. Efficiency estimation from Cobb-Douglas production functions with composed error. International Economic Review 18 (2): 435–444.
Migon, H.S., and E.V. Medici. 2001. Bayesian hierarchical models for stochastic production frontier. Working Paper, Universidade Federal do Rio de Janeiro.
Mundlak, Y. 1978. On the pooling of time series and cross section data. Econometrica 46 (1): 69–85.
Murillo-Zamorano, L.R. 2004. Economic efficiency and frontier techniques. Journal of Economic Surveys 18 (1): 33–77.
Nelsen, R.B. 2006. An introduction to copulas, 2nd ed. New York: Springer.
Nerlove, M. 1963. Returns to scale in electricity supply. In Measurement in economics: Studies in mathematical economics and econometrics, ed. C.F. Christ, M. Friedman, L.A. Goodman, Z. Griliches, A.C. Harberger, N. Liviatan, J. Mincer, Y. Mundlak, M. Nerlove, D. Patinkin, L.G. Telser, and H. Theil, 167–198. Stanford: Stanford University Press.
Nguyen, N. 2010. Estimation of technical efficiency in stochastic frontier analysis. PhD thesis, Bowling Green State University.
Oikawa, K. 2016. A microfoundation for stochastic frontier analysis. Economics Letters 139: 15–17.
Ondrich, J., and J. Ruggiero. 2001. Efficiency measurement in the stochastic frontier model. European Journal of Operational Research 129 (2): 434–442.
Orea, L., and S.C. Kumbhakar. 2004. Efficiency measurement using a latent class stochastic frontier model. Empirical Economics 29 (1): 169–183.
Pal, M. 2004. A note on a unified approach to the frontier production function models with correlated non-normal error components: The case of cross section data. Indian Economic Review 39 (1): 7–18.
Pal, M., and A. Sengupta. 1999. A model of FPF with correlated error components: An application to Indian agriculture. Sankhyā: The Indian Journal of Statistics, Series B 61 (2): 337–350.
Papadopoulos, A. 2015. The half-normal specification for the two-tier stochastic frontier model. Journal of Productivity Analysis 43 (2): 225–230.
Parmeter, C.F. 2018. Estimation of the two-tiered stochastic frontier model with the scaling property. Journal of Productivity Analysis 49 (1): 37–47.
Parmeter, C.F., and S.C. Kumbhakar. 2014. Efficiency analysis: A primer on recent advances. Foundations and Trends in Econometrics 7 (3–4): 191–385.
Pitt, M.M., and L.-F. Lee. 1981. The measurement and sources of technical inefficiency in the Indonesian weaving industry. Journal of Development Economics 9 (1): 43–64.
Polachek, S.W., and B.J. Yoon. 1987. A two-tiered earnings frontier estimation of employer and employee information in the labor market. The Review of Economics and Statistics 69 (2): 296–302.
Polachek, S.W., and B.J. Yoon. 1996. Panel estimates of a two-tiered earnings frontier. Journal of Applied Econometrics 11 (2): 169–178.
Qian, J., and R.C. Sickles. 2008. Stochastic frontiers with bounded inefficiency. Mimeo. Department of Economics, Rice University.
Reifschneider, D., and R. Stevenson. 1991. Systematic departures from the frontier: A framework for the analysis of firm inefficiency. International Economic Review 32 (3): 715–723.
Rho, S., and P. Schmidt. 2015. Are all firms inefficient? Journal of Productivity Analysis 43 (3): 327–349.
Richmond, J. 1974. Estimating the efficiency of production. International Economic Review 15 (2): 515–521.
Ritter, C., and L. Simar. 1997. Pitfalls of normal-gamma stochastic frontier models. Journal of Productivity Analysis 8 (2): 167–182.
Schmidt, P. 1976. On the statistical estimation of parametric frontier production functions. The Review of Economics and Statistics 58 (2): 238–239.
Schmidt, P., and C.A.K. Lovell. 1980. Estimating stochastic production and cost frontiers when technical and allocative inefficiency are correlated. Journal of Econometrics 13 (1): 83–100.
Schmidt, P., and R.C. Sickles. 1984. Production frontiers and panel data. Journal of Business & Economic Statistics. 2 (4): 367–374.
Self, S.G., and K.-Y. Liang. 1987. Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Journal of the American Statistical Association 82 (398): 605–610.
Shephard, R.W. 1953. Cost and production functions, 1st ed. Princeton: Princeton University Press.
Simar, L., and P.W. Wilson. 2010. Inferences from cross-sectional, stochastic frontier models. Econometric Reviews 29 (1): 62–98.
Simar, L., C.A.K. Lovell, and P. Vanden Eeckaut. 1994. Stochastic frontiers incorporating exogenous influences on efficiency. STAT Discussion Papers no. 9403, Institut de Statistique, Université Catholique de Louvain.
Sipiläinen, T., and A. Oude Lansink. 2005. Learning in switching to organic farming. NJF Report 1 (1): 169–172.
Smith, M.D. 2008. Stochastic frontier models with dependent error components. Econometrics Journal 11 (1): 172–192.
Smith, A.S.J., and P. Wheat. 2012. Estimation of cost inefficiency in panel data models with firm specific and sub-company specific effects. Journal of Productivity Analysis 37 (1): 27–40.
Stead, A.D. 2017. Regulation and efficiency in UK public utilities. PhD thesis, University of Hull.
Stead, A.D., P. Wheat, and W.H. Greene. 2018. Estimating efficiency in the presence of extreme outliers: A logistic-half normal stochastic frontier model with application to highway maintenance costs in England. In Productivity and inequality, ed W.H. Greene, L. Khalaf, P. Makdissi, R.C. Sickles, M. Veall, and M. Voia, 1–19. Springer.
Stevenson, R.E. 1980. Likelihood functions for generalized stochastic frontier estimation. Journal of Econometrics 13 (1): 57–66.
Stigler, G.J. 1976. The Xistence of X-efficiency. The American Economic Review 66 (1): 213–216.
Tancredi, A. 2002. Accounting for heavy tails in stochastic frontier models. Working Paper no. 2002.16, Department of Statistical Sciences, University of Padua.
Tchumtchoua, S., and D.K. Dey. 2007. Bayesian estimation of stochastic frontier models with multivariate skew t error terms. Communications in Statistics - Theory and Methods 36 (5): 907–916.
Terza, J.V. 2009. Parametric nonlinear regression with endogenous switching. Econometric Reviews 28 (6): 555–580.
Tibshirani, R. 1996. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B (Methodological) 58 (1): 267–288.
Timmer, C.P. 1971. Using a probabilistic frontier production function to measure technical efficiency. Journal of Political Economy 79 (4): 776–794.
Train, K.E. 2009. Discrete choice methods with simulation, 2nd ed. Cambridge: Cambridge University Press.
Tsagris, M., C. Beneki, and H. Hassani. 2014. On the folded normal distribution. Mathematics 2 (1): 12–28.
Tsionas, E.G. 2002. Stochastic frontier models with random coefficients. Journal of Applied Econometrics 17 (2): 127–147.
Tsionas, E.G. 2007. Efficiency measurement with the Weibull stochastic frontier. Oxford Bulletin of Economics and Statistics 69 (5): 693–706.
Tsionas, E.G. 2012. Maximum likelihood estimation of stochastic frontier models by the Fourier transform. Journal of Econometrics 170 (1): 234–248.
Tsionas, M.G. 2017. Microfoundations for stochastic frontiers. European Journal of Operational Research 258 (3): 1165–1170.
Tsionas, E.G., and S.C. Kumbhakar. 2014. Firm heterogeneity, persistent and transient technical inefficiency: a generalized true random-effects model. Journal of Applied Econometrics 29 (1): 110–132.
Waldman, D.M. 1984. Properties of technical efficiency estimators in the stochastic frontier model. Journal of Econometrics 25 (3): 353–364.
Wang, H.-J. 2002. Heteroscedasticity and Non-monotonic efficiency effects of a stochastic frontier model. Journal of Productivity Analysis 18 (3): 241–253.
Wang, J., and P. Schmidt. 2002. One-step and two-step estimation of the effects of exogenous variables on technical efficiency levels. Journal of Productivity Analysis 18: 129–144.
Wang, W.S., and P. Schmidt. 2009. On the distribution of estimated technical efficiency in stochastic frontier models. Journal of Econometrics 148 (1): 36–45.
Wang, W.S., C. Amsler, and P. Schmidt. 2011. Goodness of fit tests in stochastic frontier models. Journal of Productivity Analysis 35 (2): 95–118.
Weinstein, M.A. 1964. The sum of values from a normal and a truncated normal distribution. Technometrics 6 (1): 104–105.
Wheat, P., and A. Smith. 2012. Is the choice of (t−T) in Battese and Coelli (1992) type stochastic frontier models innocuous? Observations and generalisations. Economics Letters 116 (3): 291–294.
Wheat, P., W. Greene, and A. Smith. 2014. Understanding prediction intervals for firm specific inefficiency scores from parametric stochastic frontier models. Journal of Productivity Analysis 42 (1): 55–65.
Wheat, P., A.D. Stead, and W.H. Greene. 2017. Allowing for outliers in stochastic frontier models: A mixture noise distribution approach. 15th European Workshop on Efficiency and Productivity Analysis, London, UK.
Wheat, P., A.D. Stead, and W.H. Greene. 2019. Robust stochastic frontier analysis: A student’s t-half normal model with application to highway maintenance costs in England. Journal of Productivity Analysis 51 (1): 21–38. https://doi.org/10.1007/s11123-018-0541-y.
Williamson, O.E. 1963. Managerial discretion and business behavior. The American Economic Review 53 (5): 1032–1057.
Zellner, A. 1962. An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. Journal of the American Statistical Association 57 (298): 348–368.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 The Author(s)
About this chapter
Cite this chapter
Stead, A.D., Wheat, P., Greene, W.H. (2019). Distributional Forms in Stochastic Frontier Analysis. In: ten Raa, T., Greene, W. (eds) The Palgrave Handbook of Economic Performance Analysis. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-23727-1_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-23727-1_8
Published:
Publisher Name: Palgrave Macmillan, Cham
Print ISBN: 978-3-030-23726-4
Online ISBN: 978-3-030-23727-1
eBook Packages: Economics and FinanceEconomics and Finance (R0)