Abstract
We propose a Bayesian approach for inference in the stochastic ray production frontier (SRPF), which can model multiple-input–multiple-output production technologies even in case of zero output quantities, i.e., if some outputs are not produced by some of the firms. However, the econometric estimation of the SRPF—as the estimation of distance functions in general—is susceptible to endogeneity problems. To address these problems, we apply a profit-maximizing framework to derive a system of equations after incorporating technical inefficiency. As the latter enters non-trivially into the system of equations and as the Jacobian is highly complicated, we use Monte Carlo methods of inference. Using US banking data to illustrate our innovative approach, we also address the problems of missing prices and the dependence on the ordering of the outputs via model averaging.
Similar content being viewed by others
Availability of data and code
The data set and MATLAB code are available in the Electronic Research Data Archive of the University of Copenhagen (https://doi.org/10.17894/ucph.2a18b504-f8f0-471b-93f8-d631df135721).
Notes
Our approach invokes the same assumptions as dual approaches, i.e., availability of input (and output) prices and the assumption of profit maximization (or at least cost minimization). However, in certain empirical applications, our approach has the following advantages: (a) Dual approaches require notable variation in input (and output) prices between observations. However, if transaction costs are low and markets are working well, the “law of one price” approximately holds, which leaves too little variation between observations for estimating a dual approach. By our alternative approach, there is no requirement for variation in input (and output) prices. (b) To obtain estimates of output-oriented technical efficiency, a dual approach is (at best) very complicated to implement. (c) Although, in theory, primal and dual approaches should give the same result, in practice this is not assured. Hence, the preference is for a primal approach, as presented in this paper.
We refer the reader to Löthgren (1997), Gerdtham et al. (1999) and Löthgren (2000) for more details on the SRPF. The traditional specification of a Translog Shephard output distance function that corresponds to the stochastic ray output distance function specified in Eq. (1) is \(\ln D(x,y) = \ln y_m + F( \ln x, \ln ( y / y_m ) )\), where \(F(\cdot )\) denotes a quadratic functional form and m with \(1 \le m \le M\) indicates an arbitrarily chosen output that is used as numéraire to impose linear homogeneity in output quantities. Hence, the traditional Translog specification fulfills linear homogeneity through an arbitrarily chosen output and implicitly assumes that \(\ln (D(x,y))\) is a quadratic function of the logarithms of the input quantities and the logarithms of the normalized output quantities, while the Translog SRPF functional form fulfills linear homogeneity through the length of the vector of output quantities and implicitly assumes that \(\ln (D(x,y))\) is a quadratic function of the logarithms of the input quantities and the angles of the vector of output quantities.
Note that \(\partial \vartheta _{m'}/\partial \ln y_{m}=\left( \partial \vartheta _{m'}/\partial y_{m}\right) y_{m}\), \(\partial \vartheta _{m'}/\partial \ln y_{m}=0\,\forall \,m<m'\) and \(\partial \mathrm {arccos}(z)/\partial z=-1/\sqrt{1-z^{2}}\).
While this model specification does not assume that error terms \({\mathbf {v}}_{it}\) are independent of the input and output quantities, it assumes that the inefficiency term \(u_{it}\) is independent of the input and output quantities. This assumption is typical in the literature.
Our model specification assumes that the input prices w are exogenous, which is a typical assumption in many empirical analyses, e.g., in analyses based on cost minimization. However, under certain circumstances, the input prices w could be endogenous, e.g., if differences in input prices between banks reflect heterogeneous inputs rather than “true” differences in input prices (see, e.g., Quiggin and Bui-Lan 1984).
The use of parametric assumptions to approximate unavailable output prices may introduce approximation errors. The absence of output prices (or even input prices) is common in the literature. In deciding to proceed, we demonstrate that our method is applicable even if output prices are unavailable.
To measure observation-specific technical efficiency, we use \(\frac{1}{R}\sum _{j=1}^{R} \exp \left( -u_{it}^{\left( j\right) }\right) \) \(\forall \) \(i=1,\ldots ,n;t=1,\ldots ,T\), where \(\left\{ u_{it}^{\left( j\right) } \right\} _{j=1}^{R}\) is a MCMC sample drawn from the posterior.
More information on the sensitivity of our results to the ordering of the outputs is given in the Online Supplement of this article, where we present detailed statistics of the technological metrics for all 60 different orderings of the outputs.
References
Aigner D, Lovell C, Schmidt P (1977) Formulation and estimation of stochastic frontier production function models. J Econom 6(1):21–37
Akhavein JD, Swamy PAVB, Taubman SB, Singamsetti RN (1997) A general method of deriving the inefficiencies of banks from a profit function. J Prod Anal 8(1):71–93
Atkinson SE, Dorfman JH (2005) Bayesian measurement of productivity and efficiency in the presence of undesirable outputs: crediting electric utilities for reducing air pollution. J Econom 126(2):445–468
Barten AP (1969) Maximum likelihood estimation of a complete system of demand equations. Eur Econ Rev 1(1):7–73
Battese GE, Coelli TJ (1992) Frontier production functions, technical efficiency and panel data: with application to paddy farmers in India. J Prod Anal 3(1):153–169. https://doi.org/10.1007/BF00158774
Battese GE, Corra GS (1977) Estimation of a production frontier model: with application to the pastoral zone of eastern Australia. Austr J Agric Econ 21(3):169–179
Berger AN, Mester LJ (1997) Inside the black box: What explains differences in the efficiencies of financial institutions? J Bank Finance 21(7):895–947
Bhattacharyya A, Pal S (2013) Financial reforms and technical efficiency in Indian commercial banking: a generalized stochastic frontier analysis. Rev Financ Econ 22(3):109–117
Brümmer B, Glauben T, Thijssen G (2002) Decomposition of productivity growth using distance functions: the case of dairy farms in three European countries. Am J Agric Econ 84(3):628–644
Coelli T, Perelman S (1999) A comparison of parametric and non-parametric distance functions: with application to european railways. Eur J Oper Res 117(2):326–339
Coelli T, Perelman S (2000) Technical efficiency of European railways: a distance function approach. Appl Econ 32(15):1967–1976
Cuesta RA, Orea L (2002) Mergers and technical efficiency in Spanish savings banks: a stochastic distance function approach. J Bank Finance 26(12):2231–2247
DiCiccio TJ, Kass RE, Raftery A, Wasserman L (1997) Computing Bayes factors by combining simulation and asymptotic approximations. J Am Stat Assoc 92:903–915
Färe R, Grosskopf S (1994) Cost and revenue constrained production. Springer, Berlin
Färe R, Primont D (1995) Multiple-output production and duality: theory and applications. Kluwer Academic Publishers, New York
Färe R, Primont D (1996) The opportunity cost of duality. J Prod Anal 7(2):213–224
Färe R, Grosskopf S, Lovell CAK, Yaisawarng S (1993) Derivation of shadow prices for undesirable outputs: a distance function approach. Rev Econ Stat 75(2):374–380
Ferrier GD, Lovell C (1990) Measuring cost efficiency in banking: econometric and linear programming evidence. J Econom 46(1):229–245
Filippini M, Farsi M (2004) An empirical analysis of cost efficiency in non-profit and public nursing homes. Ann Public Cooper Econ 75:339–365
Gelfand AE, Dey DK (1994) Bayesian model choice: asymptotics and exact calculations. J R Stat Soc B(56):501–514
Gerdtham UG, Löthgren M, Tambour M, Rehnberg C (1999) Internal markets and health care efficiency: a multiple-output stochastic frontier analysis. Health Econ 8(2):151–164
Greene W (2005) Fixed and random effects in stochastic frontier models. J Prod Anal 23(1):7–32
Grosskopf S, Hayes KJ, Taylor LL, Weber WL (1997) Budget-constrained frontier measures of fiscal equality and efficiency in schooling. Rev Econ Stat 79(1):116–124
Henningsen A, Henning CHCA (2009) Imposing regional monotonicity on translog stochastic production frontiers with a simple three-step procedure. J Prod Anal 32(3):217–229
Henningsen G, Henningsen A, Jensen U (2015) A Monte Carlo study on multiple output stochastic frontiers: a comparison of two approaches. J Prod Anal 44(3):309–320
Henningsen A, Bělín M, Henningsen G (2017) New insights into the stochastic ray production frontier. Econ Lett 156:18–21
Huang T, Wang M (2004) Comparisons of economic inefficiency between output and input measures of technical inefficiency using the Fourier flexible cost function. J Prod Anal 22(1):123–142
Humphrey DB, Pulley LB (1997) Banks’ responses to deregulation: profits, technology, and efficiency. J Money Credit Bank 29(1):73–93
Jondrow J, Lovell CK, Materov IS, Schmidt P (1982) On the estimation of technical inefficiency in the stochastic frontier production function model. J Econom 19(2):233–238
Koop G, Steel MF (2001) Bayesian analysis of stochastic frontier models. In: Baltagi BH (ed) A companion to theoretical econometrics. Blackwell, pp 520–537
Koop G, Osiewalski J, Steel MFJ (1994) Bayesian efficiency analysis with a flexible form: the aim cost function. J Bus Econ Stat 12(3):339–346
Koop G, Steel M, Osiewalski J (1995) Posterior analysis of stochastic frontier models using Gibbs sampling. Comput Stat 10:353–373
Koop G, Osiewalski J, Steel MF (1997) Bayesian efficiency analysis through individual effects: hospital cost frontiers. J Econom 76(1):77–105
Kumbhakar SC (1990) Production frontiers, panel data, and time-varying technical inefficiency. J Econom 46(1):201–211. https://doi.org/10.1016/0304-4076(90)90055-X
Kumbhakar SC (2001) Estimation of profit functions when profit is not maximum. Am J Agric Econ 83(1):1–19
Kumbhakar SC, Bhattacharyya A (1992) Price distortions and resource-use efficiency in Indian agriculture: a restricted profit function approach. Rev Econ Stat 74(2):231–239
Kumbhakar SC, Heshmati A (1995) Efficiency measurement in Swedish dairy farms: an application of rotating panel data, 1976–88. Am J Agric Econ 77(3):660–674
Kumbhakar SC, Lovell CAK (2000) Stochastic frontier analysis. Cambridge University Press, Cambridge
Löthgren M (1997) Generalized stochastic frontier production models. Econ Lett 57:255–259
Löthgren M (2000) Specification and estimation of stochastic multiple-output production and technical inefficiency. Appl Econ 32(12):1533–1540
Lovell CAK, Travers P, Richardson S, Wood L (1994) Resources and functionings: a new view of inequality in Australia. In: Eichhorn W (ed) Models and measurement of welfare and inequality. Springer, Berlin, pp 787–807. https://doi.org/10.1007/978-3-642-79037-9_41
Malikov E, Kumbhakar SC, Tsionas MG (2015) A cost system approach to the stochastic directional technology distance function with undesirable outputs: the case of US banks in 2001–2010. J Appl Econom 31(7):1407–1429
Meeusen W, van den Broeck J (1977) Efficiency estimation from cobb-douglas production functions with composed error. Int Econ Rev 18(2):435–44
Niquidet K, Nelson H (2010) Sawmill production in the interior of British Columbia: a stochastic ray frontier approach. J For Econ 16(4):257–267
O’Donnell CJ, Coelli TJ (2005) A Bayesian approach to imposing curvature on distance functions. J Econom 126(2):493–523
Orea L, Kumbhakar SC (2004) Efficiency measurement using a latent class stochastic frontier model. Empir Econ 29(1):169–183
Pitt MM, Lee LF (1981) The measurement and sources of technical inefficiency in the Indonesian weaving industry. J Dev Econ 9(1):43–64
Quiggin J, Bui-Lan A (1984) The use of cross-sectional estimates of profit functions for tests of relative efficiency: a critical review. Aust J Agric Econ 28(1):44–55
Rosko MD (2001) Cost efficiency of us hospitals: a stochastic frontier approach. Health Econ 10(6):539–551
Schmidt P, Sickles RC (1984) Production frontiers and panel data. J Bus Econ Stat 2(4):367–374
Shephard RW (1970) Theory of cost and production functions. Princeton University Press, Princeton
Sickles RC, Good DH, Getachew L (2002) Specification of distance functions using semi- and nonparametric methods with an application to the dynamic performance of eastern and western European air carriers. J Prod Anal 17(1):133–155
Terrell D (1996) Incorporating monotonicity and concavity conditions in flexible functional forms. J Appl Econom 11:179–194
Tsionas EG (2000) Full likelihood inference in normal-gamma stochastic frontier models. J Prod Anal 13(3):183–205
Tsionas EG (2002) Stochastic frontier models with random coefficients. J Appl Econom 17(2):127–147
Tsionas EG (2006) Inference in dynamic stochastic frontier models. J Appl Econom 21(5):669–676
Tsionas EG, Kumbhakar SC, Malikov E (2015) Estimation of input distance functions: a system approach. Am J Agric Econ 97(5):1478–1493
van den Broeck J, Koop G, Osiewalski J, Steel MF (1994) Stochastic frontier models: a Bayesian perspective. J Econom 61(2):273–303
Vivas AL (1997) Profit efficiency for Spanish savings banks. Eur J Oper Res 98(2):381–394
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
All authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors wish to thank Subal C. Kumbhakar and two anonymous reviewers for valuable discussions and comments. E. Paravalos gratefully acknowledges financial support from the General Secretariat for Research and Technology (GSRT) and the Hellenic Foundation for Research and Innovation (HFRI).
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Tsionas, M., Izzeldin, M., Henningsen, A. et al. Addressing endogeneity when estimating stochastic ray production frontiers: a Bayesian approach. Empir Econ 62, 1345–1363 (2022). https://doi.org/10.1007/s00181-021-02060-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00181-021-02060-0
Keywords
- Stochastic ray production frontier
- Technical inefficiency
- Endogeneity
- Bayesian inference
- Model averaging