Skip to main content

Estimation of semi- and nonparametric stochastic frontier models with endogenous regressors

Abstract

This paper considers the problem of estimating a nonparametric stochastic frontier model with shape restrictions and when some or all regressors are endogenous. We discuss three estimation strategies based on constructing a likelihood with unknown components. One approach is a three-step constrained semiparametric limited information maximum likelihood, where the first two steps provide local polynomial estimators of the reduced form and frontier equation. This approach imposes the shape restrictions on the frontier equation explicitly. As an alternative, we consider a local limited information maximum likelihood, where we replace the constrained estimation from the first approach with a kernel-based method. This means the shape constraints are satisfied locally by construction. Finally, we consider a smooth-coefficient stochastic frontier model, for which we propose a two-step estimation procedure based on local GMM and MLE. Our Monte Carlo simulations demonstrate attractive finite sample properties of all the proposed estimators. An empirical application to the US banking sector illustrates empirical relevance of these methods.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. The term “nonparametric” used in this paper refers specifically to the assumptions on the functional form of the frontier. That is, no specific functional form of the frontier is assumed.

  2. See, for example, Kumbhakar et al. (1991), Huang and Liu (1994), Battese and Coelli (1995), Caudill et al. (1995), Wang (2002) and Amsler et al. (2014) and the references therein.

  3. In the parametric stochastic frontier literature, the Cobb–Douglas specification has been used most frequently in practice.

  4. We thank the Special Issue editors for bringing these points to our attention.

  5. We thank an anonymous referee for pointing out this estimator to us.

  6. We would like to thank an anonymous referee for pointing these out.

  7. For conservation of space, we do not report these results here but they are available from the authors upon request.

  8. Let \(x_{(1)} \le x_{(2)} \le \cdots \le x_{(n)}\) be rank ordered values. We say that the function \(g(\cdot )\) violates the monotonically increasing condition at x(i) if \(g(x_{(i)}) < g(x_{(i-1)})\). Moreover, we say that the function \(g(\cdot )\) violates the concavity condition at \(x_{i}\) if its Hessian matrix is positive definite.

References

  • Amsler C, Prokhorov A, Schmidt P (2014) Using copulas to model time dependence in stochastic frontier models. Econ Rev 33(5–6):497–522

    Article  Google Scholar 

  • Amsler C, Prokhorov A, Schmidt P (2016) Endogeneity in stochastic frontier models. J Econom 190:280–288

    Article  Google Scholar 

  • Amsler C, Prokhorov A, Schmidt P (2017) Endogenous environmental variables in stochastic frontier models. J Econom 199:131–140

    Article  Google Scholar 

  • Battese GE, Coelli TJ (1995) A model for technical inefficiency effects in a stochastic frontier production function for panel data. Empirical Econ 20:325–332

    Article  Google Scholar 

  • Caudill SB, Ford JM, Gropper DM (1995) Frontier estimation and firm-specific inefficiency measures in the presence of heteroscedasticity. J Bus Econ Stat 13(1):105–111

  • Du P, Parmeter CF, Racine JS (2013) Nonparametric kernal regression with multiple predictors and multiple shape constraints. Stat Sin 23:1347–1371

    Google Scholar 

  • Fan Y, Li Q, Weersink A (1996) Semiparametric estimation of stochastic production frontier models. J Bus Econ Stat 14:460–68

    Google Scholar 

  • Freyberger J, Horowitz JL (2015) Identification and shape restrictions in nonparametric instrumental variables estimation. J Econom 189:41–53

    Article  Google Scholar 

  • Färe R, Grosskopf S, Noh D-W, Weber W (2005) Characteristics of a polluting technology: theory and practice. J Econom 126:469–492

    Article  Google Scholar 

  • Gozalo P, Linton O (2000) Local nonlinear least squares: using parametric information in nonparametric regression. J Econom 99:63–106

    Article  Google Scholar 

  • Griffiths WE, Hajargasht G (2016) Some models for stochastic frontiers with endogeneity. J Econom 190:341–348

    Article  Google Scholar 

  • Hall P, Huang L-S (2001) Nonparametric kernel regression subject to monotonicity constraints. Ann Stat 624–647

  • Henderson DJ, List JA, Millimet DL, Parmeter CF, Price MK (2012) Empirical implementation of nonparametric first-price auction models. J Econom 168:17–28

    Article  Google Scholar 

  • Henderson DJ, Parmeter CF (2009) Imposing economic constraints in nonparametric regression: survey, implementation, and extension. Adv Econom 25:433–69

    Article  Google Scholar 

  • Henderson DJ, Parmeter CF (2015) Model averaging over nonparametric estimators. In: Essays in Honor of Aman Ullah, advances in econometrics, vol 36

  • Huang CJ, Liu J (1994) Estimation of a non-neutral stochastic frontier production function. J Prod Anal 5:171–180

    Article  Google Scholar 

  • Jondrow J, Lovell CK, Materov IS, Schmidt P (1982) On the estimation of technical inefficiency in the stochastic frontier production function model. J Econom 19:233–238

    Article  Google Scholar 

  • Karakaplan M, Kutlu L (2015) Handling endogeneity in stochastic frontier analysis. Available at SSRN 2607276

  • Kumbhakar SC, Ghosh S, McGuckin JT (1991) A generalized production frontier approach for estimating determinants of inefficiency in U.S. dairy farms. J Bus Econ Stat 9(3):279–286

    Google Scholar 

  • Kumbhakar SC, Park BU, Simar L, Tsionas EG (2007) Nonparametric stochastic frontiers: a local maximum likelihood approach. J Econom 137:1–27

    Article  Google Scholar 

  • Kumbhakar SC, Sun K, Zhang R (2016) Semiparametric smooth coefficient estimation of a production system. Pac Econ Rev 21:464–482

    Article  Google Scholar 

  • Kutlu L (2010) Battese–Coelli estimator with endogenous regressors. Econ Lett 109:79–81

    Article  Google Scholar 

  • Kutlu L, Tran KC (2019) Heterogeneity and endogeneity in panel stochastic frontier models. In: Panel data econometrics. Elsevier, pp 131–146

  • Lin X, Carroll RJ (2000) Nonparametric function estimation for clustered data when the predictor is measured without/with error. J Am Stat Assoc 95:520–534

    Article  Google Scholar 

  • Malikov E, Kumbhakar SC, Sun Y (2016a) Varying coefficient panel data model in the presence of endogenous selectivity and fixed effects. J Econom 190:233–251

    Article  Google Scholar 

  • Malikov E, Kumbhakar SC, Tsionas MG (2016b) 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 31:1407–1429

    Article  Google Scholar 

  • Martins-Filho C, Yao F (2007) Nonparametric frontier estimation via local linear regression. J Econom 141:283–319

    Article  Google Scholar 

  • Martins-Filho C, Yao F (2009) Nonparametric regression estimation with general parametric error covariance. J Multivar Anal 100:309–333

    Article  Google Scholar 

  • McManus DA (1994) Making the Cobb–Douglas functional form an efficient nonparametric estimator through localization

  • Parmeter C, Racine J (2019) Nonparametric estimation and inference for panel data models

  • Parmeter CF, Racine JS (2013) Smooth constrained frontier analysis. Springer, New York, pp 463–488

    Google Scholar 

  • Restrepo-Tobón D, Kumbhakar SC (2015) Nonparametric estimation of returns to scale using input distance functions: an application to large US banks. Empir Econ 48:143–168

    Article  Google Scholar 

  • Simar L, Van Keilegom I, Zelenyuk V (2017) Nonparametric least squares methods for stochastic frontier models. J Prod Anal 47:189–204

    Article  Google Scholar 

  • Su L, Ullah A (2008) Local polynomial estimation of nonparametric simultaneous equations models. J Econom 144:193–218

    Article  Google Scholar 

  • Su L, Ullah A, Wang Y (2013) Nonparametric regression estimation with general parametric error covariance: a more efficient two-step estimator. Empir Econ 45:1009–1024

    Article  Google Scholar 

  • Sun K (2015) Constrained nonparametric estimation of input distance function. J Prod Anal 43:85–97

    Article  Google Scholar 

  • Sun K, Kumbhakar SC (2013) Semiparametric smooth-coefficient stochastic frontier model. Econ Lett 120:305–309

    Article  Google Scholar 

  • Tran KC, Tsionas EG (2009) Local GMM estimation of semiparametric panel data with smooth coefficient models. Econom Rev 29:39–61

    Article  Google Scholar 

  • Tran KC, Tsionas EG (2013) GMM estimation of stochastic frontier model with endogenous regressors. Econ Lett 118:233–236

    Article  Google Scholar 

  • Tran KC, Tsionas EG (2015) Endogeneity in stochastic frontier models: copula approach without external instruments. Econ Lett 133:85–88

    Article  Google Scholar 

  • Wang H (2002) Heteroscedasticity and non-monotonic efficiency effects of a stochastic frontier model. J Prod Anal 18:241–253

    Article  Google Scholar 

Download references

Acknowledgements

We would like to thank the Editors: Subal Kumbhakar, Christopher Parmeter and Amir Malikov, and two anonymous referees to helpful comments and suggestions that led to substantial improvement of the paper. All the remaining errors are our responsibilities. Artem Prokhorov’s research for this paper was supported by a grant from the Russian Science Foundation (Project No.20-18-00365).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Artem Prokhorov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Prokhorov, A., Tran, K.C. & Tsionas, M.G. Estimation of semi- and nonparametric stochastic frontier models with endogenous regressors. Empir Econ 60, 3043–3068 (2021). https://doi.org/10.1007/s00181-020-01941-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00181-020-01941-0

Keywords

  • Constrained semiparametric limited information MLE
  • Efficiency
  • Endogeneity
  • Local limited information MLE
  • Smooth coefficient
  • Stochastic frontier

JEL Classification

  • C13
  • C14
  • C36