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


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.

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  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.


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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).

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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).

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

JEL Classification

  • C13
  • C14
  • C36