Abstract
This paper proposes a tail-truncated stochastic frontier model that allows for the truncation of technical efficiency from below. The truncation bound implies the inefficiency threshold for survival. Specifically, this paper assumes a uniform distribution of technical inefficiency and derives the likelihood function. Even though this distributional assumption imposes a strong restriction that technical inefficiency has a uniform probability density over [0, θ], where θ is the threshold parameter, this model has two advantages: (1) the reduction in the number of parameters compared with more complicated tail-truncated models allows better performance in numerical optimization; and (2) it is useful for empirical studies of the distribution of efficiency or productivity, particularly the truncation of the distribution. The Monte Carlo simulation results support the argument that this model approximates the distribution of inefficiency precisely, as the data-generating process not only follows the uniform distribution but also the truncated half-normal distribution if the inefficiency threshold is small.
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Notes
We would like to thank one of the reviewers for pointing this out.
Greene (2008) also pointed out that \(\lambda\) gives a misleading picture of the ratio of variances in the half-normal model since \(\sigma_{u}^{2}\) is about three times larger than \(Var(u_{i} )\).
We report the estimates of \(\theta\) only for the true model, since the same \(\theta\) in the uniform distribution and the truncated half normal distribution are not the same parameter. Suppose DGP follows the uniform distribution (DGP1). THN estimates \(\theta\) that is the bound in the truncated half normal. Therefore, the differences between the estimates of \(\theta\) of THN and the true value of \(\theta\) in the uniform distribution are not informative.
According to our pilot study, we experienced stoppage in the process of numerical maximization of the logged likelihood function of THN too many times when N = 100. Thus, we failed to report the estimation performance of THN when N = 100.
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Acknowledgments
Young Hoon Lee would like to acknowledge the supports by the National Research Foundation of Korea Grant funded by Korean Government (NRF-2010-330-B00069) and Sogang University Research Grant.
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An earlier draft of this paper was based on the thesis of Sungwon Lee.
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Lee, S., Lee, Y.H. Stochastic frontier models with threshold efficiency. J Prod Anal 42, 45–54 (2014). https://doi.org/10.1007/s11123-013-0364-9
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DOI: https://doi.org/10.1007/s11123-013-0364-9
Keywords
- Stochastic frontier
- Technical efficiency
- Threshold inefficiency
- Uniform distribution
- Productivity distribution