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A note on the assumed distributions in stochastic frontier models

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Stochastic frontier models all need an assumption on the distributional form of the (in)efficiency component. Generally this efficiency component is assumed to be half normally, truncated normally, or exponentially distributed. This paper shows that the exponential distribution is, just like the half normal distribution, a special case of the truncated normal distribution. Moreover, this paper discusses the implications that this finding has on estimation.

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  1. The proof that \(C\) indeed goes to one and thus that the likelihood is properly defined if \(\mu \rightarrow -\infty\) and \(\sigma \rightarrow \infty\) is given in the “Appendix”.

  2. The other possibility, a converged truncated SF model on exponential data is left for future research. Since the parameters are at the bound of the parameter space and the nesting of the two models is based on a ratio it is not certain that standard tests are appropriate.


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I would like to thank one anonymous referee and the editor for their suggestions. This research was supported by the European Commission, Research Directorate General as part of the 7th Framework Programme, Theme 8, “Socio-Economic Sciences and Humanities” and is part of the project “Indicators for evaluating international performance in service sectors” (INDICSER).

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Correspondence to Aljar Meesters.

Appendix: Proof of \(C\rightarrow 1\)

Appendix: Proof of \(C\rightarrow 1\)

The \(C\) parameter in Eq. (2) can be written as:

$$\begin{aligned} C\left( \mu ,\lambda \right)&= \frac{\exp \left( \frac{\mu }{2\lambda }\right) \sqrt{-\mu \lambda }}{\varPhi \left( -\sqrt{\frac{-\mu }{\lambda }}\right) \left( -\mu \right) \sqrt{2\pi }}\nonumber \\&= \frac{\exp \left( \frac{\mu }{2\lambda }\right) \sqrt{\lambda }}{\varPhi \left( -\sqrt{\frac{-\mu }{\lambda }}\right) \sqrt{-\mu }\sqrt{2\pi }}\nonumber \\&= \frac{\frac{1}{\sqrt{2\pi }}\exp \left( \frac{\mu }{2\lambda }\right) }{\varPhi \left( -\sqrt{\frac{-\mu }{\lambda }}\right) }\sqrt{\frac{\lambda }{-\mu }}\nonumber \\&= \frac{\frac{1}{\sqrt{2\pi }}\exp \left( -\frac{1}{2}\frac{-\mu }{\lambda }\right) }{\varPhi \left( -\sqrt{\frac{-\mu }{\lambda }}\right) }\sqrt{\frac{\lambda }{-\mu }}\nonumber \\&= \frac{\phi \left( -\sqrt{\frac{-\mu }{\lambda }}\right) }{\varPhi \left( -\sqrt{\frac{-\mu }{\lambda }}\right) }\sqrt{\frac{\lambda }{-\mu }} \end{aligned}$$

Where \(\phi \left( \cdot \right)\) and \(\varPhi \left( \cdot \right)\) are the probability and the cumulative density function of the standard normal distribution respectively. The \(\phi \left( \cdot \right) /\varPhi \left( \cdot \right)\) term in Eq. (3) is also known as the inverse Mills ratio. Now, let \(\gamma =-\sqrt{-\mu /\lambda }\) and taking the limit gives:

$$\begin{aligned} \lim _{\gamma \rightarrow -\infty }C\left( \gamma \right)&= \lim _{\gamma \rightarrow -\infty }\frac{\phi \left( \gamma \right) }{\varPhi \left( \gamma \right) }\left( -\frac{1}{\gamma }\right) \\&= -\lim _{\gamma \rightarrow -\infty }\frac{\phi \left( \gamma \right) \frac{1}{\gamma }}{\varPhi \left( \gamma \right) }\\&\hbox {using l'Hopital's rule}\\&= -\lim _{\gamma \rightarrow -\infty }\frac{\phi '\left( \gamma \right) \frac{1}{\gamma }-\phi \left( \gamma \right) \frac{1}{\gamma ^{2}}}{\varPhi '\left( \gamma \right) }\\&= -\lim _{\gamma \rightarrow -\infty }\frac{-\gamma \phi \left( \gamma \right) \frac{1}{\gamma }-\phi \left( \gamma \right) \frac{1}{\gamma ^{2}}}{\phi \left( \gamma \right) }\\&= -\lim _{\gamma \rightarrow -\infty }\left( -1-\frac{1}{\gamma ^{2}}\right) \\&= 1 \end{aligned}$$

and thus \(C\) goes to one if \(\mu\) and \(\sigma\) go to minus infinity and infinity respectively.

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Meesters, A. A note on the assumed distributions in stochastic frontier models. J Prod Anal 42, 171–173 (2014).

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