A note on the assumed distributions in stochastic frontier models

Abstract

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.

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

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

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

(3)

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.