Skip to main content

Estimation and efficiency evaluation of stochastic frontier models with interval dependent variables

Abstract

This paper considers the maximum likelihood estimation of a stochastic frontier production function with an interval outcome. We derive an analytical formula for calculating the likelihood function of interval stochastic frontier models. Monte Carlo experiments reveal that the finite sample performance of our method is promising even when the sample size is relatively moderate. We also provide an exact formula for evaluating technical efficiency with interval outcome and apply our method to measure information inefficiency in the labor market for newly graduated college students in Taiwan.

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

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

Notes

  1. 1.

    Other distribution assumptions for u have also been discussed in the literature, including truncated normal distribution (Stevenson 1980), gamma distribution (Greene 1990), and Weibull distribution (Tsionas 2007). Recently, Badade and Ramanathan (2020) propose a probabilistic frontier model using a logit model specification.

  2. 2.

    For discussion and evaluation of c1 and c2, readers are referred to page 261 of Tsay et al. (2013).

  3. 3.

    The original survey conducted by Peng (2005) has nine categories for worker’s monthly wage: (1) less than 15,840, (2) between 15,840 and 22,800, (3) between 22,800 and 28,800, (4) between 28,800 and 36,300, (5) between 36,300 and 45,800, (6) between 45,800 and 57,800, (7) between 57,800 and 72,800, (8) between 72,800 and 83,900, and (9) more than 83,900. However, some categories contain only small amount of observations. For example, there are only 39 observations in the first category and only 213 observations in total in categories (6)–(9). To keep the number of observations in each category more balanced, we merge the first two categories, and we combine categories (5)–(9) into one.

  4. 4.

    The efficiency estimates certainly depend on the specification of model used for empirical analysis and the distribution assumption imposed on composite error. Therefore, we treat our empirical results as a starting point of analyzing the market inefficiency of recent Taiwanese college graduates. More empirical works and data collection are useful to enhance our knowledge about this important issue.

References

  1. Abramowitz M, Stegun IA (1970) Handbook of mathematical functions. Dover, New York

    Google Scholar 

  2. Aigner D, Lovell CAK, Schmidt P (1977) Formulation and estimation of stochastic frontier production function models. J Econom 6:21–37

    Article  Google Scholar 

  3. Amsler C, Papadopoulos A, Schmidt P (2020) Evaluating the CDF of the skew normal distribution. Empir Econ. https://doi.org/10.1007/s00181-020-01868-6

  4. Amsler C, Schmidt P, Tsay WJ (2019) Evaluating the CDF of the distribution of the stochastic frontier composed error. J Prod Anal 52(1):29–35

    Article  Google Scholar 

  5. Badade M, Ramanathan TV (2020) Probabilistic frontier regression model for multinomial ordinal type output data. J Prod Anal 53:339–354

    Article  Google Scholar 

  6. Card D, Krueger A (1992) Does school quality matter? Returns to education and the characteristics of public schools in the United States. J Political Econ 100(1):1–40

    Article  Google Scholar 

  7. Carson R (2012) Contingent valuation: a comprehensive bibliography and history. Edward Elgar Publishing.

  8. Flecher C, Allard D, Naveau P (2009) Truncated skew-normal distributions: estimation by weighted moments and application to climatic data. Metron 68:331–345

    Article  Google Scholar 

  9. Fu T-T (2011) School quality, operational efficiency, and optimal size: an analysis of higher education institutions in Taiwan. J Res Educ Sci 56(3):181–213

    Google Scholar 

  10. Greene WH (1990) A Gamma-distributed stochastic frontier model. J Econom 46:141–163

    Article  Google Scholar 

  11. Griffiths W, Zhang X, Zhao X (2014) Estimation and efficiency measurement in stochastic production frontiers with ordinal outcomes. J Prod Anal 42(1):67–84

    Article  Google Scholar 

  12. Hofler RA, Polachek SW (1985) A new approach for measuring wage ignorance in the labor market. J Econ Bus 37(3):267–276

    Article  Google Scholar 

  13. Jondrow J, Lovell CAK, 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 

  14. Kumbhakar SC, Lothgren M (1998) Monte Carlo analysis of technical inefficiency predictors. Working paper series in Economics and Finance, No. 229, Stockholm School of Economics

  15. Olson JA, Schmidt P, Waldman DM (1980) A Monte Carlo study of estimators of stochastic frontier production functions. J Econom 13(1):67–82

    Article  Google Scholar 

  16. Peng SM (2005) Taiwan higher education data system and its applications. National Science Council research report, National Tsing-Hua University, Taiwan

  17. Polachek SW, Yoon BJ (1996) Panel estimates of a two-tiered earnings frontier. J Appl Econom 11(2):169–178

    Article  Google Scholar 

  18. Stevenson RE (1980) Likelihood functions for generalized stochastic frontier estimation. J Econom 13:57–66

    Article  Google Scholar 

  19. Tsay W-J, Huang CJ, Fu T-T, Ho I-L (2013) A simple closed-form approximation for the cumulative distribution function of the composite error of stochastic frontier models. J Prod Anal 39:259–269

    Article  Google Scholar 

  20. Tsionas EG (2007) Efficiency measurement with the Weibull stochastic frontier. Oxf Bull Econ Stat 69:693–706

    Article  Google Scholar 

Download references

Acknowledgements

We thank for the valuable comments from Peter Schmidt of Michigan State University and the participants of the DEAIC2017 conference at Hefei, China.

Author information

Affiliations

Authors

Corresponding authors

Correspondence to Shih-Tang Hwu, Tsu-Tan Fu or Wen-Jen Tsay.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

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

Appendix

Appendix

For convenience of exposition, we suppress subindex throughout the Appendix, and two equations from (Abramowitz and Stegun, 1970, Eqs. 7.11 and 7.4.32) are given:

$$Erf(z)=\frac{2}{\sqrt{\pi }}{\displaystyle\int\nolimits_{0}^{z}}\exp (-{t}^{2})=2{\displaystyle\int\nolimits_{0}^{\sqrt{2}z}}\phi (t)dt,$$
$$\displaystyle\int \exp -\left(k{x}^{2}+2mx+n\right)dx=\frac{1}{2}\sqrt{\frac{\pi }{k}}\exp \left(\frac{{m}^{2}-kn}{k}\right)Erf\left(\sqrt{k}x+\frac{m}{\sqrt{k}}\right)+C,\ k\,\ne \,0,$$

where C denotes a finite constant.

Derivation of Proposition 1

Given that (Q, a, b) ∈ R, b > 0, Erf(− x) = − Erf(x), and define \(\varepsilon =\sqrt{2}v\), we have:

$$\begin{array}{ll}I(Q,a,b)&=\sqrt{2}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\left({\displaystyle\int\nolimits_{-\infty }^{\sqrt{2}av}}\phi \left(\zeta \right)d\zeta \right)\phi \left(\sqrt{2}bv\right)dv\\ &=\frac{\sqrt{2}}{2}{\displaystyle\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\end{array}$$

Tsay et al. (2013) show that Erf(x) can be well approximated by a function, \(g(x)=1-\exp \left({c}_{1}x+{c}_{2}{x}^{2}\right)\) for x ≥ 0, where c1 and c2 are discussed in “Interval stochastic frontier model.” We divide the derivation into four cases: (Q ≥ 0, a ≥ 0), (Q ≤ 0, a ≥ 0), (Q ≥ 0, a ≤ 0), and (Q ≤ 0, a ≤ 0).

Case 1. (Q ≥ 0, a ≥ 0):

$$\begin{array}{ll}I(Q,a,b)\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\quad\quad\,\,=\,\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{0}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\quad\quad+\,\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{0}^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\quad\quad\,\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{0}^{\infty }}\left(1-Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\qquad+\,\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{0}^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\quad\quad\,\,\approx \frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{0}^{\infty }}\exp \left({c}_{1}av+{c}_{2}{a}^{2}{v}^{2}\right)\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\qquad+\,\sqrt{2}{\displaystyle\int\nolimits_{0}^{\frac{Q}{\sqrt{2}}}}\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\qquad-\,\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{0}^{\frac{Q}{\sqrt{2}}}}\exp \left({c}_{1}av+{c}_{2}{a}^{2}{v}^{2}\right)\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\quad\quad\,\,=\frac{1}{2\sqrt{\pi }}{\displaystyle\int\nolimits_{0}^{\infty }}\exp \left(-({b}^{2}-{a}^{2}{c}_{2}){v}^{2}+a{c}_{1}v\right)dv\\ \qquad\qquad\qquad+\,\frac{1}{\sqrt{\pi }}{\displaystyle\int\nolimits_{0}^{\frac{Q}{\sqrt{2}}}}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\qquad-\,\frac{1}{2\sqrt{\pi }}{\displaystyle\int\nolimits_{0}^{\frac{Q}{\sqrt{2}}}}\exp \left(-({b}^{2}-{a}^{2}{c}_{2}){v}^{2}+a{c}_{1}v\right)dv.\end{array}$$

When we use (7.4.32) of Abramowitz and Stegun (1970), I(Q, a, b) in this case can be approximated by:

$$\begin{array}{ll}I(Q,a,b)\,\approx \frac{\exp \left(\frac{{a}^{2}{c}_{1}^{2}}{4{b}^{2}-4{a}^{2}{c}_{2}}\right)\left[1-Erf\left(\frac{-a{c}_{1}+\sqrt{2}Q({b}^{2}-{a}^{2}{c}_{2})}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)\right]}{4\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}+\frac{1}{2b}Erf\left(\frac{bQ}{\sqrt{2}}\right).\end{array}$$

Case 2. (Q ≤ 0, a ≥ 0):

$$\begin{array}{ll}I(Q,a,b)\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\,\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\frac{Q}{\sqrt{2}}}^{\infty }}\left(1-Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\,\,\approx \frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\frac{Q}{\sqrt{2}}}^{\infty }}\exp \left({c}_{1}av+{c}_{2}{a}^{2}{v}^{2}\right)\frac{1}{\sqrt{2\pi }}\exp \left(-{b}^{2}{v}^{2}\right)dv\\ \qquad\qquad\,\,=\frac{1}{2\sqrt{\pi }}{\displaystyle\int\nolimits_{-\frac{Q}{\sqrt{2}}}^{\infty }}\exp \left(-({b}^{2}-{a}^{2}{c}_{2}){v}^{2}+a{c}_{1}v\right)dv.\end{array}$$

When we use (7.4.32) of Abramowitz and Stegun (1970), I(Q, a, b) in this case can be approximated by:

$$I(Q,a,b)\approx \frac{\exp \left(\frac{{a}^{2}{c}_{1}^{2}}{4{b}^{2}-4{a}^{2}{c}_{2}}\right)\left[1-Erf\left(\frac{-a{c}_{1}-\sqrt{2}Q({b}^{2}-{a}^{2}{c}_{2})}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)\right]}{4\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}.$$

Case 3. (Q ≥ 0, a ≤ 0):

$$\begin{array}{ll}I(Q,a,b)\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\,\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{0}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\qquad+\,\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{0}^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\,\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{0}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\qquad+\,\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\frac{Q}{\sqrt{2}}}^{0}}\left(1-Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\,\,\approx -\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{0}}\exp \left({c}_{1}av+{c}_{2}{a}^{2}{v}^{2}\right)\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\qquad+\,\sqrt{2}{\displaystyle\int\nolimits_{-\infty }^{0}}\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\qquad+\,\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\frac{Q}{\sqrt{2}}}^{0}}\exp \left({c}_{1}av+{c}_{2}{a}^{2}{v}^{2}\right)\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\,\,=-\frac{1}{2\sqrt{\pi }}{\displaystyle\int\nolimits_{-\infty }^{0}}\exp \left(-({b}^{2}-{a}^{2}{c}_{2}){v}^{2}+a{c}_{1}v\right)dv\\ \qquad\qquad\qquad+\,\frac{1}{\sqrt{\pi }}{\displaystyle\int\nolimits_{-\infty }^{0}}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\qquad+\,\frac{1}{2\sqrt{\pi }}{\displaystyle\int\nolimits_{-\frac{Q}{\sqrt{2}}}^{0}}\exp \left(-({b}^{2}-{a}^{2}{c}_{2}){v}^{2}+a{c}_{1}v\right)dv.\end{array}$$

When we use (7.4.32) of Abramowitz and Stegun (1970), I(Q, a, b) in this case can be approximated by:

$$\begin{array}{ll}I(Q,a,b)\,\approx -\frac{\exp \left(\frac{{a}^{2}{c}_{1}^{2}}{4{b}^{2}-4{a}^{2}{c}_{2}}\right)\left[1+Erf\left(\frac{\sqrt{{a}^{2}}{c}_{1}}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)\right]}{4\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}+\frac{1}{2b}\\ \qquad\qquad\quad\,\,-\frac{\exp \left(\frac{{a}^{2}{c}_{1}^{2}}{4{b}^{2}-4{a}^{2}{c}_{2}}\right)\left[Erf\left(\frac{\sqrt{{a}^{2}}{c}_{1}}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)+Erf\left(\frac{-a{c}_{1}-\sqrt{2}Q({b}^{2}-{a}^{2}{c}_{2})}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)\right]}{4\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}.\end{array}$$

Case 4. (Q ≤ 0, a ≤ 0):

$$\begin{array}{ll}I(Q,a,b)\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\,\,\approx -\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\exp \left({c}_{1}av+{c}_{2}{a}^{2}{v}^{2}\right)\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\qquad+\,\sqrt{2}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\,\,=-\frac{1}{2\sqrt{\pi }}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\exp \left(-({b}^{2}-{a}^{2}{c}_{2}){v}^{2}+a{c}_{1}v\right)dv\\ \qquad\qquad\qquad+\,\frac{1}{\sqrt{\pi }}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\exp (-{b}^{2}{v}^{2})dv.\end{array}$$

When we use (7.4.32) of Abramowitz and Stegun (1970), I(Q, a, b) in this case can be approximated by:

$$\begin{array}{ll}I(Q,a,b)\,\approx -\frac{\exp \left(\frac{{a}^{2}{c}_{1}^{2}}{4{b}^{2}-4{a}^{2}{c}_{2}}\right)\left[1+Erf\left(\frac{\sqrt{{a}^{2}}{c}_{1}}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)\right]}{4\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}+\frac{1}{2b}\\ \qquad\qquad\quad\,\,-\frac{\exp \left(\frac{{a}^{2}{c}_{1}^{2}}{4{b}^{2}-4{a}^{2}{c}_{2}}\right)\left[Erf\left(\frac{\sqrt{{a}^{2}}{c}_{1}}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)+Erf\left(\frac{-a{c}_{1}+\sqrt{2}Q({b}^{2}-{a}^{2}{c}_{2})}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)\right]}{4\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}+\frac{1}{2b}Erf\left(\frac{bQ}{\sqrt{2}}\right).\end{array}$$

Combining the results in Cases 1–4, we obtain the result in Proposition 1.

Derivation of Proposition 3

$$\begin{array}{ll}&E\left(u| a\,<\,\varepsilon \,<\, b\right)\\ &=E\left(E\left(u| \varepsilon \right)| a\,<\,\varepsilon \,<\, b\right)\\ &=\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}{\displaystyle\int\nolimits_{a}^{b}}E\left(u| \varepsilon \right)f\left(\varepsilon \right)d\varepsilon \\ &=\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}{\displaystyle\int\nolimits_{a}^{b}}\left(S\frac{{\sigma }_{u}^{2}}{{\sigma }^{2}}\varepsilon +\frac{{\sigma }_{u}{\sigma }_{v}}{\sigma }\frac{\phi \left(\lambda \frac{\varepsilon }{\sigma }\right)}{{{\Phi }}\left(S\lambda \frac{\varepsilon }{\sigma }\right)}\right)\left(\frac{2}{\sigma }\phi \left(\frac{\varepsilon }{\sigma }\right){{\Phi }}\left(S\lambda \frac{\varepsilon }{\sigma }\right)\right)d\varepsilon \\ &=\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}\left[{\displaystyle\int\nolimits_{a}^{b}}S\frac{{\sigma }_{u}^{2}}{{\sigma }^{2}}\varepsilon \frac{2}{\sigma }\phi \left(\frac{\varepsilon }{\sigma }\right){{\Phi }}\left(S\lambda \frac{\varepsilon }{\sigma }\right)d\varepsilon +{\displaystyle\int\nolimits_{a}^{b}}\frac{2{\sigma }_{u}{\sigma }_{v}}{{\sigma }^{2}}\phi \left(\frac{\varepsilon }{\sigma }\right)\phi \left(\lambda \frac{\varepsilon }{\sigma }\right)d\varepsilon \right]\\ &={E}_{1}+{E}_{2}.\end{array}$$

We first rewrite E1 as:

$$\begin{array}{ll}{E}_{1}=\,\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}\frac{S{\sigma }_{u}^{2}}{{\sigma }^{2}}{\displaystyle\int\nolimits_{a}^{b}}\varepsilon \ f\left(\varepsilon \right)d\varepsilon \\ \quad\,\,=S\frac{{\sigma }_{u}^{2}}{{\sigma }^{2}}E\left(\varepsilon | a\,< \,\varepsilon \,<\, b\right).\end{array}$$

As we can see, the integral part in E1 is a censored conditional expectation of \(f\left({\varepsilon }_{}\right)\) given that ε ∈ (a, b). Using Proposition 1 in Flecher et al. (2009), we can evaluate E1 as:

$${E}_{1}=S\frac{{\sigma }_{u}^{2}}{\sigma }\left\{-\sigma \frac{{\left[f(\varepsilon )\right]}_{a}^{b}}{{\left[F(\varepsilon )\right]}_{a}^{b}}+\frac{2\lambda }{\sqrt{2\pi }{\lambda }^{* }}\frac{{\left[{{\Phi }}\left(\frac{S{\lambda }^{* }}{\sigma }\varepsilon \right)\right]}_{a}^{b}}{{\left[F(\varepsilon )\right]}_{a}^{b}}\right\},$$

where \({\lambda }^{* }=\sqrt{1+{\lambda }^{2}}\).

We can also recast E2 as:

$$\begin{array}{ll}{E}_{2}=\,\frac{2{\sigma }_{u}{\sigma }_{v}}{{\sigma }^{2}}\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}{\displaystyle\int\nolimits_{a}^{b}}\phi \left(\frac{\varepsilon }{\sigma }\right)\phi \left(\lambda \frac{\varepsilon }{\sigma }\right)d\varepsilon \\ \quad\,\,=\frac{2{\sigma }_{u}{\sigma }_{v}}{{\sigma }^{2}}\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}{\displaystyle\int\nolimits_{a}^{b}}\frac{1}{2\pi }\exp \left[-\frac{1}{2}{\left(\frac{\varepsilon }{\sigma }\right)}^{2}\right]\exp \left[-\frac{{\lambda }^{2}}{2}{\left(\frac{\varepsilon }{\sigma }\right)}^{2}\right]d\varepsilon \\ \quad\,\,=\frac{{\sigma }_{u}{\sigma }_{v}}{\pi {\sigma }^{2}}\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}{\displaystyle\int\nolimits_{a}^{b}}\exp \left(-\frac{{{\lambda }^{* }}^{2}{\varepsilon }^{2}}{2{\sigma }^{2}}\right)d\varepsilon .\end{array}$$

Using (7.4.32) of Abramowitz and Stegun (1970), we can evaluate E2 as:

$$\begin{array}{ll}{E}_{2}=\frac{{\sigma }_{u}{\sigma }_{v}}{\pi {\sigma }^{2}}\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}{\left[\frac{1}{2}\sqrt{\frac{\pi }{\frac{{{\lambda }^{* }}^{2}}{2{\sigma }^{2}}}}erf\left(\frac{{\lambda }^{* }}{\sqrt{2}\sigma }\varepsilon \right)\right]}_{a}^{b}\\ \quad\,\,=\frac{{\sigma }_{u}{\sigma }_{v}}{\sqrt{2\pi }\sigma {\lambda }^{* }}\frac{{\left[erf\left(\frac{{\lambda }^{* }}{\sqrt{2}\sigma }\varepsilon \right)\right]}_{a}^{b}}{{\left[F\left(\varepsilon \right)\right]}_{a}^{b}}.\end{array}$$

With the above results, we now obtain the JLMS interval efficiency estimate.

Proof of Corollary 1

We express the limit of Proposition 3 as:

$$\begin{array}{ll}\lim\limits_{\delta \to 0}E\left(u| c\,<\, \varepsilon \,<\,c+\delta \right)\,=\lim\limits_{\delta \to 0}S\frac{{\sigma }_{u}^{2}}{\sigma }\left\{-\sigma \frac{{\left[f\left(\varepsilon \right)\right]}_{c}^{c+\delta }}{{\left[F\left(\varepsilon \right)\right]}_{c}^{c+\delta }}+\frac{2\lambda }{\sqrt{2\pi }{\lambda }^{* }}\frac{{\left[{{\Phi }}\left(\frac{S{\lambda }^{* }}{\sigma }\varepsilon \right)\right]}_{c}^{c+\delta }}{{\left[F\left(\varepsilon \right)\right]}_{c}^{c+\delta }}\right\}\\ \qquad\qquad\qquad\qquad\qquad\qquad+\lim\limits_{\delta \to 0}\frac{\frac{{\sigma }_{u}{\sigma }_{v}}{\sqrt{2\pi }\sigma {\lambda }^{* }}{\left[erf\left(\frac{{\lambda }^{* }}{\sqrt{2}{\sigma }^{2}}\varepsilon \right)\right]}_{c}^{c+\delta }}{{\left[F\left(\varepsilon \right)\right]}_{c}^{c+\delta }}\\ \qquad\qquad\qquad\qquad\qquad\,\,=\lim\limits_{\delta \to 0}S\frac{{\sigma }_{u}^{2}}{\sigma }\frac{-\frac{2S\lambda }{\sigma }\phi \left(\frac{c+\delta }{\sigma }\right)\phi \left(\lambda \frac{c+\delta }{\sigma }\right)-\frac{2c}{{\sigma }^{2}}\phi \left(\frac{c+\delta }{\sigma }\right){{\Phi }}\left(\lambda \frac{c+\delta }{\sigma }\right)}{f\left(c+\delta \right)}\\ \qquad\qquad\qquad\qquad\qquad\qquad+\lim\limits_{\delta \to 0}S\frac{{\sigma }_{u}^{2}}{{\sigma }^{2}}\frac{2S\lambda }{\sqrt{2\pi }}\frac{\phi \left({\lambda }^{* }\frac{c+\delta }{\sigma }\right)}{f\left(c+\delta \right)}\\ \qquad\qquad\qquad\qquad\qquad\qquad+\lim\limits_{\delta \to 0}\frac{2}{\sqrt{2\pi }}\frac{{\sigma }_{u}{\sigma }_{v}}{{\sigma }^{2}}\frac{\phi \left({\lambda }^{* }\frac{c+\delta }{\sigma }\right)}{f\left(c+\delta \right)}\\ \qquad\qquad\qquad\qquad\qquad\,\,=S\frac{{\sigma }_{u}^{2}}{{\sigma }^{2}}c-\frac{\frac{2{S}^{2}\lambda }{\sigma }\phi \left(\frac{c}{\sigma }\right)\phi \left(\lambda \frac{c}{\sigma }\right)-\frac{2{S}^{2}\lambda }{\sqrt{2\pi }\sigma }\phi \left({\lambda }^{* }\frac{c}{\sigma }\right)}{f\left(c\right)}\\ \qquad\qquad\qquad\qquad\qquad\qquad+\frac{2}{\sqrt{2\pi }}\frac{{\sigma }_{u}{\sigma }_{v}}{{\sigma }^{2}}\frac{\phi \left({\lambda }^{* }\frac{c}{\sigma }\right)}{f\left(c\right)}\\ \qquad\qquad\qquad\qquad\qquad\,\,=S\frac{{\sigma }_{u}^{2}}{{\sigma }^{2}}c+\frac{{\sigma }_{u}{\sigma }_{v}}{\sigma }\frac{\phi \left(\lambda \frac{c}{\sigma }\right)}{{{\Phi }}\left(S\lambda \frac{c}{\sigma }\right)},\end{array}$$

where the second equality is based on L’hospital Rule, and the fourth equality follows from the observation that \(\phi \left(\frac{c+\delta }{\sigma }\right)\phi \left(\lambda \frac{c+\delta }{\sigma }\right)=\frac{1}{\sqrt{2\pi }}\phi \left({\lambda }^{* }\frac{c+\delta }{\sigma }\right)\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hwu, ST., Fu, TT. & Tsay, WJ. Estimation and efficiency evaluation of stochastic frontier models with interval dependent variables. J Prod Anal 56, 33–44 (2021). https://doi.org/10.1007/s11123-021-00609-w

Download citation

Keywords

  • Stochastic frontier analysis
  • Interval dependent variable
  • Technical efficiency

JEL

  • C13
  • C24