Skip to main content
SpringerLink
Log in

An alternative corrected ordinary least squares estimator for the stochastic frontier model

  • Published:
Empirical Economics Aims and scope Submit manuscript

Abstract

The corrected ordinary least squares (COLS) estimator of the stochastic frontier model exploits the higher order moments of the OLS residuals to estimate the parameters of the composed error. However, both “Type I” and “Type II” failures in COLS can result from finite sample bias that arises in the estimation of these higher order moments, especially in small samples. We propose a novel modification to COLS by using the first moment of the absolute value of the composite error term in place of the third moment for both the Normal-Half Normal and Normal-Exponential specifications. We demonstrate via simulations that this switch considerably reduces the occurrence of both Type I and Type II failures. These Monte Carlo simulations also reveal that our alternative COLS approach, in general, performs better than standard COLS.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In the production (cost) frontier, Type I failure means the OLS residuals produce a positive (negative) skewness while a negative (positive) skewness is anticipated.

  2. For example, in the Normal-Half Normal specification, Papadopoulos and Parmeter (2021) show that a Type II Failure implies that the absolute value of the estimated skewness exceeds the maximum of the absolute value of the skewness that the Normal-Half Normal composite error term could have; i.e., while the maximum of the absolute value of skewness for the Skew Normal distribution is 0.995, the estimated skewness is smaller than \(-0.995\) in the production frontier or larger than 0.995 in the cost frontier.

  3. We thank Alecos Papadopoulos for suggesting this acronym.

  4. Proposition 2.1 is equivalent to saying that the square of a Skew Normal random variable is a \(\chi ^2\) random variable with one degree of freedom, which is stated as Property H in Azzalini (1985). This proposition has also been “proven” more recently: See Jradi et al. (2019, Theorem 1), and Huang and Chen (2007, Proposition 4); both without attribution to Azzalini’s work.

  5. In Subsection A.1 of the separate Appendix A, we provide several explanations on the restriction \({\widehat{\sigma }}_u \in [0,(\frac{\pi \widehat{r}_2}{\pi -2})^{\frac{1}{2}}]\) for the Normal-Half Normal specification.

  6. See Proposition 2.1 in Zhao and Parmeter (2022) for the proof.

  7. In Subsection A.2 of the separate Appendix A, we provide further discussion on the restriction \({{\widehat{\lambda }}}_I \ge 0\) for the Normal-Exponential specification.

  8. See Appendix B for more details about estimating individual efficiency \({\widehat{E}}\left[ \exp (-u_i)\mid \widehat{\varepsilon }_i\right] \).

  9. For more details, see Hafner et al. (2018).

References

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

    Article  Google Scholar 

  • Almanidis P, Sickles RC (2011) The skewness issue in stochastic frontiers models: Fact or fiction? In: Van Keilegom I, Wilson PW (eds) Exploring research frontiers in contemporary statistics and econometrics. Springer, Berlin, pp 201–227

    Chapter  Google Scholar 

  • Almanidis P, Qian J, Sickles RC (2014) Stochastic frontier models with bounded inefficiency. In: Sickles RC, Horrace WC (eds) Festschrift in honor of peter Schmidt: econometric methods and applications. Springer, New York, pp 47–81

    Chapter  Google Scholar 

  • Azzalini A (1985) A class of distributions which includes the Normal ones. Scand J Stat 12:171–178

    Google Scholar 

  • Azzalini A (1986) Further results on a class of distributions which includes the normal ones. Statistica (Bologna) 46:199–208

    Google Scholar 

  • Badunenko O, Henderson DJ (2021) Production analysis with asymmetric error. Paper presented at Virtual North American Productivity Workshop 2021

  • Battese GE, Coelli TJ (1988) Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data. J Econom 38:387–399

    Article  Google Scholar 

  • Bonanno G, De Giovanni D, Domma F (2015) The wrong skewness problem: a re-specification of stochastic frontiers. J Prod Anal 47:49–64

    Article  Google Scholar 

  • Cai J, Feng Q, Horrace WC, Wu GL (2021) Wrong skewness and finite sample correction in the normal-half normal stochastic frontier model. Empir Econ 60:2837–2866

    Article  Google Scholar 

  • Carree MA (2002) Technological inefficiency and the skewness of the error component in stochastic frontier analysis. Econ Lett 77:101–107

    Article  Google Scholar 

  • Coelli T (1995) Estimators and hypothesis tests for a stochastic frontier function: a Monte Carlo analysis. J Prod Anal 6:247–268

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Greene WH (2007) Matching the matching estimators. Unpublished working paper. Department of Economics, Stern School of Business, New York University

  • Hafner CM, Manner H, Simar L (2018) The wrong skewness problem in stochastic frontier models: a new approach. Econom Rev 37:380–400

    Article  Google Scholar 

  • Horrace WC (2015) Moments of the truncated normal distribution. J Prod Anal 43:133–138

    Article  Google Scholar 

  • Horrace WC, Parmeter CF (2018) A Laplace stochastic frontier model. Econom Rev 37:260–280

    Article  Google Scholar 

  • Horrace WC, Wright IA (2020) Stationary points for parametric stochastic frontier models. J Bus Econ Stat 38:516–526

    Article  Google Scholar 

  • Horrace WC, Parmeter CF, Wright IA (2022) On asymmetry and quantile estimation of the stochastic frontier model. University of Miami Working Paper

  • Huang W-J, Chen Y-H (2007) Generalized skew-Cauchy distribution. Stat Prob Lett 77:1137–1147

    Article  Google Scholar 

  • Jradi S, Parmeter CF, Ruggiero J (2019) Quantile estimation of the stochastic frontier model. Econ Lett 182:15–18

    Article  Google Scholar 

  • Jradi S, Parmeter CF, Ruggiero J (2021) Quantile estimation of stochastic frontiers with the normal-exponential specification. Eur J Oper Res 295:475–483

    Article  Google Scholar 

  • Kumbhakar SC, Parmeter CF, Zelenyuk V (2020a) Stochastic frontier analysis: foundations and advances I. In: Ray SC, Chambers R, Kumbhakar S (eds) Handbook of production economics. Springer, Singapore, pp 1–39

    Google Scholar 

  • Kumbhakar SC, Parmeter CF, Zelenyuk V (2020b) Stochastic frontier analysis: foundations and advances II. In: Ray SC, Chambers R, Kumbhakar S (eds) Handbook of production economics. Springer, Singapore, pp 1–38

    Google Scholar 

  • Li Q (1996) Estimating a stochastic production frontier when the adjusted error is symmetric. Econ Lett 52:221–228

    Article  Google Scholar 

  • Meeusen W, van den Broeck J (1977) Efficiency estimation from Cobb–Douglas production functions with composed error. Int Econ Rev 18:435–444

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Papadopoulos A, Parmeter CF (2021) Type II failure and specification testing in the stochastic frontier model. Eur J Oper Res 293:990–1001

    Article  Google Scholar 

  • Schmidt P, Lin T-F (1984) Simple tests of alternative specifications in stochastic frontier models. J Econom 24:349–361

    Article  Google Scholar 

  • Simar L, Wilson PW (2009) Inferences from cross-sectional, stochastic frontier models. Econom Rev 29:62–98

    Article  Google Scholar 

  • Simar L, Wilson PW (2022) Nonparametric, stochastic frontier models with multiple inputs and outputs. J Bus Econ Stat (forthcoming)

  • Simar L, Van Keilegom I, Zelenyuk V (2017) Nonparametric least squares methods for stochastic frontier models. J Prod Anal 47:189–204

    Article  Google Scholar 

  • Wei Z, Zhu X, Wang T (2021) The extended skew-normal-based stochastic frontier model with a solution to ‘wrong skewness’ problem. Statistics (Forthcoming)

  • Zhao S (2021) Quantile estimation of stochastic frontier models with the normal-half normal specification: a cumulative distribution function approach. Econ Lett 206:109998

    Article  Google Scholar 

  • Zhao S, Parmeter CF (2022) The wrong skewness problem: moment constrained maximum likelihood estimation of the stochastic frontier model. Econom Lett 221:110901

    Article  Google Scholar 

  • Zhu X, Wei Z, Wang T (2022) Multivariate skew normal-based stochastic frontier models. J Stat Theory Pract 16:1–21

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Economics, University of Miami, Coral Gables, FL, 33146, USA

    Christopher F. Parmeter

  2. School of Finance, Dongbei University of Finance and Economics, Dalian, 116025, Liaoning, China

    Shirong Zhao

Authors
  1. Christopher F. Parmeter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shirong Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shirong Zhao.

Ethics declarations

Conflict of interest

All authors declare that they have no conflicts of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

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

We thank participants at vAPW 2021 for valuable feedback as well as Alecos Papadopoulos, William Greene, and Robin Sickles for useful comments which improved the paper. We thank the Cyber Infrastructure Technology Integration group at Clemson University for operating the Palmetto cluster used for computations. Shirong Zhao acknowledges the support from Liaoning Social Science Foundation (L22CJY011) and Liaoning Provincial Department of Education Research Fund (LJKMZ20221602). All the authors contributed equally. All errors are our own.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file 1 (pdf 224 KB)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Parmeter, C.F., Zhao, S. An alternative corrected ordinary least squares estimator for the stochastic frontier model. Empir Econ 64, 2831–2857 (2023). https://doi.org/10.1007/s00181-023-02401-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00181-023-02401-1

Keywords

JEL Classification

Search

Navigation