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Empirical Economics

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06-04-2023

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

Authors: Christopher F. Parmeter, Shirong Zhao

Published in: Empirical Economics | Issue 6/2023

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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.

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

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.

We thank Alecos Papadopoulos for suggesting this acronym.

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.

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.

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

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.

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

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

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Title: An alternative corrected ordinary least squares estimator for the stochastic frontier model
Authors: Christopher F. Parmeter
Shirong Zhao
Publication date: 06-04-2023
Publisher: Springer Berlin Heidelberg
Published in: Empirical Economics / Issue 6/2023
Print ISSN: 0377-7332
Electronic ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-023-02401-1

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