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A simple closed-form approximation for the cumulative distribution function of the composite error of stochastic frontier models

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  • Wen-Jen Tsay
  • Cliff Huang
  • Tsu-Tan Fu
  • I.-Lin Ho

Abstract

This paper derives an analytic closed-form formula for the cumulative distribution function (cdf) of the composite error of the stochastic frontier analysis (SFA) model. Since the presence of a cdf is frequently encountered in the likelihood-based analysis with limited-dependent and qualitative variables as elegantly shown in the classic book of Maddala (Limited-dependent and qualitative variables in econometrics. Cambridge University Press, Cambridge, 1983 ), the proposed methodology is useful in the framework of the stochastic frontier analysis. We apply the formula to the maximum likelihood estimation of the SFA models with a censored dependent variable. The simulations show that the finite sample performance of the maximum likelihood estimator of the censored SFA model is very promising. A simple empirical example on the modeling of reservation wage in Taiwan is illustrated as a potential application of the censored SFA. Copyright Springer Science+Business Media, LLC 2013

Suggested Citation

  • Wen-Jen Tsay & Cliff Huang & Tsu-Tan Fu & I.-Lin Ho, 2013. "A simple closed-form approximation for the cumulative distribution function of the composite error of stochastic frontier models," Journal of Productivity Analysis, Springer, vol. 39(3), pages 259-269, June.
    • Handle: RePEc:kap:jproda:v:39:y:2013:i:3:p:259-269
    DOI: 10.1007/s11123-012-0283-1
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    References listed on IDEAS

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    1. Christine Amsler & Peter Schmidt & Wen-Jen Tsay, 2019. "Evaluating the CDF of the distribution of the stochastic frontier composed error," Journal of Productivity Analysis, Springer, vol. 52(1), pages 29-35, December.
    2. Huang, Tai-Hsin & Lin, Chung-I & Chen, Kuan-Chen, 2017. "Evaluating efficiencies of Chinese commercial banks in the context of stochastic multistage technologies," Pacific-Basin Finance Journal, Elsevier, vol. 41(C), pages 93-110.
    3. Shih-Tang Hwu & Tsu-Tan Fu & Wen-Jen Tsay, 2021. "Estimation and efficiency evaluation of stochastic frontier models with interval dependent variables," Journal of Productivity Analysis, Springer, vol. 56(1), pages 33-44, August.
    4. Christine Amsler & Alecos Papadopoulos & Peter Schmidt, 2021. "Evaluating the cdf of the Skew Normal distribution," Empirical Economics, Springer, vol. 60(6), pages 3171-3202, June.
    5. Huang, Tai-Hsin & Chen, Kuan-Chen & Lin, Chung-I, 2018. "An extension from network DEA to copula-based network SFA: Evidence from the U.S. commercial banks in 2009," The Quarterly Review of Economics and Finance, Elsevier, vol. 67(C), pages 51-62.
    6. Huang, Tai-Hsin & Hu, Chu-Nan & Chang, Bao-Guang, 2018. "Competition, efficiency, and innovation in Taiwan’s banking industry — An application of copula methods," The Quarterly Review of Economics and Finance, Elsevier, vol. 67(C), pages 362-375.
    7. Tai-Hsin Huang & Nan-Hung Liu & Subal C. Kumbhakar, 2018. "Joint estimation of the Lerner index and cost efficiency using copula methods," Empirical Economics, Springer, vol. 54(2), pages 799-822, March.
    8. Huang, Tai-Hsin & Lin, Chung-I & Wu, Ruei-Cian, 2019. "Assessing the marketing and investment efficiency of Taiwan’s life insurance firms under network structures," The Quarterly Review of Economics and Finance, Elsevier, vol. 71(C), pages 132-147.
    9. T.-F. Lo & P.-H. Ke & W.-J. Tsay, 2018. "Pairwise likelihood inference for the random effects probit model," Computational Statistics, Springer, vol. 33(2), pages 837-861, June.
    10. Hung-pin Lai & Cliff Huang, 2013. "Maximum likelihood estimation of seemingly unrelated stochastic frontier regressions," Journal of Productivity Analysis, Springer, vol. 40(1), pages 1-14, August.
    11. Tai-Hsin Huang & Nan-Hung Liu, 2014. "Bank competition in transition countries: Are those markets really in equilibrium?," Empirical Economics, Springer, vol. 47(4), pages 1283-1316, December.
    12. Huang, Tai-Hsin & Chiang, Dien-Lin & Lin, Chung-I, 2017. "A new approach to estimating a profit frontier using the censored stochastic frontier model," The North American Journal of Economics and Finance, Elsevier, vol. 39(C), pages 68-77.
    13. Martini, Gianmaria & Scotti, Davide & Viola, Domenico & Vittadini, Giorgio, 2020. "Persistent and temporary inefficiency in airport cost function: An application to Italy," Transportation Research Part A: Policy and Practice, Elsevier, vol. 132(C), pages 999-1019.
    14. Tai-Hsin Huang & Yi-Chun Lin & Kuo-Jui Huang & Yu-Wei Liao, 2022. "Comparing Cost Efficiency Between Financial and Non-financial Holding Banks and Insurers in Taiwan Under the Framework of Copula Methods and Metafrontier," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 29(4), pages 735-766, December.
    15. Huang, Tai-Hsin & Chiang, Dien-Lin & Chao, Shih-Wei, 2017. "A new approach to jointly estimating the Lerner index and cost efficiency for multi-output banks under a stochastic meta-frontier framework," The Quarterly Review of Economics and Finance, Elsevier, vol. 65(C), pages 212-226.

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    More about this item

    Keywords

    Stochastic frontier analysis; Cumulative distribution function; Censored stochastic frontier model; C13; C46;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions

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