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Note on the Uniqueness of the Maximum Likelihood Estimator for a Heckman’s Simultaneous Equations Model

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  • Akashi, Kentaro
  • Horie, Tetsushi

Abstract

It is shown that the likelihood function of a Heckman’s simultaneous equations model is identified by complementing the approach of parameter transformation. Therefore, the expectation of the log-likelihood function has a single maximum. Thus, the maximum likelihood estimator becomes asymptotically consistent without an initial consistent estimator. Additionally, the approach can show the uniqueness of the log-likelihood functions for the simultaneous Tobit, sample selection (Type 2 Tobit), and simultaneous generalized selectivity models.

Suggested Citation

  • Akashi, Kentaro & Horie, Tetsushi, 2025. "Note on the Uniqueness of the Maximum Likelihood Estimator for a Heckman’s Simultaneous Equations Model," Econometrics and Statistics, Elsevier, vol. 34(C), pages 69-77.
    • Handle: RePEc:eee:ecosta:v:34:y:2025:i:c:p:69-77
    DOI: 10.1016/j.ecosta.2022.02.004
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    References listed on IDEAS

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    1. Olsen, Randall J, 1982. "Distributional Tests for Selectivity Bias and a More Robust Likelihood Estimator," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 23(1), pages 223-240, February.
    2. Heckman, James J, 1978. "Dummy Endogenous Variables in a Simultaneous Equation System," Econometrica, Econometric Society, vol. 46(4), pages 931-959, July.
    3. James Heckman, 2013. "Sample selection bias as a specification error," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 31(3), pages 129-137.
    4. Amemiya, Takeshi, 1974. "Multivariate Regression and Simultaneous Equation Models when the Dependent Variables Are Truncated Normal," Econometrica, Econometric Society, vol. 42(6), pages 999-1012, November.
    5. Blundell, Richard & Smith, Richard J., 1994. "Coherency and estimation in simultaneous models with censored or qualitative dependent variables," Journal of Econometrics, Elsevier, vol. 64(1-2), pages 355-373.
    6. Thomas W. Zuehlke, 2021. "Estimation of a type 2 Tobit model with generalized Box-Cox transformation," Applied Economics, Taylor & Francis Journals, vol. 53(17), pages 1952-1975, April.
    7. Olsen, Randall J, 1978. "Note on the Uniqueness of the Maximum Likelihood Estimator for the Tobit Model," Econometrica, Econometric Society, vol. 46(5), pages 1211-1215, September.
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    More about this item

    JEL classification:

    • C35 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium

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