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Consistency without compactness of the parameter space in spatial econometrics

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  • Liu, Tuo
  • Xu, Xingbai
  • Lee, Lung-fei

Abstract

When studying the consistency of an estimator without a closed-form solution for a spatial econometric model, we usually assume that the parameter space is compact. However, compactness assumptions are restrictive as we need to know the boundaries of parameter spaces. We establish a consistency theorem for concave objective functions. We apply this result to rebuild the consistency of the quasi maximum likelihood estimator (QMLE) of a spatial autoregressive (SAR) model and a SAR Tobit model. Their log-likelihood functions are not concave, but they can be concave after proper reparameterization as in Olsen (1978).

Suggested Citation

  • Liu, Tuo & Xu, Xingbai & Lee, Lung-fei, 2022. "Consistency without compactness of the parameter space in spatial econometrics," Economics Letters, Elsevier, vol. 210(C).
    • Handle: RePEc:eee:ecolet:v:210:y:2022:i:c:s0165176521004675
    DOI: 10.1016/j.econlet.2021.110224
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    References listed on IDEAS

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    1. Yu, Jihai & de Jong, Robert & Lee, Lung-fei, 2008. "Quasi-maximum likelihood estimators for spatial dynamic panel data with fixed effects when both n and T are large," Journal of Econometrics, Elsevier, vol. 146(1), pages 118-134, September.
    2. Newey, Whitney K, 1991. "Uniform Convergence in Probability and Stochastic Equicontinuity," Econometrica, Econometric Society, vol. 59(4), pages 1161-1167, July.
    3. Xu, Xingbai & Lee, Lung-fei, 2015. "A spatial autoregressive model with a nonlinear transformation of the dependent variable," Journal of Econometrics, Elsevier, vol. 186(1), pages 1-18.
    4. Chernozhukov, Victor & Hong, Han, 2003. "An MCMC approach to classical estimation," Journal of Econometrics, Elsevier, vol. 115(2), pages 293-346, August.
    5. Lung-Fei Lee & Jihai Yu, 2013. "Near Unit Root in the Spatial Autoregressive Model," Spatial Economic Analysis, Taylor & Francis Journals, vol. 8(3), pages 314-351, September.
    6. Liu, Tuo & Lee, Lung-fei, 2019. "A likelihood ratio test for spatial model selection," Journal of Econometrics, Elsevier, vol. 213(2), pages 434-458.
    7. Gupta, Abhimanyu, 2019. "Estimation Of Spatial Autoregressions With Stochastic Weight Matrices," Econometric Theory, Cambridge University Press, vol. 35(2), pages 417-463, April.
    8. Park, Joon Y & Phillips, Peter C B, 2001. "Nonlinear Regressions with Integrated Time Series," Econometrica, Econometric Society, vol. 69(1), pages 117-161, January.
    9. Xu, Xingbai & Lee, Lung-fei, 2015. "Maximum likelihood estimation of a spatial autoregressive Tobit model," Journal of Econometrics, Elsevier, vol. 188(1), pages 264-280.
    10. Lung-Fei Lee, 2004. "Asymptotic Distributions of Quasi-Maximum Likelihood Estimators for Spatial Autoregressive Models," Econometrica, Econometric Society, vol. 72(6), pages 1899-1925, November.
    11. 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.
    12. Shi, Wei & Lee, Lung-fei, 2017. "Spatial dynamic panel data models with interactive fixed effects," Journal of Econometrics, Elsevier, vol. 197(2), pages 323-347.
    13. Lee, Nayoung & Moon, Hyungsik Roger & Zhou, Qiankun, 2017. "Many IVs estimation of dynamic panel regression models with measurement error," Journal of Econometrics, Elsevier, vol. 200(2), pages 251-259.
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    More about this item

    Keywords

    Non-compact parameter space; MLE; Spatial autoregressive model; Spatial autoregressive Tobit model; Concave log-likelihood;
    All these keywords.

    JEL classification:

    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics
    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models

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