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Modified QML estimation of spatial autoregressive models with unknown heteroskedasticity and nonnormality

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  • Liu, Shew Fan
  • Yang, Zhenlin

Abstract

In the presence of heteroskedasticity, Lin and Lee (2010) show that the quasi-maximum likelihood (QML) estimator of the spatial autoregressive (SAR) model can be inconsistent as a ‘necessary’ condition for consistency can be violated, and thus propose robust GMM estimators for the model. In this paper, we first show that this condition may hold in certain situations and when it does the regular QML estimator can still be consistent. In cases where this condition is violated, we propose a simple modified QML estimation method robust against unknown heteroskedasticity. In both cases, asymptotic distributions of the estimators are derived, and methods for estimating robust variances are given, leading to robust inferences for the model. Extensive Monte Carlo results show that the modified QML estimator outperforms the GMM and QML estimators even when the latter is consistent. The proposed methods are then extended to the more general SARAR models.

Suggested Citation

  • Liu, Shew Fan & Yang, Zhenlin, 2015. "Modified QML estimation of spatial autoregressive models with unknown heteroskedasticity and nonnormality," Regional Science and Urban Economics, Elsevier, vol. 52(C), pages 50-70.
  • Handle: RePEc:eee:regeco:v:52:y:2015:i:c:p:50-70
    DOI: 10.1016/j.regsciurbeco.2015.02.003
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    Cited by:

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    2. Kyriacou, Maria & Phillips, Peter C.B. & Rossi, Francesca, 2023. "Continuously Updated Indirect Inference In Heteroskedastic Spatial Models," Econometric Theory, Cambridge University Press, vol. 39(1), pages 107-145, February.
    3. Yong Bao & Xiaotian Liu & Lihong Yang, 2020. "Indirect Inference Estimation of Spatial Autoregressions," Econometrics, MDPI, vol. 8(3), pages 1-26, September.
    4. Baltagi, Badi H. & Pirotte, Alain & Yang, Zhenlin, 2021. "Diagnostic tests for homoskedasticity in spatial cross-sectional or panel models," Journal of Econometrics, Elsevier, vol. 224(2), pages 245-270.
    5. Gupta, Abhimanyu & Kokas, Sotirios & Michaelides, Alexander & Minetti, Raoul, 2023. "Networks and Information in Credit Markets," Working Papers 2023-1, Michigan State University, Department of Economics.
    6. Shew Fan Liu & Zhenlin Yang, 2015. "Asymptotic Distribution and Finite Sample Bias Correction of QML Estimators for Spatial Error Dependence Model," Econometrics, MDPI, vol. 3(2), pages 1-36, May.
    7. Rossi, Francesca & Lieberman, Offer, 2023. "Spatial autoregressions with an extended parameter space and similarity-based weights," Journal of Econometrics, Elsevier, vol. 235(2), pages 1770-1798.
    8. Bao, Yong, 2024. "Estimating spatial autoregressions under heteroskedasticity without searching for instruments," Regional Science and Urban Economics, Elsevier, vol. 106(C).
    9. Pesaran, M. Hashem & Yang, Cynthia Fan, 2021. "Estimation and inference in spatial models with dominant units," Journal of Econometrics, Elsevier, vol. 221(2), pages 591-615.
    10. Jin, Fei & Lee, Lung-fei, 2020. "Asymptotically efficient root estimators for spatial autoregressive models with spatial autoregressive disturbances," Economics Letters, Elsevier, vol. 194(C).
    11. Jin, Fei & Lee, Lung-fei, 2018. "Outer-product-of-gradients tests for spatial autoregressive models," Regional Science and Urban Economics, Elsevier, vol. 72(C), pages 35-57.
    12. Jakub Olejnik & Alicja Olejnik, 2017. "Improved asymptotic analysis of Gaussian QML estimators in spatial models," Lodz Economics Working Papers 9/2017, University of Lodz, Faculty of Economics and Sociology.
    13. Bai, Jushan & Li, Kunpeng, 2021. "Dynamic spatial panel data models with common shocks," Journal of Econometrics, Elsevier, vol. 224(1), pages 134-160.
    14. Nicolas DEBARSY & Cem ERTUR, 2016. "Interaction matrix selection in spatial econometrics with an application to growth theory," LEO Working Papers / DR LEO 2172, Orleans Economics Laboratory / Laboratoire d'Economie d'Orleans (LEO), University of Orleans.
    15. Federico Martellosio & Grant Hillier, 2019. "Adjusted QMLE for the spatial autoregressive parameter," Papers 1909.08141, arXiv.org.
    16. Gupta, Abhimanyu, 2023. "Efficient closed-form estimation of large spatial autoregressions," Journal of Econometrics, Elsevier, vol. 232(1), pages 148-167.
    17. Debarsy, Nicolas & Ertur, Cem, 2019. "Interaction matrix selection in spatial autoregressive models with an application to growth theory," Regional Science and Urban Economics, Elsevier, vol. 75(C), pages 49-69.
    18. Li, Liyao & Yang, Zhenlin, 2020. "Estimation of fixed effects spatial dynamic panel data models with small T and unknown heteroskedasticity," Regional Science and Urban Economics, Elsevier, vol. 81(C).
    19. Michaelides, Alexander & Kokas, Sotirios & Gupta, Abhimanyu, 2017. "Credit Market Spillovers: Evidence from a Syndicated Loan Market Network," CEPR Discussion Papers 12424, C.E.P.R. Discussion Papers.
    20. Jin, Fei & Lee, Lung-fei, 2019. "GEL estimation and tests of spatial autoregressive models," Journal of Econometrics, Elsevier, vol. 208(2), pages 585-612.
    21. 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.
    22. Martellosio, Federico & Hillier, Grant, 2020. "Adjusted QMLE for the spatial autoregressive parameter," Journal of Econometrics, Elsevier, vol. 219(2), pages 488-506.

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    More about this item

    Keywords

    Spatial dependence; Unknown heteroskedasticity; Nonnormality; Modified QML estimator; Robust standard error; SARAR models;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models

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