Abstract
This paper studies computation, estimation, inference and specification testing in dynamic panel threshold spatial Durbin (DPTSD) model with multiple thresholds. We first develop a new Markov chain Monte Carlo (MCMC) based algorithm to jointly estimate the threshold parameters and simultaneously construct the confidence intervals for the parameters, after suggesting a within-group spatial two-stage least squares estimator. We then construct test statistics for threshold effect and the number of thresholds. Monte Carlo experiments indicate that the proposed estimator and tests have desired performance in finite samples. We finally apply the DPTSD model to investigate the relationship between financial development and green growth, and find that the empirical results based on the DPTSD model are quite different from these based on the dynamic panel threshold model.
Funding source: National Social Science Fund of China
Award Identifier / Grant number: 23BJY239
Acknowledgments
The authors would like to thank the Editor Professor Jeremy Piger and two anonymous referees for their very constructive comments and suggestions. We also thank Professor Lixiong Yang for helpful discussions and suggestions. Any remaining errors are solely our responsibility.
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Research ethics: This article does not contain any studies with human participants or animals performed by the authors.
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Competing interests: The authors declare that they have no potential competing interests.
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Research funding: This study is supported by the National Social Science Fund of China (Grant No. 23BJY239).
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Data availability: Codes and data are available from the authors on request.
References
Andrews, D. W. K., and W. Ploberger. 1994. “Optimal Tests when a Nuisance Parameter Is Present Only under the Alternative.” Econometrica: 1383–414. https://doi.org/10.2307/2951753.Search in Google Scholar
Bai, J. 1997. “Estimating Multiple Breaks One at a Time.” Econometric Theory 13 (3): 315–52. https://doi.org/10.1017/s0266466600005831.Search in Google Scholar
Baltagi, B. H. 2013. Econometric Analysis of Panel Data, 5th ed. New York: John Wiley & Sons, Inc.Search in Google Scholar
Brooks, S. P., and A. Gelman. 1998. “General Methods for Monitoring Convergence of Iterative Simulations.” Journal of Computational and Graphical Statistics 7 (4): 434–55. https://doi.org/10.2307/1390675.Search in Google Scholar
Cao, J., S. Law, A. Samad, and W. Mohamad. 2023. “Internal Mechanism Analysis of the Financial Vanishing Effect on Green Growth: Evidence from China.” Energy Economics 120: 106579. https://doi.org/10.1016/j.eneco.2023.106579.Search in Google Scholar
Chen, H., T. T. L. Chong, and J. Bai. 2012. “Theory and Applications of the Model with Two Threshold Variables.” Econometric Reviews 31 (2): 142–70. https://doi.org/10.1080/07474938.2011.607100.Search in Google Scholar
Chernozhukov, V., and H. Hong. 2004. “Likelihood Estimation and Inference in a Class of Nonregular Econometric Models.” Econometrica 72 (5): 1445–80. https://doi.org/10.1111/j.1468-0262.2004.00540.x.Search in Google Scholar
Davies, R. B. 1977. “Hypothesis Testing when a Nuisance Parameter Is Present Only under the Alternative.” Biometrika 64: 247–54. https://doi.org/10.2307/2335690.Search in Google Scholar
Davies, R. B. 1987. “Hypothesis Testing when a Nuisance Parameter Is Present Only under the Alternative.” Biometrika 74: 33–43. https://doi.org/10.1093/biomet/74.1.33.Search in Google Scholar
Deng, Y. 2018. “Estimation for the Spatial Autoregressive Threshold Model.” Economics Letters 171: 172–5. https://doi.org/10.1016/j.econlet.2018.07.041.Search in Google Scholar
Drukker, D. M., P. Egger, and I. R. Prucha. 2013. “On Two-step Estimation of a Spatial Autoregressive Model with Autoregressive Disturbances and Endogenous Regressors.” Econometric Reviews 32 (5–6): 686–733. https://doi.org/10.1080/07474938.2013.741020.Search in Google Scholar
Gelman, A., and D. B. Rubin. 1992. “Inference from Iterative Simulation Using Multiple Sequences.” Statistical Science 7: 457–511. https://doi.org/10.1214/ss/1177011136.Search in Google Scholar
Guo, J., and X. Qu. 2020. “Fixed Effects Spatial Panel Data Models with Time-Varying Spatial Dependence.” Economics Letters 196: 109531. https://doi.org/10.1016/j.econlet.2020.109531.Search in Google Scholar
Hansen, B. E. 1996. “Inference when a Nuisance Parameter Is Not Identified under the Null Hypothesis.” Econometrica: 413–30. https://doi.org/10.2307/2171789.Search in Google Scholar
Hansen, B. E. 1999. “Threshold Effects in Non-dynamic Panels: Estimation, Testing, and Inference.” Journal of Econometrics 93: 345–68. https://doi.org/10.1016/s0304-4076(99)00025-1.Search in Google Scholar
Hansen, B. E. 2017. “Regression Kink with an Unknown Threshold.” Journal of Business and Economic Statistics 35: 228–40. https://doi.org/10.1080/07350015.2015.1073595.Search in Google Scholar
Jun, S. J., J. Pinkse, and Y. Wan. 2015. “Classical Laplace Estimation for n3$\sqrt[3]{n}$ Consistent Estimators: Improved Convergence Rates and Rate-Adaptive Inference.” Journal of Econometrics 187: 201–16. https://doi.org/10.1016/j.jeconom.2015.01.005.Search in Google Scholar
Kelejian, H. H., and I. R. Prucha. 1998. “A Generalized Spatial Two-Stage Least Squares Procedure for Estimating a Spatial Autoregressive Model with Autoregressive Disturbances.” The Journal of Real Estate Finance and Economics 17: 99–121. https://doi.org/10.1023/a:1007707430416.10.1023/A:1007707430416Search in Google Scholar
Korniotis, G. M. 2010. “Estimating Panel Models with Internal and External Habit Formation.” Journal of Business and Economic Statistics 28 (1): 145–58. https://doi.org/10.1198/jbes.2009.08041.Search in Google Scholar
Kuersteiner, G. M., and I. R. Prucha. 2020. “Dynamic Spatial Panel Models: Networks, Common Shocks, and Sequential Exogeneity.” Econometrica 88 (5): 2109–46. https://doi.org/10.3982/ecta13660.Search in Google Scholar
Lee, L-F., and J. Yu. 2016. “Identification of Spatial Durbin Panel Models.” Journal of Applied Econometrics 31 (1): 133–62. https://doi.org/10.1002/jae.2450.Search in Google Scholar
Liu, J., J. Pan, Q. Xia, and Y. Xiao. 2022. “Subset Selection of Double-Threshold Moving Average Models through the Application of the Bayesian Method.” Statistics and Its Interface 15: 51–61. https://doi.org/10.4310/21-sii674.Search in Google Scholar
Lyu, C., K. Wang, F. Zhang, and X. Zhang. 2018. “GDP Management to Meet or Beat Growth Targets.” Journal of Accounting and Economics 66 (1): 318–38. https://doi.org/10.1016/j.jacceco.2018.07.001.Search in Google Scholar
Manski, C. F. 1993. “Identification of Endogenous Social Effects: The Reflection Problem.” Review of Economic Studies 60: 531–42. https://doi.org/10.2307/2298123.Search in Google Scholar
Ni, S., Q. Xia, and J. Liu. 2018. “Bayesian Subset Selection for Two-Threshold Variable Autoregressive Models.” Studies in Nonlinear Dynamics & Econometrics 22: 1–16. https://doi.org/10.1515/snde-2017-0062.Search in Google Scholar
Qu, X., Z. Xu, J. Yu, and J. Zhu. 2023. “Understanding Local Government Debt in China: A Regional Competition Perspective.” Regional Science and Urban Economics 98: 103859. https://doi.org/10.1016/j.regsciurbeco.2022.103859.Search in Google Scholar
Seo, M. H., and Y. Shin. 2016. “Dynamic Panels with Threshold Effect and Endogeneity.” Journal of Econometrics 195: 169–86. https://doi.org/10.1016/j.jeconom.2016.03.005.Search in Google Scholar
Seo, M. H., S. Kim, and Y. J. Kim. 2019. “Estimation of Dynamic Panel Threshold Model Using Stata.” Stata Journal 19 (3): 685–97. https://doi.org/10.1177/1536867x19874243.Search in Google Scholar
Vrugt, J. A., C. J. F. TerBraak, C. G. H. Diks, B. A. Robinson, J. M. Hyman, and D. Higdon. 2009. “Accelerating Markov Chain Monte Carlo Simulation by Differential Evolution with Self-Adaptive Randomized Subspace Sampling.” International Journal of Nonlinear Sciences and Numerical Simulation 10 (3): 271–88. https://doi.org/10.1515/ijnsns.2009.10.3.273.Search in Google Scholar
Wei, L., C. Zhang, J. J. Su, and L. Yang. 2021. “Panel Threshold Spatial Durbin Models with Individual Fixed Effects.” Economics Letters 201 (5): 109778. https://doi.org/10.1016/j.econlet.2021.109778.Search in Google Scholar
Yang, L. 2024. “Panel Threshold Model with Covariate-dependent Thresholds and its Application to the Cash Flow/investment Relationship.” Studies in Nonlinear Dynamics and Econometrics 28 (4): 645–59, https://doi.org/10.1515/snde-2022-0035.Search in Google Scholar
Yang, L., and M. Ni. 2022. “Is Financial Development Beneficial to Improve the Efficiency of Green Development? Evidence from the Belt and Road Countries.” Energy Economics 105: 105734. https://doi.org/10.1016/j.eneco.2021.105734.Search in Google Scholar
Yu, P., and X. Fan. 2021. “Threshold Regression with a Threshold Boundary.” Journal of Business and Economic Statistics 39: 1–59. https://doi.org/10.1080/07350015.2020.1740712.Search in Google Scholar
Zhang, X., D. Li, and H. Tong. 2024. “On the Least Squares Estimation of Multiple-Threshold-Variable Autoregressive Models.” Journal of Business & Economic Statistics 42: 215–28. https://doi.org/10.1080/07350015.2023.2174124.Search in Google Scholar
Zheng, X., K. Liang, Q. Xia, and D. Zhang. 2022. “Best Subset Selection for Double-Threshold-Variable Autoregressive Moving-Average Models: The Bayesian Approach.” Computational Economics 59: 1175–201. https://doi.org/10.1007/s10614-021-10124-7.Search in Google Scholar
Zheng, X., Q. Xia, and R. Liang. 2023. “Bayesian Inference for Order Determination of Double Threshold Variables Autoregressive Models.” Studies in Nonlinear Dynamics & Econometrics 27 (4): 567–87. https://doi.org/10.1515/snde-2020-0096.Search in Google Scholar
Zhong, J., and T. Li. 2020. “Impact of Financial Development and its Spatial Spillover Effect on Green Total Factor Productivity: Evidence from 30 Provinces in China.” Mathematical Problems in Engineering 2020: 1–12. https://doi.org/10.1155/2020/5741387.Search in Google Scholar
Zhu, Y., X. Han, and Y. Chen. 2020. “Bayesian Estimation and Model Selection of Threshold Spatial Durbin Model.” Economics Letters 188: 108956. https://doi.org/10.1016/j.econlet.2020.108956.Search in Google Scholar
Supplementary Material
This article contains supplementary material (https://doi.org/10.1515/snde-2023-0033).
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