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APPLICATIONS OF FUNCTIONAL DEPENDENCE TO SPATIAL ECONOMETRICS

Published online by Cambridge University Press:  31 May 2024

Zeqi WuOpen the ORCID record for Zeqi Wu[Opens in a new window],
Wen JiangOpen the ORCID record for Wen Jiang[Opens in a new window] and
Xingbai XuOpen the ORCID record for Xingbai Xu[Opens in a new window]
Zeqi Wu
Affiliation:
Renmin University of China
Wen Jiang
Affiliation:
Minjiang University
Xingbai Xu*
Affiliation:
Xiamen University
*
Address correspondence to Xingbai Xu, Wang Yanan Institute for Studies in Economics, Xiamen University, Xiamen, China, e-mail: xuxingbai@xmu.edu.cn. Zeqi Wu and Wen Jiang are the co-first authors.
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Abstract

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In this paper, we generalize the concept of functional dependence (FD) from time series (see Wu [2005, Proceedings of the National Academy of Sciences 102, 14150–14154]) and stationary random fields (see El Machkouri, Volný, and Wu [2013, Stochastic Processes and Their Applications 123, 1–14]) to nonstationary spatial processes. Within conventional settings in spatial econometrics, we define the concept of spatial FD measure and establish a moment inequality, an exponential inequality, a Nagaev-type inequality, a law of large numbers, and a central limit theorem. We show that the dependent variables generated by some common spatial econometric models, including spatial autoregressive (SAR) models, threshold SAR models, and spatial panel data models, are functionally dependent under regular conditions. Furthermore, we investigate the properties of FD measures under various transformations, which are useful in applications. Moreover, we compare spatial FD with the spatial mixing and spatial near-epoch dependence proposed in Jenish and Prucha ([2009, Journal of Econometrics 150, 86–98], [2012, Journal of Econometrics 170, 178–190]), and we illustrate its advantages.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 41 , Issue 5 , October 2025, pp. 1044 - 1079
Copyright
© The Author(s), 2024. Published by Cambridge University Press

Footnotes

We sincerely thank the Editor (Peter C. B. Phillips), the Co-Editor (Liangjun Su), and three anonymous referees for their helpful comments and suggestions on the earlier version of this paper. Xu gratefully acknowledges the financial support from the NSFC (Grant Nos. 72073110 and 72333001), Basic Scientific Center Project 71988101 of the NSFC, and the 111 Project (B13028).

References

REFERENCES

Andrews, D. W. (1984). Non-strong mixing autoregressive processes. Journal of Applied Probability , 21, 930–934.10.2307/3213710CrossRefGoogle Scholar
Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. Annals of Probability , 10, 1047–1050.CrossRefGoogle Scholar
Brockwell, P. J., & Davis, R. A. (1991). Time series: Theory and methods . Springer-Verlag.10.1007/978-1-4419-0320-4CrossRefGoogle Scholar
Chen, X., Xu, M., & Wu, W. B. (2013). Covariance and precision matrix estimation for high-dimensional time series. Annals of Statistics , 41, 2994–3021.10.1214/13-AOS1182CrossRefGoogle Scholar
Chow, Y. S., & Teicher, H. (2003). Probability theory: Independence, interchangeability, martingales . (3rd ed.) Springer-Verlag.Google Scholar
Conley, T. G., & Topa, G. (2002). Socio-economic distance and spatial patterns in unemployment. Journal of Applied Econometrics , 17, 303–327.CrossRefGoogle Scholar
Davidson, J. (1994). Stochastic limit theory: An introduction for econometricians . Oxford University Press.CrossRefGoogle Scholar
Dedecker, J. (1998). A central limit theorem for stationary random fields. Probability Theory and Related Fields , 110, 397–426.CrossRefGoogle Scholar
Deng, Y. (2018). Estimation for the spatial autoregressive threshold model. Economics Letters , 171, 172–175.CrossRefGoogle Scholar
Doğan, O., & Taşpınar, S. (2013). GMM estimation of spatial autoregressive models with moving average disturbances. Regional Science and Urban Economics 43, 903–926.CrossRefGoogle Scholar
Doukhan, P. (1994). Mixing: Properties and examples . Springer.CrossRefGoogle Scholar
Doukhan, P., & Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities. Stochastic Processes and their Applications , 84, 313–342.CrossRefGoogle Scholar
El Machkouri, M., Volný, D., & Wu, W. B. (2013). A central limit theorem for stationary random fields. Stochastic Processes and Their Applications , 123, 1–14.CrossRefGoogle Scholar
Fingleton, B. (2008). A generalized method of moments estimator for a spatial model with moving average errors, with application to real estate prices. Empirical Economics , 34, 35–57.CrossRefGoogle Scholar
Forchini, G., Jiang, B., & Peng, B. (2018). TSLS and LIML estimators in panels with unobserved shocks. Econometrics , 6, 19.CrossRefGoogle Scholar
Huang, J. S. (1984). The autoregressive moving average model for spatial analysis. Australian Journal of Statistics , 26, 169–178.CrossRefGoogle Scholar
Jenish, N., & Prucha, I. R. (2009). Central limit theorems and uniform laws of large numbers for arrays of random fields. Journal of Econometrics , 150, 86–98.CrossRefGoogle ScholarPubMed
Jenish, N., & Prucha, I. R. (2012). On spatial processes and asymptotic inference under near-epoch dependence. Journal of Econometrics , 170, 178–190.CrossRefGoogle ScholarPubMed
Kelejian, H. H., & Prucha, I. R. (1998). A generalized spatial two-stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances. Journal of Real Estate Finance and Economics , 17, 99–121.CrossRefGoogle Scholar
Kelejian, H. H., & Prucha, I. R. (2001). On the asymptotic distribution of the Moran I test statistic with applications. Journal of Econometrics , 104, 219–257.CrossRefGoogle Scholar
Kojevnikov, D., Marmer, V., & Song, K. (2021). Limit theorems for network dependent random variables. Journal of Econometrics , 222, 882–908.10.1016/j.jeconom.2020.05.019CrossRefGoogle Scholar
Lee, L.-f. (2004). Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models. Econometrica , 72, 1899–1925.CrossRefGoogle Scholar
Lee, L.-f. (2007). GMM and 2SLS estimation of mixed regressive, spatial autoregressive models. Journal of Econometrics , 137, 489–514.CrossRefGoogle Scholar
Lee, L.-f., & Yu, J. (2010). Estimation of spatial autoregressive panel data models with fixed effects. Journal of Econometrics , 154, 165–185.10.1016/j.jeconom.2009.08.001CrossRefGoogle Scholar
Li, K. (2022). Threshold spatial autoregressive model. The Capital University of Economics and Business. MPRA Working Paper No. 113568.Google Scholar
Liu, T., Xu, X., Lee, L.-f., & Mei, Y. (2022). Robust estimation for the spatial autoregressive model. Working Paper, Xiamen University.Google Scholar
Liu, W., Xiao, H., & Wu, W. B. (2013). Probability and moment inequalities under dependence. Statistica Sinica , 23, 1257–1272.Google Scholar
Malikov, E., & Sun, Y. (2017). Semiparametric estimation and testing of smooth coefficient spatial autoregressive models. Journal of Econometrics , 199, 12–34.CrossRefGoogle Scholar
Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Annals of Probability , 7, 745–789.CrossRefGoogle Scholar
Prakasa Rao, B. L. S. (2009). Conditional independence, conditional mixing and conditional association. Annals of the Institute of Statistical Mathematics , 61, 441–460.CrossRefGoogle Scholar
Qu, X., & Lee, L.-f. (2015). Estimating a spatial autoregressive model with an endogenous spatial weight matrix. Journal of Econometrics , 184, 209–232.CrossRefGoogle Scholar
Qu, X., Lee, L.-f., & Yu, J. (2017). QML estimation of spatial dynamic panel data models with endogenous time varying spatial weights matrices. Journal of Econometrics , 197, 173–201.CrossRefGoogle Scholar
Roussas, G. G. (2008). On conditional independence, mixing, and association. Stochastic Analysis and Applications , 26, 1274–1309.CrossRefGoogle Scholar
Su, L., Wang, W., & Xu, X. (2023). Identifying latent group structures in spatial dynamic panels. Journal of Econometrics , 235, 1955–1980.CrossRefGoogle Scholar
Sun, Y. (2016). Functional-coefficient spatial autoregressive models with nonparametric spatial weights. Journal of Econometrics , 195, 134–153.CrossRefGoogle Scholar
Wainwright, M. J. (2019). High-dimensional statistics: A non-asymptotic viewpoint . Cambridge University Press.CrossRefGoogle Scholar
White, H., & Wooldridge, J. M. (1991). Some results for sieve estimation with dependent observations. In Barnett, W. A., Powell, J., & Tauchen, G. E. (Eds.), Non-parametric and semi-parametric methods in econometrics and statistics (pp. 459–493). Cambridge University Press.Google Scholar
Wu, W. B. (2005). Nonlinear system theory: Another look at dependence. Proceedings of the National Academy of Sciences , 102, 14150–14154.CrossRefGoogle Scholar
Wu, W. B. (2011). Asymptotic theory for stationary processes. Statistics and its Interface , 4, 207–226.CrossRefGoogle Scholar
Wu, W. B., & Wu, Y. N. (2016). Performance bounds for parameter estimates of high-dimensional linear models with correlated errors. Electronic Journal of Statistics , 10, 352–379.CrossRefGoogle Scholar
Xu, X., & Lee, L.-f. (2015a). Maximum likelihood estimation of a spatial autoregressive Tobit model. Journal of Econometrics , 188, 264–280.CrossRefGoogle Scholar
Xu, X., & Lee, L.-f. (2015b). A spatial autoregressive model with a nonlinear transformation of the dependent variable. Journal of Econometrics , 186, 1–18.CrossRefGoogle Scholar
Xu, X., & Lee, L.-f. (2018). Sieve maximum likelihood estimation of the spatial autoregressive Tobit model. Journal of Econometrics , 203, 96–112.10.1016/j.jeconom.2017.10.008CrossRefGoogle Scholar
Xu, X., & Lee, L.-f. (2024). Conditions for $\alpha$ -mixing linear random fields: With applications to spatial econometric models. China Journal of Econmetrics , 4, 26–57.Google Scholar
Xu, X., Wang, W., Shin, Y., & Zheng, C. (2024). Dynamic network quantile regression model. Journal of Business & Economic Statistics , 42, 407–421.CrossRefGoogle Scholar
Yu, J., de Jong, R., & Lee, L.-f. (2008). Quasi-maximum likelihood estimators for spatial dynamic panel data with fixed effects when both $n$ and $T$ are large. Journal of Econometrics , 146, 118–134.CrossRefGoogle Scholar
Yuan, D.-M., Wei, L.-R., & Lei, L. (2014). Conditional central limit theorems for a sequence of conditional independent random variables. Journal of the Korean Mathematical Society , 51, 1–15.CrossRefGoogle Scholar
Yuan, Z., & Spindler, M. (2022). Bernstein-type inequalities and nonparametric estimation under near-epoch dependence. Working Paper, University of Hamburg.Google Scholar
Zhou, Z., & Wu, W. B. (2009). Local linear quantile estimation for nonstationary time series. Annals of Statistics , 37, 2696–2729.CrossRefGoogle Scholar
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