Hostname: page-component-76d6cb85b7-92wsb Total loading time: 0 Render date: 2026-07-12T15:33:47.700Z Has data issue: false hasContentIssue false
  • English
  • Français

CONSISTENT SPECIFICATION TESTING UNDER SPATIAL DEPENDENCE

Published online by Cambridge University Press:  11 October 2022

Abhimanyu Gupta Open the ORCID record for Abhimanyu Gupta [Opens in a new window]  and
Xi Qu Open the ORCID record for Xi Qu [Opens in a new window]
Abhimanyu Gupta*
Affiliation:
University of Essex
Xi Qu
Affiliation:
Shanghai Jiao Tong University
*
Address correspondence to Abhimanyu Gupta, Department of Economics, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK; e-mail: a.gupta@essex.ac.uk.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We propose a series-based nonparametric specification test for a regression function when data are spatially dependent, the “space” being of a general economic or social nature. Dependence can be parametric, parametric with increasing dimension, semiparametric or any combination thereof, thus covering a vast variety of settings. These include spatial error models of varying types and levels of complexity. Under a new smooth spatial dependence condition, our test statistic is asymptotically standard normal. To prove the latter property, we establish a central limit theorem for quadratic forms in linear processes in an increasing dimension setting. Finite sample performance is investigated in a simulation study, with a bootstrap method also justified and illustrated. Empirical examples illustrate the test with real-world data.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 2 , April 2024 , pp. 278 - 319
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Footnotes

We thank the Editor, the Co-Editor, and three referees for insightful comments that improved the paper. We are grateful to Swati Chandna, Miguel Delgado, Emmanuel Guerre, Fernando López Hernandéz, Hon Ho Kwok, Arthur Lewbel, Daisuke Murakami, Ryo Okui, and Amol Sasane for helpful comments, and audiences at YEAP 2018 (Shanghai University of Finance and Economics), NYU Shanghai, Carlos III Madrid, SEW 2018 (Dijon), Aarhus University, SEA 2018 (Vienna), EcoSta 2018 (Hong Kong), Hong Kong University, AFES 2018 (Cotonou), ESEM 2018 (Cologne), CFE 2018 (Pisa), University of York, Penn State, Michigan State, University of Michigan, Texas A&M, 1st Southampton Workshop on Econometrics and Statistics, and MEG 2019 (Columbus). We also thank Xifeng Wen from the Experiment and Data Center of Antai College of Economics and Management (SJTU) for expert computing assistance. The research of the first author was supported by ESRC grant ES/R006032/1. The research of the second author was supported by the National Natural Science Foundation of China (Project Nos. 72222007, 71973097, and 72031006).

References

REFERENCES

Ambrosetti, A. & Prodi, G. (1995) A Primer of Nonlinear Analysis . Cambridge University Press.Google Scholar
Anatolyev, S. (2012) Inference in regression models with many regressors. Journal of Econometrics 170, 368–382.CrossRefGoogle Scholar
Autant-Bernard, C. & LeSage, J.P. (2011) Quantifying knowledge spillovers using spatial autoregressive models. Journal of Regional Science 51, 471–496.CrossRefGoogle Scholar
Bloom, N., Schankerman, M., & van Reenen, J. (2013) Identifying technology pillovers and product market rivalry. Econometrica 81, 1347–1393.Google Scholar
Case, A.C. (1991) Spatial patterns in household demand. Econometrica 59, 953–965.CrossRefGoogle Scholar
Chen, X. (2007) Chapter 76: Large sample sieve estimation of semi-nonparametric models . In Heckman, J.J. and Leamer, E.E. (eds.), Handbook of Econometrics , vol. 6B, pp. 5549–5632. North-Holland.CrossRefGoogle Scholar
Chen, X., Hong, H., & Tamer, E. (2005) Measurement error models with auxiliary data. Review of Economic Studies 72, 343–366.CrossRefGoogle Scholar
Cliff, A.D. & Ord, J.K. (1973) Spatial Autocorrelation . Pion.Google Scholar
Conley, T.G. & Dupor, B. (2003) A spatial analysis of sectoral complementarity. Journal of Political Economy 111, 311–352.CrossRefGoogle Scholar
Cooke, R.G. (1950) Infinite Matrices & Sequence Spaces . Macmillan and Company.Google Scholar
Crespo Cuaresma, J. & Feldkircher, M. (2013) Spatial filtering, model uncertainty and the speed of income convergence in Europe. Journal of Applied Econometrics 28, 720–741.CrossRefGoogle Scholar
De Jong, R.M. & Bierens, H.J. (1994) On the limit behavior of a chi-square type test if the number of conditional moments tested approaches infinity. Econometric Theory 10, 70–90.CrossRefGoogle Scholar
De Oliveira, V., Kedem, B., & Short, D.A. (1997) Bayesian prediction of transformed Gaussian random fields. Journal of the American Statistical Association 92, 1422–1433.Google Scholar
Debarsy, N., Jin, F., & Lee, L.F. (2015) Large sample properties of the matrix exponential spatial specification with an application to FDI. Journal of Econometrics 188, 1–21.CrossRefGoogle Scholar
Delgado, M. & Robinson, P.M. (2015) Non-nested testing of spatial correlation. Journal of Econometrics 187, 385–401.CrossRefGoogle Scholar
Ertur, C. & Koch, W. (2007) Growth, technological interdependence and spatial externalities: Theory and evidence. Journal of Applied Econometrics 22, 1033–1062.CrossRefGoogle Scholar
Evans, P. & Kim, J.U. (2014) The spatial dynamics of growth and convergence in Korean regional incomes. Applied Economics Letters 21, 1139–1143.CrossRefGoogle Scholar
Fuentes, M. (2007) Approximate likelihood for large irregularly spaced spatial data. Journal of the American Statistical Association 102, 321–331.CrossRefGoogle ScholarPubMed
Gao, J. & Anh, V. (2000) A central limit theorem for a random quadratic form of strictly stationary processes. Statistics and Probability Letters 49, 69–79.CrossRefGoogle Scholar
Gneiting, T. (2002) Nonseparable, stationary covariance functions for space-time data. Journal of the American Statistical Association 97, 590–600.CrossRefGoogle Scholar
Gradshteyn, I.S. & Ryzhik, I.M. (1994) Table of Integrals, Series and Products , 5th Edition. Academic Press.Google Scholar
Gupta, A. (2018a) Autoregressive spatial spectral estimates. Journal of Econometrics 203, 80–95.CrossRefGoogle Scholar
Gupta, A. (2018b) Nonparametric specification testing via the trinity of tests. Journal of Econometrics 203, 169–185.CrossRefGoogle Scholar
Gupta, A., Kokas, S., & Michaelides, A. (2021) Credit market spillovers in a financial network. Working paper. doi:10.2139/ssrn.3063886.CrossRefGoogle Scholar
Gupta, A. & Robinson, P.M. (2015) Inference on higher-order spatial autoregressive models with increasingly many parameters. Journal of Econometrics 186, 19–31.CrossRefGoogle Scholar
Gupta, A. & Robinson, P.M. (2018) Pseudo maximum likelihood estimation of spatial autoregressive models with increasing dimension. Journal of Econometrics 202, 92–107.CrossRefGoogle Scholar
Hahn, J., Kuersteiner, G., & Mazzocco, M. (2020) Joint time-series and cross-section limit theory under mixingale assumptions. Econometric Theory, Published online 11 August 2020. doi:10.1017/S0266466620000316.CrossRefGoogle Scholar
Han, X., Lee, L.-F., & Xu, X. (2021) Large sample properties of Bayesian estimation of spatial econometric models. Econometric Theory 37, 708–746.CrossRefGoogle Scholar
Hannan, E.J. (1970) Multiple Time Series . Wiley.CrossRefGoogle Scholar
Heston, A., Summers, R., & Aten, B. (2002) Penn World Tables Verison 6.1. Downloadable Dataset. Center for International Comparisons at the University of Pennsylvania.Google Scholar
Hidalgo, J. & Schafgans, M. (2017) Inference and testing breaks in large dynamic panels with strong cross sectional dependence. Journal of Econometrics 196, 259–274.CrossRefGoogle Scholar
Hillier, G. & Martellosio, F. (2018a) Exact and higher-order properties of the MLE in spatial autoregressive models, with applications to inference. Journal of Econometrics 205, 402–422.CrossRefGoogle Scholar
Hillier, G. & Martellosio, F. (2018b) Exact likelihood inference in group interaction network models. Econometric Theory 34, 383–415.CrossRefGoogle Scholar
Ho, C.-Y., Wang, W., & Yu, J. (2013) Growth spillover through trade: A spatial dynamic panel data approach. Economics Letters 120, 450–453.CrossRefGoogle Scholar
Hong, Y. & White, H. (1995) Consistent specification testing via nonparametric series regression. Econometrica 63, 1133–1159.CrossRefGoogle Scholar
Huber, P.J. (1973) Robust regression: Asymptotics, conjectures and Monte Carlo. Annals of Statistics 1, 799–821.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
Jin, F. & Lee, L.F. (2015) On the bootstrap for Moran’s I test for spatial dependence. Journal of Econometrics 184, 295–314.CrossRefGoogle Scholar
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
Koenker, R. & Machado, J.A.F. (1999) GMM inference when the number of moment conditions is large. Journal of Econometrics 93, 327–344.CrossRefGoogle Scholar
König, M.D., Rohner, D., Thoenig, M., & Zilibotti, F. (2017) Networks in conflict: Theory and evidence from the Great War of Africa. Econometrica 85, 1093–1132.CrossRefGoogle Scholar
Kuersteiner, G.M. & Prucha, I.R. (2020) Dynamic spatial panel models: Networks, common shocks, and sequential exogeneity. Econometrica 88, 2109–2146.CrossRefGoogle Scholar
Lee, J., Phillips, P.C.B., & Rossi, F. (2020) Consistent misspecification testing in spatial autoregressive models. Cowles Foundation Discussion paper 2256.CrossRefGoogle Scholar
Lee, J. & Robinson, P.M. (2016) Series estimation under cross-sectional dependence. Journal of Econometrics 190, 1–17.CrossRefGoogle 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. & Liu, X. (2010) Efficient GMM estimation of high order spatial autoregressive models with autoregressive disturbances. Econometric Theory 26, 187–230.CrossRefGoogle Scholar
LeSage, J.P. & Pace, R. (2007) A matrix exponential spatial specification. Journal of Econometrics 140, 190–214.CrossRefGoogle Scholar
Malikov, E. & Sun, Y. (2017) Semiparametric estimation and testing of smooth coefficient spatial autoregressive models. Journal of Econometrics 199, 12–34.CrossRefGoogle Scholar
Matérn, B. (1986) Spatial Variation . Almaenna Foerlaget.CrossRefGoogle Scholar
Mohnen, M. (2022) Stars and brokers: Peer effects among medical scientists. Management Science 68, 2377–3174.CrossRefGoogle Scholar
Newey, W.K. (1997) Convergence rates and asymptotic normality for series estimators. Journal of Econometrics 79, 147–168.CrossRefGoogle Scholar
Oettl, A. (2012) Reconceptualizing stars: Scientist helpfulness and peer performance. Management Science 58, 1122–1140.CrossRefGoogle Scholar
Pinkse, J. (1999). Asymptotic Properties of Moran and Related Tests and Testing for Spatial Correlation in Probit Models. Mimeo, Department of Economics, University of British Columbia and University College London.Google Scholar
Pinkse, J., Slade, M.E., & Brett, C. (2002) Spatial price competition: A semiparametric approach. Econometrica 70, 1111–1153.CrossRefGoogle Scholar
Portnoy, S. (1984) Asymptotic behavior of $M$ -estimators of $p$ regression parameters when ${p}^2/ n$ is large. I. Consistency. Annals of Statistics 12, 1298–1309.CrossRefGoogle Scholar
Portnoy, S. (1985) Asymptotic behavior of $M$ -estimators of $p$ regression parameters when ${p}^2/n$ is large. II. Normal approximation. Annals of Statistics 13, 1403–1417.CrossRefGoogle Scholar
Robinson, P.M. (1972) Non-linear regression for multiple time-series. Journal of Applied Probability 9, 758–768.CrossRefGoogle Scholar
Robinson, P.M. (2011) Asymptotic theory for nonparametric regression with spatial data. Journal of Econometrics 165, 5–19.CrossRefGoogle Scholar
Robinson, P.M. & Rossi, F. (2015) Refined tests for spatial correlation. Econometric Theory 31, 1249–1280.CrossRefGoogle Scholar
Robinson, P.M. & Thawornkaiwong, S. (2012) Statistical inference on regression with spatial dependence. Journal of Econometrics 167, 521–542.CrossRefGoogle Scholar
Scott, D.J. (1973) Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach. Advances in Applied Probability 5, 119–137.CrossRefGoogle Scholar
Stein, M. (1999) Interpolation of Spatial Data . Springer.CrossRefGoogle Scholar
Stinchcombe, M.B. & White, H. (1998) Consistent specification testing with nuisance parameters present only under the alternative. Econometric Theory 14, 295–325.CrossRefGoogle Scholar
Su, L. & Jin, S. (2010) Profile quasi-maximum likelihood estimation of partially linear spatial autoregressive models. Journal of Econometrics 157, 18–33.CrossRefGoogle Scholar
Su, L. & Qu, X. (2017) Specification test for spatial autoregressive models. Journal of Business & Economic Statistics 35, 572–584.CrossRefGoogle Scholar
Sun, Y. (2016) Functional-coefficient spatial autoregressive models with nonparametric spatial weights. Journal of Econometrics 195, 134–153.CrossRefGoogle Scholar
Sun, Y. (2020) The LLN and CLT for U-statistics under cross-sectional dependence. Journal of Nonparametric Statistics 32, 201–224.CrossRefGoogle Scholar
Supplementary material: PDF

Gupta and Qu supplementary material

Gupta and Qu supplementary material

Download Gupta and Qu supplementary material(PDF)
PDF 287.8 KB