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HETEROSKEDASTICITY ROBUST SPECIFICATION TESTING IN SPATIAL AUTOREGRESSION

Published online by Cambridge University Press:  21 May 2024

Jungyoon Lee Open the ORCID record for Jungyoon Lee [Opens in a new window] ,
Peter C. B. Phillips Open the ORCID record for Peter C. B. Phillips [Opens in a new window]  and
Francesca Rossi Open the ORCID record for Francesca Rossi [Opens in a new window]
Jungyoon Lee*
Affiliation:
Royal Holloway, University of London
Peter C. B. Phillips
Affiliation:
Yale University, University of Auckland, and Singapore Management University
Francesca Rossi
Affiliation:
University of Verona
*
Address correspondence to Jungyoon Lee, Department of Economics, Royal Holloway, University of London, Egham, United Kingdom; e-mail: [email protected].
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Abstract

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Spatial autoregressive (SAR) and related models offer flexible yet parsimonious ways to model spatial and network interactions. SAR specifications typically rely on a particular parametric functional form and an exogenous choice of the so-called spatial weight matrix with only limited guidance from theory in making these specifications. Also, the choice of a SAR model over other alternatives, such as spatial Durbin (SD) or spatial lagged X (SLX) models, is often arbitrary, raising issues of potential specification error. To address such issues, this paper develops a new specification test within the SAR framework that can detect general forms of misspecification including that of the spatial weight matrix, the functional form and the model itself. The test is robust to the presence of heteroskedasticity of unknown form in the disturbances and the approach relates to the conditional moment test framework of Bierens ([1982, Journal of Econometrics 20, 105–134], [1990, Econometrica 58, 1443–1458]). The Bierens test is shown to be inconsistent in general against spatial alternatives and the new test introduces modifications to achieve test consistency in the spatial setting. A central element is the infinite-dimensional endogeneity induced by spatial linkages. This complexity is addressed by introducing a new component to the omnibus test that captures the effects of potential spatial matrix misspecification. With this modification, the approach leads to a simple pivotal test procedure with standard critical values that is the first test in the literature to have power against misspecifications in the spatial linkages. We derive the asymptotic distribution of the test under the null hypothesis of correct SAR specification and prove consistency. A Monte Carlo study is conducted to study its finite sample performance. An empirical illustration on the performance of the test in modeling tax competition in Finland is included.

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ARTICLES
Information
Econometric Theory , First View , pp. 1 - 49
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), 2024. Published by Cambridge University Press

Footnotes

Phillips acknowledges research support from the NSF under Grant No. SES 18-50860 and the Kelly Fund at the University of Auckland. Lee and Rossi acknowledge research support from the ESRC Grant No. ES/P011705/1. We are indebted to Teemu Lyytikäinen for sharing his dataset.

References

REFERENCES

Allers, M. A., & Elhorst, J. P. (2005). Tax mimicking and yardstick competition among local governments in the Netherlands. International Tax and Public Finance , 12, 493–513.CrossRefGoogle Scholar
Anselin, L. (2001). Rao’s score test in spatial econometrics. Journal of Statistical Planning and Inference , 97, 113–139.CrossRefGoogle Scholar
Arraiz, I., Drukker, D. M., Kelejian, H. H., & Prucha, I. R. (2010). A spatial Cliff–Ord-type-model with heteroskedastic innovations: Small and large sample results. Journal of Regional Science , 50, 592–614.CrossRefGoogle Scholar
Bailey, N., Holly, S., & Pesaran, M. H. (2016). A two-stage approach to spatio-temporal analysis with strong and weak cross-sectional dependence. Journal of Applied Econometrics , 31, 249–280.CrossRefGoogle Scholar
Baltagi, B. H., & Li, D. (2001). LM tests for functional form and spatial error correlation. International Regional Science Review , 24, 194–225.CrossRefGoogle Scholar
Beenstock, M., & Felsenstein, D. (2012). Nonparametric estimation of the spatial connectivity matrix using spatial panel data. Geographical Analysis , 44, 386–397.CrossRefGoogle Scholar
Bierens, H. J. (1982). Consistent model specification tests. Journal of Econometrics , 20, 105–134.CrossRefGoogle Scholar
Bierens, H. J. (1984). Model specification testing of time series regressions. Journal of Econometrics , 26, 323–353.CrossRefGoogle Scholar
Bierens, H. J. (1988). ARMA memory index modeling of economic time series. Econometric Theory , 4, 35–59.CrossRefGoogle Scholar
Bierens, H. J. (1990). A consistent conditional moment test of functional form. Econometrica , 58, 1443–1458.CrossRefGoogle Scholar
Bierens, H. J. (2015). Addendum to: Consistent model specification tests (1982). Mimeo.Google Scholar
Burridge, P. (1980). On the Cliff–Ord test for spatial correlation. Journal of the Royal Statistical Society: Series B (Methodological) , 42, 107–108.CrossRefGoogle Scholar
Case, A. C. (1991). Spatial patterns in household demand. Econometrica , 59, 953–965.CrossRefGoogle Scholar
Chirinko, R. S., & Wilson, D. J. (2017). Tax competition among U.S. states: Racing to the bottom or riding on a seesaw? Journal of Public Economics , 155, 147–163.CrossRefGoogle Scholar
Cliff, A., & Ord, J. (1968). The problem of spatial autocorrelation. Joint Discussion Paper. University of Bristol: Department of Economics, 26, Department of Geography.Google Scholar
Cliff, A., & Ord, J. (1981). Spatial processes: Models and applications . Pion.Google Scholar
de Jong, R. M. (1996). The Bierens test under data dependence. Journal of Econometrics , 72, 1–32.CrossRefGoogle Scholar
de Jong, R. M., & Bierens, H. (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
Debarsy, N., & Ertur, C. (2019). Interaction matrix selection in spatial autoregressive models with an application to growth theory. Regional Science and Urban Economics , 75, 49–69.CrossRefGoogle Scholar
Delgado, M. A., & Robinson, P. M. (2015). Non-nested testing of spatial correlation. Journal of Econometrics , 187, 385–401.CrossRefGoogle Scholar
Elhorst, J. P. (2010). Applied spatial econometrics: Raising the bar. Spatial Economic Analysis , 5, 9–28.CrossRefGoogle Scholar
Escanciano, J. C. (2006). Goodness-of-fit tests for linear and nonlinear time series models. Journal of the American Statistical Association , 101, 531–541.CrossRefGoogle Scholar
Escanciano, J. C. (2007). Model checks using residual marked empirical processes. Statistica Sinica , 17, 115–138.Google Scholar
Fan, Y., & Li, Q. (1996). Consistent model specification tests: Omitted variables and semiparametric functional forms. Econometrica , 64, 865–890.CrossRefGoogle Scholar
Florax, R. J., Folmer, H., & Rey, S. J. (2003). Specification searches in spatial econometrics: the relevance of Hendry’s methodology. Regional Science and Urban Economics , 33, 557–579.CrossRefGoogle Scholar
Gupta, A., & Qu, X. (2021). Consistent specification testing under spatial dependence. Mimeo.Google 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
Kasparis, I. (2010). The Bierens test for certain nonstationary models. Specification Analysis in Honor of Phoebus J. Dhrymes Journal of Econometrics , 158, 221–230.CrossRefGoogle Scholar
Kelejian, H. H. (2008). A spatial J-test for model specification against a single or a set of non-nested alternatives. Letters in Spatial and Resource Sciences , 1, 3–11.CrossRefGoogle Scholar
Kelejian, H. H., & Piras, G. (2011). An extension of Kelejian’s J-test for non-nested spatial models. Regional Science and Urban Economics , 41, 281–292.CrossRefGoogle Scholar
Kelejian, H. H., & Piras, G. (2016). An extension of the J-test to a spatial panel data framework. Journal of Applied Econometrics , 31, 387–402.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. (1999). A generalized moments estimator for the autoregressive parameter in a spatial model. International Economic Review , 40, 509–533.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
Kelejian, H. H., & Prucha, I. R. (2010). Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances. Journal of Econometrics , 157, 53–67.CrossRefGoogle ScholarPubMed
Koul, H. L., & Stute, W. (1999). Nonparametric model checks for time series. Annals of Statistics , 27, 204–236.CrossRefGoogle Scholar
Lam, C., & Souza, P. C. L. (2016). Detection and estimation of block structure in spatial weight matrix. Econometric Reviews , 35, 1347–1376.CrossRefGoogle Scholar
Lam, C., & Souza, P. C. L. (2020). Estimation and selection of spatial weight matrix in a spatial lag model. Journal of Business & Economic Statistics , 38, 693–710.CrossRefGoogle Scholar
Lee, L.-F. (2003). Best spatial two-stage least squares estimators for a spatial autoregressive model with autoregressive disturbances. Econometric Reviews , 22, 307–335.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. (2007). GMM and 2SLS estimation of mixed regressive, spatial autoregressive models. Journal of Econometrics , 137, 489–514.CrossRefGoogle Scholar
Lee, L.-F., & Yu, J. (2012). The C-type gradient test for spatial dependence in spatial autoregressive models. Letters in Spatial and Resource Sciences , 5, 119–135.CrossRefGoogle Scholar
Liu, T., & Lung-fei, L. (2019). A likelihood ratio test for spatial model selection. Journal of Econometrics , 213, 434–458.CrossRefGoogle Scholar
Liu, X., & Prucha, I. R. (2018). A robust test for network generated dependence. Journal of Econometrics , 207, 92–113.CrossRefGoogle Scholar
Lyytikäinen, T. (2012). Tax competition among local governments: Evidence from a property tax reform in Finland. Journal of Public Economics , 96, 584–595.CrossRefGoogle Scholar
Martellosio, F. (2012). Testing for spatial autocorrelation: The regressors that make the power disappear. Econometric Reviews , 31, 215–240.CrossRefGoogle Scholar
Newey, W. K. (1985). Maximum likelihood specification testing and conditional moment tests. Econometrica , 53, 1047–1070.CrossRefGoogle Scholar
Ord, K. (1975). Estimation methods for models of spatial interaction. Journal of the American Statistical Association , 70, 120–126.CrossRefGoogle Scholar
Pinkse, J., Slade, M. E., & Brett, C. (2002). Spatial price competition: A semiparametric approach. Econometrica , 70, 1111–1153.CrossRefGoogle Scholar
Ramsey, J. B. (1969). Tests for specification errors in classical linear least-squares regression analysis. Journal of the Royal Statistical Society: Series B (Methodological) , 31, 350–371.CrossRefGoogle Scholar
Resnick, S. I. (2008). Extreme values, regular variation, and point processes (Vol. 4). Springer Science & Business Media.Google Scholar
Robinson, P. M. (2008). Correlation testing in time series, spatial and cross-sectional data. Journal of Econometrics , 147, 5–16.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
Stute, W. (1997). Nonparametric model checks for regression. Annals of Statistics , 25, 613–641.CrossRefGoogle Scholar
Stute, W., & Zhu, L. (2005). Nonparametric checks for single-index models. Annals of Statistics , 33, 1048–1083.CrossRefGoogle Scholar
Su, L., & Qu, X. (2017). Specification test for spatial autoregressive models. Journal of Business & Economic Statistics , 35, 572–584.CrossRefGoogle Scholar
Vuong, Q. H. (1989). Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica , 57, 307–333.CrossRefGoogle Scholar
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